Some topics in functional analysis

Stephen Semmes Rice University

Preface

These informal notes are intended to be at least somewhat introductory, particularly near the beginning. We shall also be concerned with connections with other aspects of analysis, or other areas of mathematics, even if we do not get into it too much here. Of course, there are many texts with more information, some of which may be found in the bibliography.

The main prerequisites, at least near the beginning, are familiarity with basic analysis and linear algebra, including abstract metric spaces and vector spaces. In some places, it may be helpful for the reader to be familiar with some general topology, measure theory and integration, or complex analysis. Some familiarity with abstract algebra, including groups and rings, would be helpful in some places as well. We shall sometimes consider simpler versions of some basic notions or results, which could be treated more fully with more familiarity or review of related facts.

Of course, linear algebra deals with vector spaces and linear mappings between them. This is already very interesting and important in the finitedimensional case, and one may consider infinite-dimensional vector spaces as well. One may consider vector spaces over arbitrary fields too. Here we deal largely with vector spaces over the real and complex numbers, and norms and inner products on these vector spaces. The corresponding metrics determine topologies on these vector spaces, and we may sometimes consider other types of topologies on them as well.

Beginning in Part II, we shall consider algebras over the real and complex numbers. These are basically vector spaces on which a bilinear operation of multiplication is defined. Some basic examples include algebras of functions on some nonempty set, and algebras of linear mappings from a vector space into itself. Complex analysis can play a large role in this, and we shall often try to include real versions where possible.

The definition of a norm uses an absolute value function on the corresponding field of scalars. Here we normally use the standard absolute value functions on the real and complex numbers, but one can consider absolute value functions on arbitrary fields, as in [62, 131], Chapter V of [164], and Chapter XII of [206]. This will be discussed in Appendix A, for which the reader should have some familiarity with some basic notions related to fields. Some topics related to algebras over fields with non-archimedean absolute value functions may be found in [97, 98]. Some related references include [14, 57, 82, 156, 224, 238, 246, 248, 338].

To be more precise, Part I deals with norms on vector spaces over the real and complex numbers, as well as bounded linear mappings between them, including bounded linear functionals. The main examples of norms in the first three chapters are supremum norms and ℓ^{p} norms, and L^{p} spaces are mentioned in Chapter 4. We also consider ℓ^{p} and L^{p} spaces when 0 < p < 1, for which one does not typically have a norm, although there is a nice metric on them. Some related matters are discussed in Appendix B, for vector spaces over other fields with absolute value functions too.

Algebras with norms that are usually asked to be submultiplicative are discussed beginning in Part II, as before. Some notions and results that are important for vector spaces with norms more broadly are mentioned as well, such as quotient spaces, the open mapping and closed graph theorems, and projections on vector spaces. Sometimes the algebras of interest will also have an involution, such as the complex-conjugate of a complex-valued function, the transpose of a matrix, or the adjoint of a bounded linear mapping from a real or complex Hilbert space into itself. The involution may be compatible with a suitable norm on the algebra, or have other important properties, and these and related matters will be discussed in several places here.

Contents

I Norms and bounded linear mappings 1

1 Norms and Banach Spaces 2 1.1 Norms on vector spaces . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Seminorms and convex sets . . . . . . . . . . . . . . . . . . . . . 3 1.3 Some norms on R^{n}, C^{n} . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Linear subspaces and continuous functions . . . . . . . . . . . . . 6 1.5 Sequences, series, and Banach spaces . . . . . . . . . . . . . . . . 7 1.6 Completeness and subsets . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.8 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . 12 1.9 Inner products and Hilbert spaces . . . . . . . . . . . . . . . . . 12 1.10 Sums of orthogonal vectors . . . . . . . . . . . . . . . . . . . . . 14 1.11 Arbitrary norms on R^{n}, C^{n} . . . . . . . . . . . . . . . . . . . . . 15 1.12 Functions with finite support . . . . . . . . . . . . . . . . . . . . 16 1.13 Vanishing at infinity . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.14 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.15 Some continuous extensions . . . . . . . . . . . . . . . . . . . . . 21

2 Lipschitz and bounded linear mappings 23 2.1 Lipschitz mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Bounded linear mappings . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Some linear mappings . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Nonnegative sums . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 p-Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 ℓ^{p} Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7 Generalized convergence of sums . . . . . . . . . . . . . . . . . . 33 2.8 More on generalized convergence . . . . . . . . . . . . . . . . . . 36 2.9 Bounded finite sums . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.10 Some related linear mappings . . . . . . . . . . . . . . . . . . . . 40 2.11 Distances to sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.12 Distances and orthogonality . . . . . . . . . . . . . . . . . . . . . 44 2.13 Orthonormal families of vectors . . . . . . . . . . . . . . . . . . . 46 2.14 Orthonormal bases . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.15 Minimizing distances . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Bounded linear functionals 52

3.1 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Duals of ℓ^{1} spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Duals of c_{0} spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4 Hölder’s inequality for sums . . . . . . . . . . . . . . . . . . . . . 56 3.5 Hilbert space adjoints . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6 Sublinear functions . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.7 Convex cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.8 Minkowski functionals . . . . . . . . . . . . . . . . . . . . . . . . 64 3.9 Some one-step extensions . . . . . . . . . . . . . . . . . . . . . . 65 3.10 The Hahn–Banach theorem . . . . . . . . . . . . . . . . . . . . . 67 3.11 Some separation results . . . . . . . . . . . . . . . . . . . . . . . 69 3.12 Complex linear functionals . . . . . . . . . . . . . . . . . . . . . . 71 3.13 Dual linear mappings . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.14 Second dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.15 Second duals of linear mappings . . . . . . . . . . . . . . . . . . 75

4 Uniform boundedness and related topics 77 4.1 The Baire category theorem . . . . . . . . . . . . . . . . . . . . . 77 4.2 Pointwise and uniform boundedness . . . . . . . . . . . . . . . . 79 4.3 The Banach–Steinhaus theorem . . . . . . . . . . . . . . . . . . . 80 4.4 Pointwise convergence . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 A sequential compactness theorem . . . . . . . . . . . . . . . . . 82 4.6 A criterion for density . . . . . . . . . . . . . . . . . . . . . . . . 83 4.7 Pointwise convergence in ℓ^{q} spaces . . . . . . . . . . . . . . . . . 84 4.8 Weak^{∗} convergence in ℓ^{q} spaces . . . . . . . . . . . . . . . . . . . 85 4.9 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.10 Some multiplication operators . . . . . . . . . . . . . . . . . . . . 87 4.11 Convergence of multiplication operators . . . . . . . . . . . . . . 89 4.12 L^{p} Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.13 A measure-theoretic lemma . . . . . . . . . . . . . . . . . . . . . 93 4.14 Some more multiplication operators . . . . . . . . . . . . . . . . 94 4.15 Convergence in measure . . . . . . . . . . . . . . . . . . . . . . . 96

5 Some more spaces and mappings 99 5.1 Compact support and local compactness . . . . . . . . . . . . . . 99 5.2 Vanishing at infinity and compactness . . . . . . . . . . . . . . . 100 5.3 Uniform convergence on compact sets . . . . . . . . . . . . . . . 101 5.4 Some related multiplication operators . . . . . . . . . . . . . . . 102 5.5 Some separation conditions . . . . . . . . . . . . . . . . . . . . . 104 5.6 Point evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.7 Some remarks about Borel measures . . . . . . . . . . . . . . . . 107 5.8 Approximation by continuous functions . . . . . . . . . . . . . . 109 5.9 Real and complex measures . . . . . . . . . . . . . . . . . . . . . 111 5.10 Real and complex Borel measures . . . . . . . . . . . . . . . . . . 113 5.11 Some remarks about product spaces . . . . . . . . . . . . . . . . 114 5.12 Some remarks about direct sums . . . . . . . . . . . . . . . . . . 116 5.13 Bilinear mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.14 Separate continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.15 Bilinear and sesquilinear forms . . . . . . . . . . . . . . . . . . . 120

II Algebras, norms, and operators 123

6 Algebras and norms 124 6.1 Algebras in the strict sense . . . . . . . . . . . . . . . . . . . . . 124 6.2 Norms on algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.3 Algebra homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 126 6.4 Opposite algebra homomorphisms . . . . . . . . . . . . . . . . . 128 6.5 Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.6 More on invertible elements . . . . . . . . . . . . . . . . . . . . . 130 6.7 Some additional properties of G(A) . . . . . . . . . . . . . . . . . 132 6.8 The spectrum of an element . . . . . . . . . . . . . . . . . . . . . 133 6.9 Homomorphisms into R, C . . . . . . . . . . . . . . . . . . . . . 135 6.10 Quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.11 Two-sided ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.12 Ideals in commutative algebras . . . . . . . . . . . . . . . . . . . 139 6.13 More on invertibility . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.14 More on r_{A}(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.15 Identity elements and essential ranges . . . . . . . . . . . . . . . 144

7 Algebras, norms, and linear mappings 146 7.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.2 Some additional properties of r_{A}(x) . . . . . . . . . . . . . . . . 147 7.3 Invertibility and subalgebras . . . . . . . . . . . . . . . . . . . . . 148 7.4 Some Volterra integral operators . . . . . . . . . . . . . . . . . . 149 7.5 Self-adjointness and involutions . . . . . . . . . . . . . . . . . . . 151 7.6 Some remarks about idempotents . . . . . . . . . . . . . . . . . . 152 7.7 The C^{∗} identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.8 Continuous vector-valued functions . . . . . . . . . . . . . . . . . 155 7.9 Algebra-valued functions . . . . . . . . . . . . . . . . . . . . . . . 157 7.10 Bounded Lipschitz functions . . . . . . . . . . . . . . . . . . . . . 158 7.11 Bilipschitz conditions and invertibility . . . . . . . . . . . . . . . 160 7.12 The contraction mapping theorem . . . . . . . . . . . . . . . . . 162 7.13 The open mapping theorem . . . . . . . . . . . . . . . . . . . . . 164 7.14 Open mappings and quotient spaces . . . . . . . . . . . . . . . . 166 7.15 The closed graph theorem . . . . . . . . . . . . . . . . . . . . . . 167 8 Bilinear forms and linear mappings 170

8.1 Invertible mappings and product spaces . . . . . . . . . . . . . . 170 8.2 Projections on vector spaces . . . . . . . . . . . . . . . . . . . . . 172 8.3 Orthogonal projections . . . . . . . . . . . . . . . . . . . . . . . . 173 8.4 Projections and distances . . . . . . . . . . . . . . . . . . . . . . 175 8.5 Symmetric bilinear forms . . . . . . . . . . . . . . . . . . . . . . 178 8.6 Self-adjoint linear operators . . . . . . . . . . . . . . . . . . . . . 179 8.7 Some related continuity arguments . . . . . . . . . . . . . . . . . 181 8.8 Bounded symmetric bilinear forms . . . . . . . . . . . . . . . . . 183 8.9 Bounded symmetric linear mappings . . . . . . . . . . . . . . . . 185 8.10 Nonnegative bilinear forms and operators . . . . . . . . . . . . . 186 8.11 Some remarks about normal operators . . . . . . . . . . . . . . . 188 8.12 Positivity and invertibility . . . . . . . . . . . . . . . . . . . . . . 190 8.13 Polynomials and associative algebras . . . . . . . . . . . . . . . . 192 8.14 Polynomials of self-adjoint operators . . . . . . . . . . . . . . . . 194 8.15 Some functions on associative algebras . . . . . . . . . . . . . . . 195

9 Algebras, operators, and power series 197 9.1 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . 197 9.2 Eigenvalues of bounded linear mappings . . . . . . . . . . . . . . 198 9.3 Approximate eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 200 9.4 More on approximate eigenvalues . . . . . . . . . . . . . . . . . . 201 9.5 Inner products and σ_{ap}(T) . . . . . . . . . . . . . . . . . . . . . . 202 9.6 Invertibility and dual linear mappings . . . . . . . . . . . . . . . 204 9.7 Subadditivity of r_{A}(x) . . . . . . . . . . . . . . . . . . . . . . . . 205 9.8 Sums of nonnegative sums . . . . . . . . . . . . . . . . . . . . . . 207 9.9 Sums of generalized sums . . . . . . . . . . . . . . . . . . . . . . 209 9.10 Cauchy products . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 9.11 Cauchy products and bilinear mappings . . . . . . . . . . . . . . 212 9.12 Radius of convergence . . . . . . . . . . . . . . . . . . . . . . . . 213 9.13 Cauchy products and Banach algebras . . . . . . . . . . . . . . . 215 9.14 Power series and Banach algebras . . . . . . . . . . . . . . . . . . 217 9.15 More on f̃(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

10 Some more algebras, power series 221 10.1 Power series and Lipschitz conditions . . . . . . . . . . . . . . . . 221 10.2 Another property of power series . . . . . . . . . . . . . . . . . . 223 10.3 Taking w to be fixed . . . . . . . . . . . . . . . . . . . . . . . . . 224 10.4 The exponential function . . . . . . . . . . . . . . . . . . . . . . 226 10.5 Algebra derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 227 10.6 Commutators and conjugations . . . . . . . . . . . . . . . . . . . 228 10.7 A criterion for local invertibility . . . . . . . . . . . . . . . . . . . 230 10.8 More on local invertibility . . . . . . . . . . . . . . . . . . . . . . 232 10.9 More on the exponential function . . . . . . . . . . . . . . . . . . 234 10.10The subgroup G_{1}(A) of G(A) . . . . . . . . . . . . . . . . . . . . 235 10.11Some induced homomorphisms . . . . . . . . . . . . . . . . . . . 236 10.12Some continuity conditions . . . . . . . . . . . . . . . . . . . . . 237 10.13Some remarks about homotopies . . . . . . . . . . . . . . . . . . 239 10.14More on G_{1}(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 10.15Some additional properties of G_{1}(A) . . . . . . . . . . . . . . . . 242

III Algebras, norms, and operators, 2 244

11 Algebras, norms, and square roots 245 11.1 Exponentials of nilpotent elements . . . . . . . . . . . . . . . . . 245 11.2 One-sided inverses . . . . . . . . . . . . . . . . . . . . . . . . . . 246 11.3 Topological divisors of zero . . . . . . . . . . . . . . . . . . . . . 248 11.4 Another condition on a norm . . . . . . . . . . . . . . . . . . . . 250 11.5 Some properties of the spectrum . . . . . . . . . . . . . . . . . . 251 11.6 Commutativity and norms . . . . . . . . . . . . . . . . . . . . . . 253 11.7 More on commutativity and norms . . . . . . . . . . . . . . . . . 254 11.8 A continuity property . . . . . . . . . . . . . . . . . . . . . . . . 255 11.9 More on self-adjoint operators . . . . . . . . . . . . . . . . . . . . 256 11.10Nonnegative square roots . . . . . . . . . . . . . . . . . . . . . . 258 11.11Polar decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 258 11.12More on polar decompositions . . . . . . . . . . . . . . . . . . . . 260 11.13Commuting with adjoints . . . . . . . . . . . . . . . . . . . . . . 262 11.14Similar normal operators . . . . . . . . . . . . . . . . . . . . . . . 263 11.15More on square roots . . . . . . . . . . . . . . . . . . . . . . . . . 264

12 Complexifications, nets, and C^{∗} algebras 267 12.1 Complexifying real vector spaces . . . . . . . . . . . . . . . . . . 267 12.2 Complexifying inner products . . . . . . . . . . . . . . . . . . . . 268 12.3 Convergence of nets . . . . . . . . . . . . . . . . . . . . . . . . . 270 12.4 More on convergence of nets . . . . . . . . . . . . . . . . . . . . . 271 12.5 Nets and approximate eigenvalues . . . . . . . . . . . . . . . . . 273 12.6 C^{∗} Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 12.7 Some remarks about involutions . . . . . . . . . . . . . . . . . . 276 12.8 Self-adjoint elements . . . . . . . . . . . . . . . . . . . . . . . . . 278 12.9 Some maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . 280 12.10Commutative Banach algebras . . . . . . . . . . . . . . . . . . . 282 12.11More on the Gelfand map . . . . . . . . . . . . . . . . . . . . . . 284 12.12Bounded continuous functions . . . . . . . . . . . . . . . . . . . . 285 12.13Some remarks about weak topologies . . . . . . . . . . . . . . . . 287 12.14Some remarks about Sp(A) . . . . . . . . . . . . . . . . . . . . . 289 12.15Some more remarks about Sp(A) . . . . . . . . . . . . . . . . . . 291 13 Algebras, polynomials, and Sp(A) 293

13.1 Some remarks about continuous functions . . . . . . . . . . . . . 293 13.2 Some conditions related to compactness . . . . . . . . . . . . . . 295 13.3 Polynomials in n variables . . . . . . . . . . . . . . . . . . . . . . 296 13.4 Polynomials and homomorphisms . . . . . . . . . . . . . . . . . . 298 13.5 More on polynomials, homomorphisms . . . . . . . . . . . . . . . 299 13.6 Polynomials, homomorphisms, and Sp(A) . . . . . . . . . . . . . 300 13.7 A class of Banach algebras . . . . . . . . . . . . . . . . . . . . . . 301 13.8 Polynomial convexity . . . . . . . . . . . . . . . . . . . . . . . . . 303 13.9 A class of C^{∗} algebras . . . . . . . . . . . . . . . . . . . . . . . . 304 13.10Spectral permanence and C^{∗} algebras . . . . . . . . . . . . . . . 306 13.11Another continuity property . . . . . . . . . . . . . . . . . . . . . 307 13.12Holomorphic functions in the plane . . . . . . . . . . . . . . . . . 308 13.13Disk algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 13.14Polydisk algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 13.15Continuity and semisimplicity . . . . . . . . . . . . . . . . . . . . 313

14 Algebras, norms, and involutions 315 14.1 Some more remarks about involutions . . . . . . . . . . . . . . . 315 14.2 Involutions and ideals . . . . . . . . . . . . . . . . . . . . . . . . 316 14.3 Involutions and square roots . . . . . . . . . . . . . . . . . . . . . 318 14.4 Self-adjoint square roots . . . . . . . . . . . . . . . . . . . . . . . 319 14.5 Centralizers in associative algebras . . . . . . . . . . . . . . . . . 321 14.6 Spectra of sums and products . . . . . . . . . . . . . . . . . . . . 323 14.7 Invertibility of L_{a}, R_{a} . . . . . . . . . . . . . . . . . . . . . . . . 324 14.8 Involutions and centralizers . . . . . . . . . . . . . . . . . . . . . 325 14.9 Another criterion for self-adjointness . . . . . . . . . . . . . . . . 326 14.10A nonnegativity condition . . . . . . . . . . . . . . . . . . . . . . 327 14.11More on nonnegativity . . . . . . . . . . . . . . . . . . . . . . . . 328 14.12Sums in C^{∗} algebras . . . . . . . . . . . . . . . . . . . . . . . . . 330 14.13Nonnegativity of a^{∗} a . . . . . . . . . . . . . . . . . . . . . . . . . 331 14.14A proof of a^{∗} a ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 332 14.15Some remarks about related arguments . . . . . . . . . . . . . . 334

15 Norms, weights, and power series 335 15.1 Norms, isometries, and G(A) . . . . . . . . . . . . . . . . . . . . 335 15.2 Some remarks about unitary elements . . . . . . . . . . . . . . . 336 15.3 Weighted ℓ^{p} spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 338 15.4 Polynomials with vector coefficients . . . . . . . . . . . . . . . . . 340 15.5 Formal power series . . . . . . . . . . . . . . . . . . . . . . . . . 341 15.6 Invertibility in A[[T]] . . . . . . . . . . . . . . . . . . . . . . . . . 343 15.7 Some spaces of power series . . . . . . . . . . . . . . . . . . . . . 344 15.8 Absolute convergence of power series . . . . . . . . . . . . . . . . 346 15.9 Products and absolute convergence . . . . . . . . . . . . . . . . . 347 15.10Differentiating power series . . . . . . . . . . . . . . . . . . . . . 348 15.11Coefficients in R, C . . . . . . . . . . . . . . . . . . . . . . . . . 350 15.12Some formal compositions . . . . . . . . . . . . . . . . . . . . . . 351 15.13Some more formal compositions . . . . . . . . . . . . . . . . . . . 352 15.14Polynomials of power series . . . . . . . . . . . . . . . . . . . . . 353 15.15Some compositions of power series . . . . . . . . . . . . . . . . . 355

IV Algebras, norms, and operators, 3 358

16 Algebras, norms, and power series 359 16.1 More on differentiation . . . . . . . . . . . . . . . . . . . . . . . . 359 16.2 Some norms of derivatives . . . . . . . . . . . . . . . . . . . . . . 360 16.3 More on compositions . . . . . . . . . . . . . . . . . . . . . . . . 361 16.4 Some differential equations . . . . . . . . . . . . . . . . . . . . . 363 16.5 More on log(1 + T) . . . . . . . . . . . . . . . . . . . . . . . . . . 364 16.6 Some more logarithms . . . . . . . . . . . . . . . . . . . . . . . . 365 16.7 A basic type of composition . . . . . . . . . . . . . . . . . . . . . 367 16.8 Some more compositions . . . . . . . . . . . . . . . . . . . . . . . 368 16.9 Some compositions with convergence . . . . . . . . . . . . . . . . 370 16.10Some more convergence conditions . . . . . . . . . . . . . . . . . 371 16.11Some more exponentials of logarithms . . . . . . . . . . . . . . . 372 16.12The binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . 374 16.13Binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 16.14Using the ratio test . . . . . . . . . . . . . . . . . . . . . . . . . . 377 16.15Some related estimates . . . . . . . . . . . . . . . . . . . . . . . . 379

17 Norms, power series, and involutions 380 17.1 Another convergence property . . . . . . . . . . . . . . . . . . . . 380 17.2 Binomial series and Banach algebras . . . . . . . . . . . . . . . . 381 17.3 Some related formal power series . . . . . . . . . . . . . . . . . . 382 17.4 Logarithms and binomial series . . . . . . . . . . . . . . . . . . . 384 17.5 More on logarithms, binomial series . . . . . . . . . . . . . . . . 386 17.6 Power series and involutions . . . . . . . . . . . . . . . . . . . . . 388 17.7 Coefficients in A . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 17.8 Convergence conditions and involutions . . . . . . . . . . . . . . 390 17.9 Square roots from binomial series . . . . . . . . . . . . . . . . . . 392 17.10Some more self-adjointness arguments . . . . . . . . . . . . . . . 393 17.11Involutions and r_{A}(x) . . . . . . . . . . . . . . . . . . . . . . . . 394 17.12Identity elements and r_{A}(·) . . . . . . . . . . . . . . . . . . . . . 394 17.13Limits at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 17.14Some weighted ℓ^{∞} spaces . . . . . . . . . . . . . . . . . . . . . . 398 17.15Continuous weights . . . . . . . . . . . . . . . . . . . . . . . . . . 400 18 Algebras, involutions, and positivity 402

18.1 One-sided multiplicative identity elements . . . . . . . . . . . . . 402 18.2 Some subalgebras of L(A) . . . . . . . . . . . . . . . . . . . . . . 404 18.3 Another condition on L_{a} . . . . . . . . . . . . . . . . . . . . . . . 405 18.4 An involution on B_{1} . . . . . . . . . . . . . . . . . . . . . . . . . 407 18.5 Multiplication operators on C^{∗} algebras . . . . . . . . . . . . . . 408 18.6 Positive linear functionals . . . . . . . . . . . . . . . . . . . . . . 409 18.7 Positive functionals and identity elements . . . . . . . . . . . . . 411 18.8 Positive linear functionals on A_{1} . . . . . . . . . . . . . . . . . . 412 18.9 Another nonnegativity condition . . . . . . . . . . . . . . . . . . 413 18.10Nonnegativity and local compactness . . . . . . . . . . . . . . . . 414 18.11Positivity on Banach algebras . . . . . . . . . . . . . . . . . . . . 416 18.12Another approach to estimating λ . . . . . . . . . . . . . . . . . 418 18.13Some related properties of λ . . . . . . . . . . . . . . . . . . . . . 419 18.14Positivity and homomorphisms . . . . . . . . . . . . . . . . . . . 420 18.15Positivity and commutativity . . . . . . . . . . . . . . . . . . . . 421

19 Algebras over fields 423 19.1 Algebras and homomorphisms . . . . . . . . . . . . . . . . . . . . 423 19.2 Ideals and identity elements . . . . . . . . . . . . . . . . . . . . . 424 19.3 Invertible elements and homomorphisms . . . . . . . . . . . . . . 425 19.4 Left and right multiplication operators . . . . . . . . . . . . . . . 427 19.5 Centralizers and invertibility . . . . . . . . . . . . . . . . . . . . 428 19.6 Some algebras of polynomials . . . . . . . . . . . . . . . . . . . . 430 19.7 Subalgebras of finite dimension . . . . . . . . . . . . . . . . . . . 431 19.8 An invariant linear subspace . . . . . . . . . . . . . . . . . . . . . 431 19.9 Algebraic duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 19.10Algebraic duals and linear subspaces . . . . . . . . . . . . . . . . 434 19.11Bilinear functionals . . . . . . . . . . . . . . . . . . . . . . . . . . 436 19.12Nondegenerate bilinear forms . . . . . . . . . . . . . . . . . . . . 437 19.13More on bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . 438 19.14Involutions and conjugations . . . . . . . . . . . . . . . . . . . . 440 19.15Nondegenerate sesquilinear forms . . . . . . . . . . . . . . . . . . 442

20 Representations of associative algebras 446 20.1 Representations and modules . . . . . . . . . . . . . . . . . . . . 446 20.2 Some examples and remarks . . . . . . . . . . . . . . . . . . . . . 447 20.3 Opposite algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 20.4 Invariant linear subspaces . . . . . . . . . . . . . . . . . . . . . . 449 20.5 More on invariant linear subspaces . . . . . . . . . . . . . . . . . 451 20.6 Quotient representations . . . . . . . . . . . . . . . . . . . . . . . 452 20.7 Homomorphisms between representations . . . . . . . . . . . . . 453 20.8 Direct sums of representations . . . . . . . . . . . . . . . . . . . . 454 20.9 Nondegenerate representations . . . . . . . . . . . . . . . . . . . 456 20.10Representations and identity elements . . . . . . . . . . . . . . . 457 20.11Dual modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 20.12More on quotient spaces . . . . . . . . . . . . . . . . . . . . . . . 460 20.13Cyclic representations . . . . . . . . . . . . . . . . . . . . . . . . 462 20.14Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 20.15Schur’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

V Algebras, norms, and operators, 4 468

21 Some more representations and modules 469 21.1 Spaces of linear mappings . . . . . . . . . . . . . . . . . . . . . . 469 21.2 More on module homomorphisms . . . . . . . . . . . . . . . . . . 471 21.3 Division algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 21.4 T -Cyclic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 21.5 Representations and ∗-homomorphisms . . . . . . . . . . . . . . . 474 21.6 Involutions and bounded linear mappings . . . . . . . . . . . . . 476 21.7 Invariant subspaces and Hilbert spaces . . . . . . . . . . . . . . . 477 21.8 Some closed invariant linear subspaces . . . . . . . . . . . . . . . 478 21.9 Self-adjoint collections of operators . . . . . . . . . . . . . . . . . 480 21.10A helpful inner product . . . . . . . . . . . . . . . . . . . . . . . 481 21.11Nondegeneracy on Hilbert spaces . . . . . . . . . . . . . . . . . . 483 21.12Some topologically cyclic subspaces . . . . . . . . . . . . . . . . . 485 21.13Direct sums and products . . . . . . . . . . . . . . . . . . . . . . 487 21.14Some norms on direct sums . . . . . . . . . . . . . . . . . . . . . 488 21.15More on direct sums . . . . . . . . . . . . . . . . . . . . . . . . . 489

22 Algebras, norms, and representations 491 22.1 Direct sums of Hilbert spaces . . . . . . . . . . . . . . . . . . . . 491 22.2 Direct sums of linear subspaces . . . . . . . . . . . . . . . . . . . 492 22.3 Orthogonal direct sums . . . . . . . . . . . . . . . . . . . . . . . 493 22.4 Sums of topologically cyclic subspaces . . . . . . . . . . . . . . . 494 22.5 Boundedness of some homomorphisms . . . . . . . . . . . . . . . 496 22.6 Boundedness and identity elements . . . . . . . . . . . . . . . . . 497 22.7 Boundedness of ∗-representations . . . . . . . . . . . . . . . . . . 498 22.8 Some positive linear functionals . . . . . . . . . . . . . . . . . . . 498 22.9 Positivity and identity elements . . . . . . . . . . . . . . . . . . . 500 22.10A result about unitary equivalence . . . . . . . . . . . . . . . . . 500 22.11More on this unitary mapping . . . . . . . . . . . . . . . . . . . . 502 22.12Some more positive linear functionals . . . . . . . . . . . . . . . . 503 22.13Kernels of bilinear mappings . . . . . . . . . . . . . . . . . . . . 505 22.14Kernels of sesquilinear functionals . . . . . . . . . . . . . . . . . 506 22.15Bilinear forms and algebra involutions . . . . . . . . . . . . . . . 507 23 Linear functionals and representations 510

23.1 Some more isomorphisms . . . . . . . . . . . . . . . . . . . . . . 510 23.2 More on this isomorphism . . . . . . . . . . . . . . . . . . . . . . 512 23.3 Linear functionals from bilinear forms . . . . . . . . . . . . . . . 513 23.4 Bilinear forms from linear functionals . . . . . . . . . . . . . . . 514 23.5 Using positive linear functionals . . . . . . . . . . . . . . . . . . . 516 23.6 Some analogous arguments using A_{1} . . . . . . . . . . . . . . . . 518 23.7 Positivity and topological cyclicity . . . . . . . . . . . . . . . . . 519 23.8 Some remarks about V_{λ}, W_{λ} . . . . . . . . . . . . . . . . . . . . . 521 23.9 Some remarks about convex sets . . . . . . . . . . . . . . . . . . 522 23.10Some bounded linear functionals . . . . . . . . . . . . . . . . . . 524 23.11Positivity on C^{∗} algebras . . . . . . . . . . . . . . . . . . . . . . 525 23.12Another argument to get positivity . . . . . . . . . . . . . . . . . 526 23.13Getting some positive linear functionals . . . . . . . . . . . . . . 528 23.14Some remarks about positivity . . . . . . . . . . . . . . . . . . . 528 23.15Some related ∗-representations . . . . . . . . . . . . . . . . . . . 530

24 Involutions and representations 532 24.1 Some subsets of algebras . . . . . . . . . . . . . . . . . . . . . . . 532 24.2 Some separability conditions . . . . . . . . . . . . . . . . . . . . . 533 24.3 Comparing positive linear functionals . . . . . . . . . . . . . . . . 534 24.4 Linear functionals from linear mappings . . . . . . . . . . . . . . 535 24.5 Linear mappings from linear functionals . . . . . . . . . . . . . . 536 24.6 Showing that μ = μ_{R} . . . . . . . . . . . . . . . . . . . . . . . . . 538 24.7 More on positive linear functionals . . . . . . . . . . . . . . . . . 540 24.8 More on M(λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 24.9 Some related convex sets . . . . . . . . . . . . . . . . . . . . . . . 543 24.10Extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 24.11Irreducibility from extreme points . . . . . . . . . . . . . . . . . 546 24.12Showing that λ is not an extreme point . . . . . . . . . . . . . . 547 24.13Multiplicativity and extreme points . . . . . . . . . . . . . . . . . 549 24.14Commutativity and extreme points . . . . . . . . . . . . . . . . . 550 24.15Weak^{∗} compactness of P . . . . . . . . . . . . . . . . . . . . . . . 552

25 Algebras, matrices, and norms 554 25.1 Algebras of matrices . . . . . . . . . . . . . . . . . . . . . . . . . 554 25.2 Matrices and linear mappings . . . . . . . . . . . . . . . . . . . . 555 25.3 More on mappings and matrices . . . . . . . . . . . . . . . . . . 558 25.4 Some norms on V^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 559 25.5 Some norms on matrices . . . . . . . . . . . . . . . . . . . . . . . 560 25.6 Some estimates for T_{a} . . . . . . . . . . . . . . . . . . . . . . . . 562 25.7 Some estimates for R_{b} . . . . . . . . . . . . . . . . . . . . . . . . 563 25.8 Some norms of products . . . . . . . . . . . . . . . . . . . . . . . 564 25.9 Some estimates for T_{a}^{V} . . . . . . . . . . . . . . . . . . . . . . . . 566 25.10Transposes and involutions . . . . . . . . . . . . . . . . . . . . . 568 25.11Traces of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 569 25.12Traces of linear mappings . . . . . . . . . . . . . . . . . . . . . . 571 25.13Traces and linear functionals . . . . . . . . . . . . . . . . . . . . 573 25.14A continuity result . . . . . . . . . . . . . . . . . . . . . . . . . . 575 25.15Some norms on C^{2} . . . . . . . . . . . . . . . . . . . . . . . . . . 577

Appendices 581

A Absolute values on fields 581 A.1 Metrics and ultrametrics . . . . . . . . . . . . . . . . . . . . . . . 581 A.2 Absolute value functions . . . . . . . . . . . . . . . . . . . . . . . 582 A.3 The archimedean property . . . . . . . . . . . . . . . . . . . . . . 584 A.4 Equivalent absolute value functions . . . . . . . . . . . . . . . . . 585 A.5 Norms and seminorms . . . . . . . . . . . . . . . . . . . . . . . . 585 A.6 Some norms on k^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 587 A.7 Sequences and series . . . . . . . . . . . . . . . . . . . . . . . . . 588 A.8 Discrete absolute value functions . . . . . . . . . . . . . . . . . . 590 A.9 Supremum seminorms and c_{0} spaces . . . . . . . . . . . . . . . . 591 A.10 Lipschitz and operator seminorms . . . . . . . . . . . . . . . . . 593 A.11 ℓ^{r} Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 A.12 Some bounded linear mappings . . . . . . . . . . . . . . . . . . . 596 A.13 Sums of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 A.14 Some related spaces of functions . . . . . . . . . . . . . . . . . . 600

B More on metrics and norms 603 B.1 q-Metrics and q-semimetrics . . . . . . . . . . . . . . . . . . . . . 603 B.2 More on q-metrics and q-semimetrics . . . . . . . . . . . . . . . . 604 B.3 q-Absolute value functions . . . . . . . . . . . . . . . . . . . . . . 605 B.4 Quasimetric absolute value functions . . . . . . . . . . . . . . . . 606 B.5 q-Norms and q-seminorms . . . . . . . . . . . . . . . . . . . . . . 608 B.6 ℓ^{r} Spaces and q-seminorms . . . . . . . . . . . . . . . . . . . . . 609 B.7 More on ℓ^{r} spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 610 B.8 q-Absolute convergence . . . . . . . . . . . . . . . . . . . . . . . 611 B.9 Some more Lipschitz conditions . . . . . . . . . . . . . . . . . . . 612 B.10 Operator q-seminorms . . . . . . . . . . . . . . . . . . . . . . . . 614 B.11 Some more bounded linear mappings . . . . . . . . . . . . . . . . 615 B.12 More on sums of vectors . . . . . . . . . . . . . . . . . . . . . . . 617

Bibliography 618

Index 639

Part I

Chapter 1

Norms and bounded linear mappings Norms and Banach Spaces

1.1 Norms on vector spaces

Let V be a vector space over the real or complex numbers. A nonnegative realvalued function N on V is said to be a norm on V if it satisfies the following three conditions. First, for each v ∈ V,

(1.1.1) N(v) = 0 if and only if v = 0.

Second, N(t v) = |t| N(v) (1.1.2)

for every v ∈ V and t ∈ R or C, as appropriate. Here |t| denotes the usual absolute value of the real or complex number t. Third,

N(v + w) ≤ N(v) + N(w) (1.1.3)

for every v, w ∈ V. This is known as the triangle inequality for N.

Of course, the real line R and complex plane C may be considered as onedimensional vector spaces over themselves. The usual absolute value functions on R and C may be considered as norms on these spaces. More precisely, the only norms on R or C, as vector spaces over themselves, are positive constant multiples of the usual absolute value functions.

If V is a vector space over the complex numbers, then V may be considered as a vector space over the real numbers, by simply restricting scalar multiplication on V to real numbers. We may use

V_{R} (1.1.4)

to denote V considered as a vector space over R in this way. If N is a norm on V as a vector space over C, then N may also be considered as a norm on V_{R}. However, a norm on V_{R} may not be a norm on V, as a vector space over the complex numbers.

Let X be a nonempty set, and note that the spaces of all real or complexvalued functions on X are vector spaces over the real and complex numbers, respectively, with respect to pointwise addition and scalar multiplication of functions on X. Also let

ℓ^{∞}(X, R), ℓ^{∞}(X, C) (1.1.5)

be the spaces of bounded real and complex-valued functions on X, respectively. It is well known and not difficult to check that these are linear subspaces of the spaces of all real and complex-valued functions on X, respectively. In particular, ℓ^{∞}(X, R) and ℓ^{∞}(X, C) are vector spaces over the real and complex numbers, respectively, with respect to pointwise addition and scalar multiplication of functions on X.

If f is a bounded real or complex-valued function on X, then put

∥f∥_{∞} = sup{|f(x)| : x ∈ X}. (1.1.6)

One can verify that this defines a norm on each of ℓ^{∞}(X, R) and ℓ^{∞}(X, C), as vector spaces over the real and complex numbers, respectively. This is known as the supremum norm, and we may sometimes use the notation

∥f∥_{sup} (1.1.7)

instead. We may also use the notation

∥f∥_{ℓ^{∞}(X,R)}, ∥f∥_{ℓ^{∞}(X,C)} (1.1.8)

as well, to be more precise.

Let V be a vector space over the real or complex numbers again, and let N be a norm on V. It is easy to see that

d_{N}(v, w) = N(v − w) (1.1.9)

defines a metric on V. The metric on ℓ^{∞}(X, R) or ℓ^{∞}(X, C) associated to the supremum norm in this way is the usual supremum metric, for instance.

In particular, all of the usual notions and results about metric spaces can be used in this situation. Of course, there are a lot of additional properties in this case.

1.2 Seminorms and convex sets

Let V be a vector space over the real numbers. A subset E of V is said to be convex if for every v, w ∈ E and real number t with 0 ≤ t ≤ 1, we have that

(1 − t) v + t w ∈ E. (1.2.1)

If V is a vector space over the complex numbers, then one can use the same definition, which basically corresponds to considering V as a vector space over the real numbers.

Now let V be a vector space over the real or complex numbers. A nonnegative real-valued function N on V is said to be a seminorm or pseudonorm if it satisfies the homogeneity condition (1.1.2) and triangle inequality (1.1.3). Note that (1.1.2) implies that N(0) = 0, by taking t = 0. Thus a seminorm N on V is a norm if and only if N(v) > 0 for every v ∈ V with v ≠ 0.

Let N be a seminorm on V. If v ∈ V and r is a positive real number, then the open ball in V centered at v with radius r may be defined by

B(v, r) = B_{N}(v, r) = {w ∈ V : N(v − w) < r}. (1.2.2)

Similarly, if r is a nonnegative real number, then the closed ball in V centered at v with radius r may be defined by

\overline{B}(v, r) = \overline{B}_{N}(v, r) = \{w ∈ V : N(v − w) ≤ r\}. (1.2.3)

One can check that these are convex sets in V.

If N is a norm on V, then (1.2.2) and (1.2.3) are the same as the open and closed balls in V determined by the metric d_{N}(·, ·) on V associated to N. If N is a seminorm on V, then (1.1.9) defines a semimetric or pseudometric on V. This means that it satisfies the same conditions as a metric, except that d_{N}(v, w) = 0 does not necessarily imply that v = w.

Let N be a nonnegative real-valued function on V that satisfies the homogeneity condition (1.1.2). We may still use the notation (1.2.2) and (1.2.3) in this case, even if (1.1.9) is not necessarily a metric or semimetric on V. If the open unit ball B_{N}(0, 1) in V with respect to N is convex, then it is well known that N satisfies the triangle inequality, and is thus a seminorm on V.

To see this, let v, w ∈ V be given, and let r, t be real numbers with

N(v) < r, N(w) < t. (1.2.4)

This implies that r^{−1} v, t^{−1} w ∈ B_{N}(0, 1). (1.2.5)

If B_{N}(0, 1) is convex, then it follows that

(r + t)^{−1} (v + w) = (r (r + t)^{−1}) (r^{−1} v) + (t (r + t)^{−1}) (t^{−1} w) (1.2.6)

is an element of B_{N}(0, 1) too. This means that

N(v + w) < r + t. (1.2.7)

One can use this to get (1.1.3).

Similarly, if the closed unit ball \overline{B}_{N}(0, 1) is convex, then N satisfies the triangle inequality on V. More precisely, it would be enough to know that (1.2.5) implies that (1.2.6) is an element of \overline{B}_{N}(0, 1). In this case, the inequality in (1.2.7) might not be strict, but the non-strict version would be sufficient to get (1.1.3).

If N is a seminorm on V, then

N(v) ≤ N(w) + N(v − w) (1.2.8)

1.3. SOME NORMS ON R^{N}, C^{N} 5

for every v, w ∈ V, and similarly with the roles of v and w interchanged. One can use this to check that

|N(v) − N(w)| ≤ N(v − w) (1.2.9)

for all v, w ∈ V, using the ordinary absolute value of a real number on the left side. If N is a norm on V, then this implies that N is uniformly continuous as a real-valued function on V, with respect to the metric d_{N}(·, ·) associated to N on V, and the standard Euclidean metric on R.

1.3 Some norms on R^{n}, C^{n}

Let n be a positive integer, and let R^{n}, C^{n} be the usual spaces of n-tuples of real and complex numbers, respectively. These are vector spaces over the real and complex numbers, respectively, with respect to coordinatewise addition and scalar multiplication.

If v ∈ R^{n} or C^{n}, then put

∥v∥_{∞} = max_{1≤j≤n} |v_{j}|. (1.3.1)

One can check that this defines a norm on each of R^{n} and C^{n}. This corresponds to the suremum norm defined in Section 1.1, with X equal to the set of integers from 1 to n.

Let p be a positive real number, and put

∥v∥_{p} = ( ∑_{j=1}^{n} |v_{j}|^{p})^{1/p} (1.3.2)

for each v ∈ R^{n} or C^{n}. Observe that

(1.3.3) ∥v∥_{p} = 0 if and only if v = 0.

We also have that ∥t v∥_{p} = |t| ∥v∥_{p} (1.3.4)

for each v ∈ R^{n} or C^{n} and t ∈ R or C, as appropriate.

If p ≥ 1, then it is well known that (1.3.2) satisfies the triangle inequality on R^{n} and C^{n}, and thus defines a norm. This is Minkowski’s inequality for finite sums. Of course, this follows easily from the triangle inequality for the standard absolute value functions on R and C when p = 1.

If p > 1, then the triangle inequality can be obtained from the convexity of the open or closed unit ball, as in the previous section. This also uses the convexity of r^{p} as a function of a nonnegative real number r. More precisely, if x, y, and t are nonnegative real numbers, and t ≤ 1, then

((1 − t) x + t y)^{p} ≤ (1 − t) x^{p} + t y^{p}. (1.3.5)

If p = 2, then (1.3.2) is the standard Euclidean norm on R^{n}, C^{n}. One can also look at this in terms of inner products, and we shall return to that in Section 1.9.

If n = 1, then (1.3.2) is the same as the standard asolute value function on R, C. If n ≥ 2 and 0 < p < 1, then one can check that (1.3.2) does not satisfy the triangle inequality on R^{n} or C^{n}. However, it is well known that

∥v + w∥_{p}^{p} ≤ ∥v∥_{p}^{p} + ∥w∥_{p}^{p} (1.3.6)

for all v, w ∈ R^{n} or C^{n} when 0 < p ≤ 1. This implies that

∥v − w∥_{p}^{p} (1.3.7)

defines a metric on each of R^{n} and C^{n} when 0 < p ≤ 1.

Clearly ∥v∥_{∞} ≤ ∥v∥_{p} (1.3.8)

for every v ∈ R^{n} or C^{n} and positive real number p. It is also easy to see that

∥v∥_{p} ≤ n^{1/p} ∥v∥_{∞} (1.3.9)

for all v ∈ R^{n} or C^{n} and 0 < p < ∞. It is well known that n^{1/p} → 1 as p → ∞. This implies that

lim_{p→∞} ∥v∥_{p} = ∥v∥_{∞} (1.3.10)

for all v ∈ R^{n} or C^{n}.

1.4 Linear subspaces and continuous functions

Let V be a vector space over the real or complex numbers, and let W be a linear subspace of V. If N is a norm on V, then the restriction of N to W is a norm on W. Similarly, if N is a seminorm on V, then the restriction of N to W is a seminorm on W.

Let N be a norm on V, and let d_{N}(u, v) be the corresponding metric on V. Note that the restriction of d_{N}(u, v) to u, v ∈ W is the same as the metric on W associated to the restriction of N to W.

One can check that the closure \overline{W} of W in V with respect to d_{N}(·, ·) is a linear subspace of V as well. This can be viewed in terms of convergent sequences, which will be discussed further in the next section.

Let X be a nonempty metric space, or topological space. Consider the spaces C(X, R), C(X, C) of continuous real and complex-valued functions on X. More precisely, this uses the standard metrics or topologies on R, C, respectively. It is well known that C(X, R), C(X, C) are vector spaces over the real and complex numbers, respectively, with respect to pointwise addition and scalar multiplication of functions. Equivalently, these are linear subspaces of the corresponding spaces of all real and complex-valued functions on X, respectively.

Consider the spaces

C_{b}(X, R) = C(X, R) ∩ ℓ^{∞}(X, R), C_{b}(X, C) = C(X, C) ∩ ℓ^{∞}(X, C) (1.4.1)

of all bounded continuous real and complex-valued functions on X, respectively. These are linear subspaces of the spaces of all continuous real and complexvalued functions on X, respectively, as well as the spaces of all bounded real and complex-valued functions on X. In particular, the restriction of the supremum norm to each of these spaces is a norm, that we shall normally denote ∥ · ∥_{sup}.

It is well known that

(1.4.2) C_{b}(X, R), C_{b}(X, C) are closed subsets of ℓ^{∞}(X, R), ℓ^{∞}(X, C),

respectively, with respect to the supremum metric. This basically corresponds to the fact that if {f_{j}}_{j=1}^{∞} is a sequence of continuous real or complex-valued functions on X that converges uniformly to a real or complex-valued function f on X, then f is continuous on X.

If X is compact, then every continuous real or complex-valued function f on X is bounded. More precisely, f(X) is a compact subset of the real line or complex plane, as appropriate, with respect to the standard Euclidean metric in this case. In particular, this implies that f(X) is a bounded subset of R or C, as appropriate.

If X is equipped with the discrete metric or topology, then every function on X is continuous. This means that C_{b}(X, R), C_{b}(X, C) are the same as ℓ^{∞}(X, R), ℓ^{∞}(X, C) , respectively, in this case.

1.5 Sequences, series, and Banach spaces

Let V be a vector space over the real or complex numbers, let N be a norm on V, and let d_{N}(·, ·) be the corresponding metric on V. Suppose that {v_{j}}_{j=1}^{∞} and {w_{j}}_{j=1}^{∞} are sequences of elements of V that converge to elements v and w of V, respectively, with respect to d_{N}(·, ·). Under these conditions, it is easy to see that {v_{j} + w_{j}}_{j=1}^{∞} converges to v + w with respect to d_{N}(·, ·), so that

(1.5.1)  lim_{j→∞}(v_{j} + w_{j}) = ( lim_{j→∞} v_{j}) + ( lim_{j→∞} w_{j}).

This is analogous to the corresponding statement for convergent sequences of real or complex numbers. Similarly, if t ∈ R or C, as appropriate, then {t v_{j}}_{j=1}^{∞} converges to t v with respect to d_{N}(·, ·) on V, so that

lim_{j→∞}(t v_{j}) = t ( lim_{j→∞} v_{j}). (1.5.2)

If {t_{j}}_{j=1}^{∞} is a sequence of real or complex numbers that converges to t ∈ R or C, as appropriate, then one can check that {t_{j} v_{j}}_{j=1}^{∞} converges to t v with respect to d_{N}(·, ·) on V, so that

(1.5.3)  lim_{j→∞}(t_{j} v_{j}) = ( lim_{j→∞} t_{j}) ( lim_{j→∞} v_{j}).

More precisely, one can verify that

lim_{j→∞}(t_{j} v_{j}) = 0 (1.5.4)

when {t_{j}}_{j=1}^{∞} is a bounded sequence of real or complex numbers, as appropriate, and {v_{j}}_{j=1}^{∞} converges to 0 in V. Similarly, (1.5.4) holds when {v_{j}}_{j=1}^{∞} is a bounded sequence in V, and {t_{j}}_{j=1}^{∞} converges to 0 in R or C, as appropriate. One can obtain (1.5.3) from these other properties of limits in V, in essentially the same way as for products of convergent sequences of real and complex numbers.

An infinite series ∑_{j=1}^{∞} v_{j} with terms in V is said to converge in V if the corresponding sequence of partial sums ∑_{j=1}^{n} v_{j} converges in V with respect to d_{N}(·, ·). In this case, the value of the sum is defined by

∑_{j=1}^{∞} v_{j} = lim_{n→∞} ∑_{j=1}^{n} v_{j}, (1.5.5)

as usual. If t ∈ R or C, as appropriate, then it is easy to see that ∑_{j=1}^{∞} t v_{j} converges, with

∑_{j=1}^{∞}(t v_{j}) = t ∑_{j=1}^{∞} v_{j}. (1.5.6)

If ∑_{j=1}^{∞} w_{j} is another convergent series with terms in V, then ∑_{j=1}^{∞}(v_{j} + w_{j}) converges, with

∑_{j=1}^{∞}(v_{j} + w_{j}) = ∑_{j=1}^{∞} v_{j} + ∑_{j=1}^{∞} w_{j}. (1.5.7)

It is sometimes convenient to consider infinite series starting at j = 0, for which there are analogous statements.

It is well known that convergent sequences in metric spaces are Cauchy sequences. Let ∑_{j=1}^{∞} v_{j} be an infinite series with terms in V again. One can check that the corresponding sequence of partial sums is a Cauchy sequence if and only if for every ϵ > 0 there is a positive integer L(ϵ) such that

N( ∑_{j=l}^{n} v_{j}) < ϵ (1.5.8)

for all positive integers l, n with n ≥ l ≥ L(ϵ). In particular, this implies that

N(v_{l}) < ϵ (1.5.9)

for all l ≥ L(ϵ), by taking n = l. This means that

lim_{l→∞} v_{l} = 0 (1.5.10)

in V, with respect to d_{N}(·, ·).

If V is complete as a metric space with respect to d_{N}(·, ·), then V is said to be a Banach space with respect to N. In this case, the criterion for the sequence of partial sums to be a Cauchy sequence in the preceding paragraph implies the convergence of the infinite series. This is analogous to another classical fact about convergent series of real and complex numbers.

1.6 Completeness and subsets

Let (M, d(·, ·)) be a metric space, and let E be a subset of M. The restriction of d(·, ·) to E defines a metric on E, so that E may be considered as a metric space as well. If M is complete as a metric space with respect to d(·, ·), and if E is a closed set in M, then

(1.6.1) E is complete as a metric space,

with respect to the restriction of d(·, ·) to E. This uses the fact that a sequence of elements of E is a Cauchy sequence in E if and only if it is a Cauchy sequence as a sequence of elements of M.

If E is complete as a metric space, then one can check that

(1.6.2) E is a closed set in M.

More precisely, this does not use completeness of M. It suffices to show that if a sequence of elements of E converges to an element of M, then the limit of the sequence is in E. Such a sequence may be considered as a Cauchy sequence in E, which converges to an element of E, by hypothesis. This is the same as the limit of the sequence in M, by uniqueness of the limit.

Let V be a real or complex vector space with a norm N, and let W be a linear subspace of V. If V is a Banach space with respect to N, and if W is a closed set in V with respect to the metric d_{N}(·, ·) associated to N, then it follows that (1.6.3) W is a Banach space,

with respect to the restriction of N to W. If W is a Banach space with respect to the restriction of N to W, then W is a closed set in V with respect to d_{N}(·, ·), as in the preceding paragraph.

If X is a nonempty set, then it is well known that

(1.6.4) ℓ^{∞}(X, R) and ℓ^{∞}(X, C) are Banach spaces,

with respect to the supremum norm. Indeed, let {f_{j}}_{j=1}^{∞} be a Cauchy sequence of bounded real or complex-valued functions on X with respect to the supremum metric. It is easy to see that for each x ∈ X,

(1.6.5) {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence

in R or C, as appropriate, with respect to the standard Euclidean metric. This implies that {f_{j}(x)}_{j=1}^{∞} converges in R or C, as appropriate, because the real line and complex plane are complete with respect to their standard metrics. Put

f(x) = lim_{j→∞} f_{j}(x) (1.6.6)

for each x ∈ X, which defines f as a real or complex-valued function on X, as appropriate. Thus {f_{j}}_{j=1}^{∞} converges to f pointwise on X, by construction. One can use this and the fact that {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to the supremum metric to get that

(1.6.7) {f_{j}}_{j=1}^{∞} converges to f uniformly on X.

Using this, it is easy to see that f is bounded on X, because f_{j} is bounded on X for each j, by hypothesis. This means that f ∈ ℓ^{∞}(X, R) or ℓ^{∞}(X, C), as appropriate, and one can verify that {f_{j}}_{j=1}^{∞} converges to f with respect to the supremum metric.

Suppose now that X is a nonempty metric space, or topological space. Remember that C_{b}(X, R), C_{b}(X, C) are closed sets in ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively, with respect to the supremum metric, as in Section 1.4. It follows that (1.6.8) C_{b}(X, R) and C_{b}(X, C) are Banach spaces,

with respect to the supremum norm too.

1.7 Absolute convergence

It is well known that an infinite series ∑_{j=1}^{∞} a_{j} of nonnegative real numbers converges if and only if the corresponding sequence of partial sums is bounded. In this case,

∑_{j=1}^{∞} a_{j} = sup_{n≥1} ∑_{j=1}^{n} a_{j}. (1.7.1)

Let V be a vector space over the real or complex numbers, and let N be a norm on V. An infinite series ∑_{j=1}^{∞} v_{j} with terms in V is said to converge absolutely with respect to N if

∑_{j=1}^{∞} N(v_{j}) (1.7.2)

converges as an infinite series of nonnegative real numbers.

If l, n are positive integers with l ≤ n, then

N( ∑_{j=l}^{n} v_{j}) ≤ ∑_{j=l}^{n} N(v_{j}), (1.7.3)

by the triangle inequality. If ∑_{j=1}^{∞} v_{j} converges absolutely with respect to N, then the corresponding sequence of partial sums is a Cauchy sequence with respect to the metric d_{N} on V associated to N. This follows from the characterization of the Cauchy condition mentioned in Section 1.5.

If V is a Banach space with respect to N, then we get that ∑_{j=1}^{∞} v_{j} converges in V. One can also check that

N( ∑_{j=1}^{∞} v_{j}) ≤ ∑_{j=1}^{∞} N(v_{j}) (1.7.4)

under these conditions.

More precisely, it is well known that the completeness of V with respect to d_{N} is characterized by the condition that absolute convergence of an infinite series with respect to N imply convergence. To see this, let {w_{j}}_{j=1}^{∞} be a Cauchy sequence of elements of V with respect to d_{N}. One can find a subsequence {w_{j_{l}}}_{l=1}^{∞} of {w_{j}}_{j=1}^{∞} such that

∑_{l=1}^{∞} N(w_{j_{l}} − w_{j_{l+1}}) (1.7.5)

converges as an infinite series of nonnegative real numbers. Indeed, one can choose the j_{l}’s so that

N(w_{j_{l}} − w_{j_{l+1}}) ≤ 2^{−l} (1.7.6)

for each l, for instance.

Thus we get that

∑_{l=1}^{∞}(w_{j_{l}} − w_{j_{l+1}}) (1.7.7)

converges in V, by hypothesis. Note that

∑_{l=1}^{n}(w_{j_{l}} − w_{j_{l+1}}) = w_{j_{1}} − w_{j_{n+1}} (1.7.8)

for every positive integer n. This means that the convergence of (1.7.7) is equivalent to the convergence of {w_{j_{l}}}_{l=1}^{∞} to an element of V.

It is well known and not difficult to check that if a Cauchy sequence of elements of a metric space has a subsequence that converges to an element of the metric space, then the whole Cauchy sequence converges to the same limit. It follows that {w_{j}}_{j=1}^{∞} converges to an element of V with respect to d_{N}.

Let X be a nonempty set, and suppose that V is the space of bounded real or complex-valued functions on X, equipped with the supremum norm. In this case, the absolute convergence of an infinite series with terms in V with respect to the supremum norm corresponds to a well-known criterion of Weierstrass for the uniform convergence of the corresponding sequence of partial sums.

1.8 Holomorphic functions

Let U be a nonempty open subset of the complex plane, with respect to the standard Euclidean metric on C. Consider the space H(U) of all complexvalued functions on U that are complex-analytic, or holomorphic. This is a linear subspace of the space C(U, C) of all continuous complex-valued functions on U, with respect to the restriction of the standard Euclidean metric on C to U.

If {f_{j}}_{j=1}^{∞} is a sequence of holomorphic functions on U that converges uniformly to a complex-valued function f on U, then it is well known that f is holomorphic on U as well. Indeed, one can use the Cauchy integral formula for the f_{j}’s to get that f can be expressed in the same way. This also works when {f_{j}}_{j=1}^{∞} converges to f uniformly on compact subsets of U.

Let H^{∞}(U) = H(U) ∩ C_{b}(U, C) (1.8.1)

be the space of all bounded holomorphic functions on U. This is a linear subspace of each of the spaces of all holomorphic functions on U, and all bounded continuous complex-valued functions on U. In fact, H^{∞}(U) is a closed set in C_{b}(U, C), with respect to the supremum metric, as in the preceding paragraph. Thus H^{∞}(U) is a Banach spaces with respect to the supremum norm.

Let \overline{U} be the closure of U in C, with respect to the standard Euclidean metric. Consider the space A(U) of continuous complex-valued functions on \overline{U} that are holomorphic on U. This is a linear subspace of the space C(\overline{U}, C) of all continuous complex-valued functions on \overline{U}. Similarly, let

A_{b}(U) = A(U) ∩ C_{b}(\overline{U}, C) (1.8.2)

be the space of bounded continuous complex-valued functions on \overline{U} that are holomorphic on U. Of course, if U is a bounded open set in C, then \overline{U} is compact, and A_{b}(U) = A(U). As before, A_{b}(U) is a closed linear subspace of C_{b}(\overline{U}, C), with respect to the supremum metric. This means that A_{b}(U) is a Banach space too, with respect to the supremum norm.

There are many examples of Banach spaces related to complex analysis like these. There are also analogous spaces of harmonic functions on open sets in R^{n}, and spaces of holomorphic functions of several complex variables, for instance. Of course, Banach spaces and related notions are involved in the study of complex analysis and differential equations in other ways too. See [190, 230, 267] for some nice introductions to several complex variables, as well as [191] for more information.

1.9 Inner products and Hilbert spaces

Let V and W be vector spaces over the complex numbers. It is sometimes convenient to refer to a linear mapping from V into W, as vector spaces over C, as being complex-linear. Similarly, it is sometimes convenient to refer to a linear 1.9. INNER PRODUCTS AND HILBERT SPACES 13 mapping from V into W, considered as vector spaces over the real numbers, as being real-linear. Thus a real-linear mapping T from V into W is complex-linear if and only if T (i v) = i T (v) (1.9.1) for every v ∈ V.

A real-linear mapping T from V into W is said to be conjugate-linear if T (i v) = −i T (v) (1.9.2) for every v ∈ V. This implies that T (a v) = \overline{a} T (v) (1.9.3) for every v ∈ V and a ∈ C. Here \overline{a} is the complex-conjugate of a ∈ C, as usual.

Now let V be a vector space over the real or complex numbers. An inner product on V is a real or complex-valued function ⟨v, w⟩ = ⟨v, w⟩_{V}, as appropriate, defined for v, w ∈ V, that satisfies the following three conditions. First, for each w ∈ V, ⟨v, w⟩ should be linear as a function of v. Second, for each v, w ∈ V, we should have that ⟨v, w⟩ = ⟨w, v⟩ (1.9.4) in the real case, and ⟨v, w⟩ = \overline{⟨w, v⟩} (1.9.5) in the complex case. This implies that for each v ∈ V, ⟨v, w⟩ is linear as a function of w in the real case, and conjugate-linear as a function of w in the complex case. This also implies that ⟨v, v⟩ ∈ R (1.9.6) for each v ∈ V in the complex case. The third condition is that ⟨v, v⟩ > 0 (1.9.7) for each v ∈ V with v ≠ 0. Of course, ⟨v, v⟩ = 0 when v = 0.

If ⟨·, ·⟩ is an inner product on V, then we put ∥v∥ = ∥v∥_{V} = ⟨v, v⟩^{1/2} (1.9.8) for each v ∈ V, using the nonnegative square root on the right side. Note that ∥v∥ = 0 if and only if v = 0, and that ∥t v∥ = |t| ∥v∥ (1.9.9) for each v ∈ V and t ∈ R or C, as appropriate. The Cauchy–Schwarz inequality states that |⟨v, w⟩| ≤ ∥v∥ ∥w∥ (1.9.10) for all v, w ∈ V. This can be obtained from the fact that

⟨v + t w, v + t w⟩ ≥ 0 (1.9.11)

for every t ∈ R or C, as appropriate. One can use the Cauchy–Schwarz inequality to get that ∥v + w∥ ≤ ∥v∥ + ∥w∥ (1.9.12)

for all v, w ∈ V, so that (1.9.8) defines a norm on V.

If V is a complex vector space, then

⟨v, w⟩_{V_{R}} = Re⟨v, w⟩_{V} (1.9.13)

defines an inner product on V as a vector space over the real numbers, where Re a is the real part of a complex number a. Clearly the norm associated to (1.9.13) is the same as before.

If n is a positive integer, then the standard inner product on R^{n} is defined by

⟨v, w⟩ = ⟨v, w⟩_{R^{n}} = ∑_{j=1}^{n} v_{j} w_{j}. (1.9.14)

Similarly, the standard inner product on C^{n} is defined by

⟨v, w⟩ = ⟨v, w⟩_{C^{n}} = ∑_{j=1}^{n} v_{j} \overline{w_{j}}. (1.9.15)

The norms associated to these inner products are the same as the standard Euclidean norms, as in Section 1.3.

Let V be a real or complex vector space with an inner product ⟨·, ·⟩ again. If V is complete with respect to the metric associated to the corresponding norm ∥ · ∥, then V is said to be a Hilbert space. In particular, this means that V is a Banach space with respect to ∥ · ∥.

1.10 Sums of orthogonal vectors

Let ∑_{j=1}^{∞} a_{j} be a convergent series of nonnegative real numbers. It is well known that ∑_{j=1}^{∞} a_{j}^{2} also converges under these conditions. This uses the fact that the sequence of a_{j}’s is bounded in this case. More precisely, we have that

∑_{j=1}^{∞} a_{j}^{2} ≤ ( sup_{l≥1} a_{l}) ∑_{j=1}^{∞} a_{j}. (1.10.1)

Let V be a vector space over the real or complex numbers with an inner product ⟨·, ·⟩, and let ∥ · ∥ be the corresponding norm on V, as in the previous section. We say that v, w ∈ V are orthogonal with respect to ⟨·, ·⟩ if

⟨v, w⟩ = 0. (1.10.2)

1.11. ARBITRARY NORMS ON R^{N}, C^{N} 15

Note that this condition is symmetric in v and w. In this case, we get that

∥v + w∥^{2} = ∥v∥^{2} + ∥w∥^{2}. (1.10.3)

Suppose that ∑_{j=1}^{∞} v_{j} is an infinite series of pairwise-orthogonal vectors in V. This means that ⟨v_{j}, v_{l}⟩ = 0 (1.10.4)

when j ≠ l. If l, n are positive integers with l ≤ n, then

∥∑_{j=l}^{n} v_{j}∥^{2} = ∑_{j=l}^{n} ∥v_{j}∥^{2}. (1.10.5)

If we take l = 1 in (1.10.5), then we get that the sequence of partial sums

∑_{j=1}^{n} v_{j} (1.10.6)

is bounded in V with respect to ∥ · ∥ if and only if the sequence of partial sums

∑_{j=1}^{n} ∥v_{j}∥^{2} (1.10.7)

is bounded in R. Of course, this happens if and only if

∑_{j=1}^{∞} ∥v_{j}∥^{2} (1.10.8)

converges as an infinite series of nonnegative real numbers. If this series converges, then the right side of (1.10.5) is as small as we want when l is large enough. This implies that the sequence of partial sums (1.10.6) is a Cauchy sequence in V with respect to the metric associated to ∥ · ∥, as in Section 1.5.

Suppose now that V is a Hilbert space. If (1.10.8) converges, then it follows that ∑_{j=1}^{∞} v_{j} converges in V. In this case, we also get that

∥∑_{j=1}^{∞} v_{j}∥^{2} = ∑_{j=1}^{∞} ∥v_{j}∥^{2}. (1.10.9)

Of course, if the sequence of partial sums (1.10.6) is a Cauchy sequence in V with respect to the metric associated to ∥ · ∥, then it is bounded. In particular, this happens when ∑_{j=1}^{∞} v_{j} converges in V.

1.11 Arbitrary norms on R^{n}, C^{n}

Let n be a positive integer, and let N be a seminorm on R^{n} or C^{n}. Also let e_{1}, …, e_{n} be the standard basis vectors in R^{n} or C^{n}, so that the lth coordinate of e_{j} is equal to 1 when j = l, and to 0 otherwise. It is easy to see that

N(v) ≤ ∑_{j=1}^{n} N(e_{j}) |v_{j}| (1.11.1)

for all v ∈ R^{n} or C^{n}, as appropriate, by expressing v as a linear combination of the e_{j}’s, and using the triangle inequality. This implies that

N(v) ≤ ( ∑_{j=1}^{n} N(e_{j})^{2})^{1/2} ∥v∥_{2} (1.11.2)

for each v ∈ R^{n} or C^{n}, as appropriate, where ∥v∥_{2} is the standard Euclidean norm of v, as in Section 1.3. This also uses the Cauchy–Schwarz inequality.

It follows that

|N(v) − N(w)| ≤ N(v − w) ≤ ( ∑_{j=1}^{n} N(e_{j})^{2})^{1/2} ∥v − w∥_{2} (1.11.3)

for all v, w ∈ R^{n} or C^{n}, as appropriate, using (1.2.9) in the first step. In particular, this means that N is uniformly continuous as a real-valued function on R^{n} or C^{n}, as appropriate, with respect to the corresponding Euclidean metric on the domain and range.

Suppose now that N is a norm on R^{n} or C^{n}, as appropriate. The extreme value theorem implies that N attains its minimum on the unit sphere in R^{n} or C^{n}, as appropriate, with respect to the standard Euclidean metric. The minimum of N on the unit sphere is positive, because N is a norm. This implies that there is a positive real number c such that

N(u) ≥ c (1.11.4)

for each v ∈ R^{n} or C^{n}, as appropriate, with ∥v∥_{2} = 1. Using this, one can check that c ∥v∥_{2} ≤ N(v) (1.11.5)

for all v ∈ R^{n} or C^{n}, as appropriate.

It is well known that R^{n} and C^{n} are complete with respect to their standard Euclidean metrics. One can use this and (1.11.2), (1.11.5) to get the completeness of R^{n} or C^{n}, as appropriate, with respect to the metric associated to N.

If V is any vector space over the real or complex numbers of positive finite dimension n, then V is isomorphic to R^{n} or C^{n}, as appropriate, as a vector space. If N_{V} is a norm on V, then it follows that V is complete with respect to the metric associated to N_{V}, as before.

1.12 Functions with finite support

Let X be a nonempty set, and let f be a real or complex-valued function on X. The support of f in X may be defined by

supp f = {x ∈ X : f(x) ≠ 0}. (1.12.1)

Let c_{00}(X, R), c_{00}(X, C) be the spaces of real and complex-valued functions f on X such that

(1.12.2) supp f has only finitely many elements,

respectively. These are linear subspaces of the spaces of all real and complexvalued functions on X, as vector spaces over R and C, respectively, with respect to pointwise addition and scalar multiplication. Of course, if X has only finitely many elements, then every real or complex-valued function on X has finite support.

A real or complex-valued function on X with finite support is obviously bounded, so that c_{00}(X, R), c_{00}(X, C) may be considered as linear subspaces of ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively. In particular, the supremum norm defines a norm on each of c_{00}(X, R), c_{00}(X, C).

If f is a real or complex-valued function on X with finite support, then

∑_{x∈X} f(x) (1.12.3)

may be defined as a real or complex number, as appropriate. This is the same as the sum of f(x) over any nonempty finite subset of X that contains the support of f. Similarly, if p is a positive real number and f is a real or complex-valued function on X with finite support, then |f(x)|^{p} is a nonnegative real-valued function on X with finite support, so that

∑_{x∈X} |f(x)|^{p} (1.12.4)

may be defined as a nonnegative real number, as before. Under these conditions, we put

∥f∥_{p} = ( ∑_{x∈X} |f(x)|^{p})^{1/p}. (1.12.5)

This corresponds to the definition of ∥ · ∥_{p} on R^{n}, C^{n} in Section 1.3 when X is the set of integers from 1 to n.

Clearly ∥f∥_{p} = 0 if and only if f = 0 on X. It is easy to see that

∥t f∥_{p} = |t| ∥f∥_{p} (1.12.6)

for every t ∈ R or C, as appropriate. If p ≥ 1, then

∥f + g∥_{p} ≤ ∥f∥_{p} + ∥g∥_{p} (1.12.7)

for all real and complex-valued functions f, g on X with finite support. This can be obtained from the analogous statement on R^{n}, C^{n} in Section 1.3, or shown in essentially the same way. Thus (1.12.5) defines a norm on each of c_{00}(X, R) and c_{00}(X, C) when p ≥ 1.

As before, ∥f∥_{∞} ≤ ∥f∥_{p} (1.12.8) for every p > 0. If 0 < p_{1} ≤ p_{2} < +∞, then

(1.12.9)  ∥f∥_{p_{2}}^{p_{2}} = ∑_{x∈X} |f(x)|^{p_{2}} ≤ ∥f∥_{∞}^{p_{2}−p_{1}} ∑_{x∈X} |f(x)|^{p_{1}}
= ∥f∥_{∞}^{p_{2}−p_{1}} ∥f∥_{p_{1}}^{p_{1}} ≤ ∥f∥_{p_{1}}^{p_{2}}.

This means that ∥f∥_{p_{2}} ≤ ∥f∥_{p_{1}}. (1.12.10)

If 0 < p ≤ 1, then it is well known that

(a + b)^{p} ≤ a^{p} + b^{p} (1.12.11)

for all nonnegative real numbers a, b. This can be obtained from (1.12.10), with p_{1} = p, and p_{2} = 1. Using this, one can check that

∥f + g∥_{p}^{p} ≤ ∥f∥_{p}^{p} + ∥g∥_{p}^{p} (1.12.12)

for all real or complex-valued functions on on X finite support. This implies that ∥f − g∥_{p}^{p} (1.12.13)

defines a metric on each of c_{00}(X, R) and c_{00}(X, C).

It is easy to see that

⟨f, g⟩ = ⟨f, g⟩_{c_{00}(X,R)} = ∑_{x∈X} f(x) g(x) (1.12.14)

and

⟨f, g⟩ = ⟨f, g⟩_{c_{00}(X,C)} = ∑_{x∈X} f(x) \overline{g(x)} (1.12.15)

define inner products on c_{00}(X, R) and c_{00}(X, C), respectively. The norms associated to these inner products are equal to (1.12.5), with p = 2. These inner products correspond to the standard inner products on R^{n} and C^{n} when X is the set of integers from 1 to n.

If x ∈ X, then let δ_{x} be the real-valued function on X defined by

(1.12.16)  δ_{x}(y) = 1 when y = x
= 0 when y ≠ x.

It is easy to see that the collection of δ_{x}’s, x ∈ X, is a basis for each of c_{00}(X, R) and c_{00}(X, C), as vector spaces over the real and complex numbers, respectively. Of course, ∥δ_{x}∥_{p} = 1 (1.12.17)

for every x ∈ X and 0 < p ≤ ∞. The δ_{x}’s are also pairwise orthogonal with respect to the inner products defined in the preceding paragraph.

1.13 Vanishing at infinity

Let X be a nonempty set, and let f be a real or complex-valued function on X again. We say that f vanishes at infinity on X if for every ϵ > 0,

|f(x)| < ϵ (1.13.1)

for all but finitely many x ∈ X. This holds automatically when f has finite support in X, and in particular when X has only finitely many elements. If X is the set Z_{+} of all positive integers, then f vanishes at infinity if and only if

lim_{j→∞} f(j) = 0. (1.13.2)

If f is a real or complex-valued function on X that vanishes at infinity, then

(1.13.3) supp f has at most finitely or countably many elements.

Indeed, for each positive integer n,

{x ∈ X : |f(x)| ≥ 1/n} (1.13.4)

has only finitely many elements, and the support of f is the same as the union of these sets.

Suppose that E is a countably infinite subset of X, and let {x_{j}}_{j=1}^{∞} be a sequence of elements of E such that every element of E occurs in the sequence exactly once. Let f be a real or complex-valued function on X with support contained in E. Under these conditions, f vanishes at infinity on X if and only if

lim_{j→∞} f(x_{j}) = 0. (1.13.5)

Let c_{0}(X, R), c_{0}(X, C) (1.13.6)

be the spaces of real and complex-valued functions on X that vanish at infinity, respectively. If f is a real or complex-valued function on X that vanishes at infinity, then it is easy to see that f is bounded on X. Thus

c_{00}(X, R) ⊆ c_{0}(X, R) ⊆ ℓ^{∞}(X, R) (1.13.7)

and c_{00}(X, C) ⊆ c_{0}(X, C) ⊆ ℓ^{∞}(X, C). (1.13.8)

More precisely, one can check that c_{0}(X, R), c_{0}(X, C) are linear subspaces of ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively.

One can also verify that

(1.13.9) c_{0}(X, R), c_{0}(X, C) are closed sets in ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively, with respect to the supremum metric. Equivalently, if a real or complex-valued function f on X can be approximated uniformly on X by functions that vanish at infinity on X, then f vanishes at infinity on X as well. Thus (1.13.10) c_{0}(X, R), c_{0}(X, C) are Banach spaces

with respect to the supremum norm.

If f is a real or complex-valued function on X that vanishes at infinity, then it is easy to see that f can be approximated by functions with finite support on X, uniformly on X. This means that c_{0}(X, R), c_{0}(X, C) are the same as the closures of c_{00}(X, R), c_{00}(X, C) in ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively, with respect to the supremum metric.

Suppose that {x_{j}}_{j=1}^{∞} is a sequence of distinct elements of X, and let δ_{x_{j}} be as in (1.12.16) for each j. Also let a_{1}, a_{2}, a_{3}, … be an infinite sequence of real or complex numbers. One can check that

∑_{j=1}^{∞} a_{j} δ_{x_{j}} (1.13.11)

converges with respect to the supremum metric if and only if {a_{j}}_{j=1}^{∞} converges to 0. More precisely, it is easy to see that the sequence of partial sums

∑_{j=1}^{n} a_{j} δ_{x_{j}} (1.13.12)

converges pointwise on X to the function that is equal to a_{j} at x_{j} for every j, and to 0 at all other points in X. The previous statement is basically the same as saying that this sequence of partial sums converges uniformly on X to the function just mentioned if and only if {a_{j}}_{j=1}^{∞} converges to 0.

1.14 Separability

A metric or topological space X is said to be separable if there is a dense set E in X such that E has only finitely or countably many elements. It is well known that R^{n} and C^{n} are separable with respect to their standard Euclidean metrics for each positive integer n.

Let V be a vector space over the real or complex numbers with a norm N. If V has finite dimension, then one can check that V is separable, with respect to the metric associated to N.

Similarly, suppose that v_{1}, v_{2}, v_{3}, … is a sequence of elements of V whose linear span is dense in V, with respect to the metric associated to N. One can verify that

(1.14.1) V is separable with respect to the metric associated to N

under these conditions. More precisely, one can get a countable dense set in V by taking linear combinations of the v_{j}’s with rational coefficients in the real case, and with coefficients that have rational real and imaginary parts in the complex case.

If X is an infinite set, then it is well known and not too difficult to show that ℓ^{∞}(X, R) is not separable with respect to the supremum metric. If X is countably infinite, then c_{00}(X, R), c_{00}(X, C) are separable with respect to the metric associated to any norm, as in the preceding paragraph. Similarly, c_{0}(X, R) and c_{0}(X, C) are separable with respect to the supremum metric in this case.

A collection B of open subsets of a metric or topological space X is said to be a base for the topology of X if every open set in X can be expressed as a union of elements of B. If there is a base B for the topology of X such that B has only finitely or countably many elements, then X is said to satisfy the second countability condition. It is well known that X is separable in this case. Topological spaces that satisfy the second countability condition are also sometimes said to be completely separable. It is well known that separable metric spaces satisfy the second countability condition.

Let Y be a subset of X, equipped with the induced topology. If X satisfies the second countability condition, then it is easy to see that Y satisfies the second countability condition too. In particular, this implies that Y is separable, as before.

If d(·, ·) is a metric on X, then the restriction of d(·, ·) to Y is a metric on Y. It is well known that the topology determined on Y by the restriction of d(·, ·) to Y is the same as the topology induced on Y by the topology determined on X by d(·, ·). If X is separable, then one can show more directly that Y is separable in this case.

1.15 Some continuous extensions

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces. Suppose for the moment that f is a uniformly continuous mapping from X into Y. If {x_{j}}_{j=1}^{∞} is a Cauchy sequence of elements of X, then one can check that

(1.15.1) {f(x_{j})}_{j=1}^{∞} is a Cauchy sequence in Y.

Of course, if Y is complete, then it follows that {f(x_{j})}_{j=1}^{∞} converges to an element of Y.

Suppose now that E is a dense set in X, and that f is a uniformly continuous mapping from E into Y, with respect to the restriction of d_{X}(·, ·) to E. If Y is complete, then it is well known that

(1.15.2) f has a unique extension to a uniformly continuous mapping from X into Y.

More precisely, uniqueness of the extension only requires ordinary continuity on X.

To get the existence of the extension, let x ∈ X be given, and let {x_{j}}_{j=1}^{∞} be a sequence of elements of E that converges to x. Thus {x_{j}}_{j=1}^{∞} may be considered as a Cauchy sequence in E, so that

(1.15.3) {f(x_{j})}_{j=1}^{∞} converges to an element of Y,

as before. One can check that the limit only depends on x, and not the particular sequence {x_{j}}_{j=1}^{∞}. One can also verify that this defines a uniformly continuous extension of f to X.

A mapping ϕ from X into Y is said to be an isometry if

d_{Y} (ϕ(x), ϕ(w)) = d_{X}(x, w) (1.15.4)

for every x, w ∈ X. If ϕ is an isometry from X into Y, and X is complete, then

(1.15.5) ϕ(X) is a closed set in Y.

More precisely, ϕ(X) is complete with respect to the restriction of d_{Y} to ϕ(X) in this case. This implies that ϕ(X) is a closed set in Y, as in Section 1.6.

A completion of X as a metric space may be defined as an isometric embedding of X onto a dense subset of a complete metric space. If X is already complete, then one can simply use the identity mapping on X. It is well known that every metric space has a completion. One can also show that completions are unique, up to a sutiable isometric equivalence, using the previous remarks.

Let V be a vector space over the real or complex numbers with a norm N. If V is not already complete with respect to the metric associated to N, then it is well known that V has a completion that is a Banach space. More precisely, this may be considered as an isometric linear mapping from V onto a dense linear subspace of a Banach space. If one uses a standard abstract construction of the completion, then one can check that the completion has these additional properties in this case. Alternatively, one can show that the vector space operations and norm on V can be extended to any completion of V as a metric space in a natural way.

Similarly, if V is an inner product space that it not already complete with respect to the metric associated to the corresponding norm, then it is well known that V has a completion that is a Hilbert space. As before, this can be shown using a standard abstract construction of the completion, or by extending the inner product to any completion of V in a natural way.

Lipschitz and bounded linear mappings

2.1 Lipschitz mappings

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces. A mapping f from X into Y is said to be Lipschitz if d_{Y} (f(x), f(w)) ≤ C d_{X}(x, w) (2.1.1)

for some nonnegative real number C and all x, w ∈ X. In this case, we may say that f is Lipschitz with constant C, to be more precise. Note that

(2.1.2) Lipschitz mappings are uniformly continuous.

Clearly a mapping f from X into Y is Lipschitz with constant C = 0 if and only if f is constant on X.

One can check that a real-valued function f on X is Lipschitz with constant C ≥ 0 if and only if f(x) ≤ f(w) + C d_{X}(x, w) (2.1.3)

for every x, w ∈ X. This uses the standard Euclidean metric on the real line, as the range of f. In particular, if p ∈ X, then

f_{p}(x) = d_{X}(x, p) (2.1.4)

is Lipschitz with constant 1 on X.

Let f be a continuous mapping from X into Y, and let E be a dense set in X. Suppose that f is Lipschitz with constant C on E, with respect to the restriction of d_{X} to E. Under these conditions, one can check that

(2.1.5) f is Lipschitz with constant C on all of X.

The space of all Lipschitz mappings from X into Y may be denoted

Lip(X, Y). (2.1.6)

Suppose that X ≠ ∅, and that f is a Lipschitz mapping from X into Y. If X has at least two elements, then put

(2.1.7) Lip(f) = Lip_{X,Y} (f) = sup\{\frac{d_{Y} (f(x), f(w))}{d_{X}(x, w)} : x, w ∈ X, x ≠ w\}.

Otherwise, if X has only one element, then this may be interpreted as being equal to 0. It is easy to see that f is Lipschitz with constant Lip(f). More precisely, Lip(f) is the smallest nonnegative real number C such that f is Lipschitz with constant C.

Let (Z, d_{Z}) be another metric space, and suppose that f is a Lipschitz mapping from X into Y, and that g is a Lipschitz mapping from Y into Z. It is easy to see that the compostion g ∘ f of f and g is a Lipschitz mapping from X into Z, with (2.1.8) Lip_{X,Z}(g ∘ f) ≤ Lip_{X,Y} (f) Lip_{Y,Z}(g).

Suppose now that Y is a vector space over the real or complex numbers. The space of all functions on X with values in Y is a vector space over R or C as well, as appropriate, with respect to pointwise addition and scalar multiplication of functions. Let N_{Y} be a norm on Y, and suppose that d_{Y} = d_{N_{Y}} is the metric on Y associated to N_{Y}. One can check that

(2.1.9) Lip(X, Y) is a linear subspace of the space of all functions on X with values in Y,

and that (2.1.10) Lip(f) is a seminorm on Lip(X, Y).

Let us take Y = R with the standard Euclidean metric again, and let f_{p} be as in (2.1.4) for each p ∈ X. One can verify that f_{p} − f_{q} is bounded on X for every p, q ∈ X, with

sup_{x∈X} |f_{p}(x) − f_{q}(x)| = d_{X}(p, q). (2.1.11)

If X is bounded with respect to d_{X}, then f_{p} is bounded on X for each p ∈ X. This means that p ↦ f_{p} (2.1.12)

defines an isometric embedding of X into the space C_{b}(X, R) of all bounded continuous real-valued functions on X, with respect to the supremum metric.

If X is not bounded, then let us fix a basepoint p_{0} ∈ X. Thus f_{p} − f_{p_{0}} is bounded on X for each p ∈ X, as in the preceding paragraph. It is easy to see that p ↦ f_{p} − f_{p_{0}} (2.1.13)

is an isometric embedding of X into C_{b}(X, R), with respect to the supremum metric.

2.2 Bounded linear mappings

Let V, W be vector spaces, both real or both complex. The space of all linear mappings from V into W may be denoted

L(V, W). (2.2.1)

This is a linear subspace of the space of all functions on V with values in W.

Let ∥ · ∥_{V}, ∥ · ∥_{W} be norms on V, W, respectively. A linear mapping T from V into W is said to be bounded with respect to these norms if

∥T (v)∥_{W} ≤ C ∥v∥_{V} (2.2.2)

for some nonnegative real number C and all v ∈ V. This implies that

∥T (u) − T (v)∥_{W} = ∥T (u − v)∥_{W} ≤ C ∥u − v∥_{V} (2.2.3)

for all u, v ∈ V, so that T is Lipschitz with constant C with respect to the metrics on V, W associated to the norms.

If T is a linear mapping from V into W, and if ∥T (v)∥_{W} is bounded on a ball of positive radius in V centered at 0 with respect to the metric associated to ∥ · ∥_{V}, then one can check that

(2.2.4) T is a bounded linear mapping.

In particular, this holds when T is continuous at 0 with respect to the metrics on V, W associated to the norms.

Let BL(V, W) (2.2.5)

be the space of all bounded linear mappings from V into W. One can verify that this is a linear subspace of L(V, W).

If T is a bounded linear mapping from V into W, then put

∥T ∥_{op} = ∥T ∥_{op,V W} = sup{∥T (v)∥_{W} : v ∈ V, ∥v∥_{V} ≤ 1}. (2.2.6)

This is the same as the smallest nonnegative real number C such that (2.2.2) holds. This is also the same as the Lipschitz constant Lip(T) of T as in the previous section, with respect to the metrics on V, W associated to their norms. One can check that

(2.2.7) ∥ · ∥_{op} defines a norm on BL(V, W),

which is the operator norm associated to the norms on V, W.

Let Z be another vector space over the real or complex numbers, as appropriate, and with a norm ∥ · ∥_{Z}. If T_{1} is a bounded linear mapping from V into W, and T_{2} is a bounded linear mapping from W into Z, then their composition T_{2} ∘ T_{1} is a bounded linear mapping from V into Z, with

∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}. (2.2.8)

If W is complete with respect to the metric associated to ∥ · ∥_{W}, then it is well known that (2.2.9) BL(V, W) is complete

with respect to the metric asociated to the operator norm. To see this, let {T_{j}}_{j=1}^{∞} be a Cauchy sequence in BL(V, W) with respect to the metric associated to the operator norm. This means that for each ϵ > 0 there is a positive integer L(ϵ) such that ∥T_{j} − T_{l}∥_{op} < ϵ (2.2.10)

for every j, l ≥ L(ϵ). It follows that

∥T_{j}(v) − T_{l}(v)∥_{W} ≤ ϵ ∥v∥_{V} (2.2.11)

for every v ∈ V and j, l ≥ L(ϵ). In particular, this implies that

(2.2.12) {T_{j}(v)}_{j=1}^{∞} is a Cauchy sequence in W

for each v ∈ V, with respect to the metric associated to ∥·∥_{W}. Thus {T_{j}(v)}_{j=1}^{∞} converges to an element of W for each v ∈ V, by completeness. If we put

T (v) = lim_{j→∞} T_{j}(v) (2.2.13)

for each v ∈ V, then it is easy to see that T defines a linear mapping from V into W.

It is well known and easy to see that a Cauchy sequence in any metric space is bounded. In this case, this means that {T_{j}}_{j=1}^{∞} is bounded with respect to the operator norm, so that {∥T_{j}∥_{op}}_{j=1}^{∞} is a bounded sequence of nonnegative real numbers. If v ∈ V, then one can check that

∥T (v)∥_{W} ≤ sup_{j≥1} ∥T_{j}(v)∥_{W} ≤ ( sup_{j≥1} ∥T_{j}∥_{op}) ∥v∥_{V} . (2.2.14)

This means that T is a bounded linear mapping from V into W, with

∥T ∥_{op} ≤ sup_{j≥1} ∥T_{j}∥_{op}. (2.2.15)

Similarly, one can verify that

∥T (v) − T_{l}(v)∥_{W} ≤ ϵ ∥v∥_{V} (2.2.16)

for every v ∈ V and l ≥ L(ϵ). This implies that

∥T − T_{l}∥_{op} ≤ ϵ (2.2.17)

for every l ≥ L(ϵ). This shows that {T_{l}}_{l=1}^{∞} converges to T with respect to the metric on BL(V, W) associated to the operator norm, as desired.

Let V_{0} be a linear subspace of V that is dense in V, with respect to the metric associated to ∥ · ∥_{V}. Also let T_{0} be a bounded linear mapping from V_{0} 2.3. SOME LINEAR MAPPINGS 27 into W, with respect to the restriction of ∥ · ∥_{V} to V_{0}. If W is complete with respect to the metric associated to ∥ · ∥_{W}, then there is a unique extension of T_{0} to a uniformly continuous mapping T from V into W, as in Section 1.15. In this case, one can check that (2.2.18) T is a linear mapping from V into W. More precisely, (2.2.19) T is a bounded linear mapping from V into W, with operator norm equal to the operator norm of T_{0} on V_{0}.

2.3 Some linear mappings

Let X be a nonempty set, and let W be a vector space over the real or complex numbers. If f is a function on X with values in W, then the support of f may be defined as the set supp f of x ∈ X such that f(x) ≠ 0, (2.3.1) just as for real or complex-valued functions on X, as in Section 1.12. Let c_{00}(X, W) be the space of W -valued functions f on X such that (2.3.2) supp f has only finitely many elements. This is a linear subspace of the space of all W -valued functions on X.

If f ∈ c_{00}(X, W), then

∑_{x∈X} f(x) (2.3.3)

may be defined as an element of W. This is the same as the sum of f(x) over any nonempty finite subset of X that contains the support of f, as before. This defines a linear mapping from c_{00}(X, W) into W.

It is convenient to put V = c_{00}(X, R) or c_{00}(X, C) (2.3.4) for the moment, depending on whether W is a vector space over R or C. Let a be any W -valued function on X. If f ∈ V, then a f ∈ c_{00}(X, W), (2.3.5) and we put

T_{a}(f) = ∑_{x∈X} a(x) f(x). (2.3.6)

This defines a linear mapping from V into W.

28 CHAPTER 2. LIPSCHITZ AND BOUNDED LINEAR MAPPINGS

If y ∈ X, then let δ_{y} be the real-valued function on X equal to 1 at x and 0 otherwise, as in Section 1.12. Note that T_{a}(δ_{y}) = a(y) (2.3.7) for every y ∈ X. It is easy to see that (2.3.8) every linear mapping T from V into W is of the form T_{a} for a unique W -valued function a on X. This uses the fact that the collection of δ_{y}’s, y ∈ X, is a basis for V as a vector space over R or C, as appropriate, as in Section 1.12. More precisely, this holds with a(y) = T (δ_{y}) (2.3.9) for every y ∈ X.

Let ∥ · ∥_{W} be a norm on W. A W -valued function a on X is said to be bounded on X with respect to ∥ · ∥_{W} if (2.3.10) ∥a(x)∥_{W} is bounded as a real-valued function on X. Let ℓ^{∞}(X, W) (2.3.11) be the space of W -valued functions on X that are bounded with respect to ∥ · ∥_{W}. This is a linear subspace of the space of all W -valued functions on X.

If a ∈ ℓ^{∞}(X, W), then put ∥a∥_{∞} = ∥a∥_{sup} = ∥a∥_{ℓ^{∞}(X,W)} = sup{∥a(x)∥_{W} : x ∈ X}. (2.3.12) One can check that this defines a norm on ℓ^{∞}(X, W), which is another version of the supremum norm. The metric associated to the supremum norm on ℓ^{∞}(X, W) is the same as the supremum metric corresponding to the metric on W associated to ∥ · ∥_{W}. If W is a Banach space with respect to ∥ · ∥_{W}, then (2.3.13) ℓ^{∞}(X, W) is a Banach space with respect to (2.3.12), by standard arguments. This is analogous to the cases of bounded real and complex-valued functions on X, as in Section 1.6.

Let a be a W -valued function on X again, and let T_{a} be as in (2.3.6). Also let ∥ · ∥_{1} be the norm defined on V as in Section 1.12, with p = 1. Note that ∥δ_{y}∥_{1} = 1 (2.3.14) for every y ∈ X. If T_{a} is bounded as a linear mapping from V into W, with respect to ∥ · ∥_{1} and ∥ · ∥_{W}, respectively, then ∥a(y)∥_{W} = ∥T_{a}(δ_{y})∥_{W} ≤ ∥T_{a}∥_{op} ∥δ_{y}∥_{1} = ∥T_{a}∥_{op} (2.3.15) for every y ∈ X. This means that a ∈ ℓ^{∞}(X, W), with

∥a∥_{∞} ≤ ∥T_{a}∥_{op}. (2.3.16)

If a is any element of ℓ^{∞}(X, W) and f ∈ V, then

∥T_{a}(f)∥_{W} ≤ ∑_{x∈X} ∥a(x)∥_{W} |f(x)| ≤ ∥a∥_{∞} ∥f∥_{1}. (2.3.17)

This implies that T_{a} is a bounded linear mapping from V into W, with respect to ∥ · ∥_{1} and ∥ · ∥_{W}, respectively, with

∥T_{a}∥_{op} ≤ ∥a∥_{∞}. (2.3.18)

It follows that ∥T_{a}∥_{op} = ∥a∥_{∞} (2.3.19)

under these conditions.

2.4 Nonnegative sums

Let X be a nonempty set, and let f be a nonnegative real-valued function on X. If A is a nonempty finite subset of X, then the sum

∑_{x∈A} f(x) (2.4.1)

is defined as a nonnegative real number in the usual way. This may be interpreted as being equal to 0 when A = ∅.

The sum

∑_{x∈X} f(x) (2.4.2)

may be defined as a nonnegative extended real number as the supremum of the finite subsums (2.4.1) over all nonempty finite subsets A of X. Of course, if X has only finitely many elements, then this is the same as the usual sum over X. This definition of the sum over an arbitrary nonempty set X is the same as the Lebesgue integral of f over X, with respect to counting measure on X.

If t is a positive real number, then

∑_{x∈X} t f(x) = t ∑_{x∈X} f(x), (2.4.3)

where the right side is interpreted as being equal to +∞ when (2.4.2) is equal to +∞, as usual. This also works when t = 0, with the right side interpreted as being equal to 0 even when (2.4.2) is +∞.

If g is another nonnegative real-valued function on X, then one can check that

∑_{x∈X}(f(x) + g(x)) = ∑_{x∈X} f(x) + ∑_{x∈X} g(x). (2.4.4)

The right side should be interpreted as being equal to +∞ when either of the sums is +∞.

If (2.4.2) is finite, then f is said to be summable on X. It is easy to see that

(2.4.5) f vanishes at infinity on X

in this case, in the sense of Section 1.13. In particular, this implies that

(2.4.6) the support of f has only finitely or countably many elements,

as before.

Let {x_{j}}_{j=1}^{∞} be a sequence of distinct elements of X, and suppose for the moment that the support of f is contained in the set of x_{j}’s. Under these conditions, one can verify that

∑_{x∈X} f(x) = ∑_{j=1}^{∞} f(x_{j}). (2.4.7)

More precisely, the sum on the right should be interpreted as being +∞ when the series does not converge.

If E is any subset of X, then ∑_{x∈E} f(x) may be defined as a nonnegative extended real number, as before. If E_{1} and E_{2} are disjoint subsets of X, then

∑_{x∈E_{1}∪E_{2}} f(x) = ∑_{x∈E_{1}} f(x) + ∑_{x∈E_{2}} f(x), (2.4.8)

as in (2.4.4).

Suppose for the moment that f is summable on X, and let ϵ > 0 be given. Observe that there is a finite subset A(ϵ) of X such that

∑_{x∈X} f(x) < ∑_{x∈A(ϵ)} f(x) + ϵ, (2.4.9)

by the definition of the sum (2.4.2). This means that

∑_{x∈X\A(ϵ)} f(x) < ϵ. (2.4.10)

Let {f_{j}}_{j=1}^{∞} be a sequence of nonnegative real-valued functions on X that converges to f pointwise on X. Suppose that there is a nonnegative real-number C such that

∑_{x∈X} f_{j}(x) ≤ C (2.4.11)

for each j. Under these conditions, it is well known that

∑_{x∈X} f(x) ≤ C. (2.4.12)

Indeed, if A is any finite subset of X, then

∑_{x∈A} f(x) = lim_{j→∞} ∑_{x∈A} f_{j}(x) ≤ C. (2.4.13)

2.5. P -SUMMABILITY 31

2.5 p-Summability

Let X be a nonempty set, and let p be a positive real number. A nonnegative real-valued function f on X is said to be p-summable on X if

(2.5.1) f(x)^{p} is summable on X.

If g is another nonnegative real-valued function on X, and f, g are both psummable on X, then one can check that

(2.5.2) f + g is p-summable on X

too.

More precisely, if 0 < p ≤ 1, then

∑_{x∈X}(f(x) + g(x))^{p} ≤ ∑_{x∈X}(f(x)^{p} + g(x)^{p}) = ∑_{x∈X} f(x)^{p} + ∑_{x∈X} g(x)^{p}, (2.5.3)

using (1.12.11) in the first step. If p ≥ 1, and f, g are p-summable on X, then

( ∑_{x∈X}(f(x) + g(x))^{p})^{1/p} ≤ ( ∑_{x∈X} f(x)^{p})^{1/p} + ( ∑_{x∈X} g(x)^{p})^{1/p}. (2.5.4)

This is Minkowski’s inequality for arbitrary sums. This can be obtained from Minkowski’s inequality for finite sums, as in Section 1.3, or shown using similar arguments.

If f is p-summable on X for some p > 0, then f is bounded on X, with

sup_{x∈X} f(x) ≤ ( ∑_{x∈X} f(x)^{p})^{1/p}. (2.5.5)

We also have that f vanishes at infinity on X, because f(x)^{p} vanishes at infinity on X, as in the previous section. In fact, for each ϵ > 0, we have that

ϵ^{p} (#{x ∈ X : f(x) ≥ ϵ}) ≤ ∑_{x∈X} f(x)^{p}. (2.5.6)

Here #E denotes the number of elements of a set E.

Suppose that 0 < p_{1} ≤ p_{2} < ∞, and that f is p_{1}-summable on X. Under these conditions, f is p_{2}-summable on X, with

∑_{x∈X} f(x)^{p_{2}} ≤ ( sup_{x∈X} f(x))^{p_{2}−p_{1}} ∑_{x∈X} f(x)^{p_{1}} ≤ ( ∑_{x∈X} f(x)^{p_{1}})^{p_{2}/p_{1}}, (2.5.7)

using (2.5.5) with p = p_{1} in the second step. This implies that

(2.5.8) ( ∑_{x∈X} f(x)^{p_{2}})^{1/p_{2}} ≤ ( sup_{x∈X} f(x))^{1−(p_{1}/p_{2})} ( ∑_{x∈X} f(x)^{p_{1}})^{1/p_{2}}

≤ ( ∑_{x∈X} f(x)^{p_{1}})^{1/p_{1}}.

32 CHAPTER 2. LIPSCHITZ AND BOUNDED LINEAR MAPPINGS

One can use (2.5.5) and the first inequality in (2.5.8) to get that

lim_{p→∞} ( ∑_{x∈X} f(x)^{p})^{1/p} = sup_{x∈X} f(x) (2.5.9)

in this case.

2.6 ℓ^{p} Spaces

Let X be a nonempty set again, and let p be a positive real number. Also let W be a vector space over the real or complex numbers, with a norm ∥ · ∥_{W}. Consider the space ℓ^{p}(X, W) (2.6.1) of W -valued functions f on X such that (2.6.2) ∥f(x)∥_{W} is p-summable as a nonnegative real-valued function on X. In particular, we can take W = R or C, considered as one-dimensional vector spaces over themselves, with the usual absolute value function as the norm.

One can check that ℓ^{p}(X, W) is a linear subspace of the space of all W -valued functions on X. If f ∈ ℓ^{p}(X, W), then put

∥f∥_{p} = ∥f∥_{ℓ^{p}(X,W)} = ( ∑_{x∈X} ∥f(x)∥_{W}^{p} )^{1/p}. (2.6.3)

This defines a norm on ℓ^{p}(X, W) when p ≥ 1. If 0 < p ≤ 1, then this satisfies the usual homogeneity property of a norm, as well as ∥f + g∥_{p}^{p} ≤ ∥f∥_{p}^{p} + ∥g∥_{p}^{p} (2.6.4) for all f, g ∈ ℓ^{p}(X, W). This implies that ∥f − g∥_{p}^{p} (2.6.5) defines a metric on ℓ^{p}(X, W) when p ≤ 1, as usual.

If 0 < p_{1} ≤ p_{2} ≤ +∞, then ℓ^{p_{1}}(X, W) ⊆ ℓ^{p_{2}}(X, W), (2.6.6) as in the previous section. We also have that ∥f∥_{p_{2}} ≤ ∥f∥_{p_{1}} (2.6.7) for all f ∈ ℓ^{p_{1}}(X, W) in this case.

Let us say that a W -valued function f on X vanishes at infinity on X with respect to ∥ · ∥_{W} on W if (2.6.8) ∥f(x)∥_{W} vanishes at infinity as a

nonnegative real-valued function on X.

Let c_{0}(X, W) be the space of W -valued functions on X that vanish at infinity on X with respect to ∥ · ∥_{W} on W. It is easy to see that

(2.6.9) c_{0}(X, W) is a linear subspace of ℓ^{∞}(X, W).

More precisely, one can check that

(2.6.10) c_{0}(X, W) is the same as the closure of c_{00}(X, W) in ℓ^{∞}(X, W)

with respect to the supremum metric, as in Section 1.13.

If 0 < p < ∞, then

c_{00}(X, W) ⊆ ℓ^{p}(X, W) ⊆ c_{0}(X, W). (2.6.11)

One can verify that

(2.6.12) c_{00}(X, W) is dense in ℓ^{p}(X, W)

when p < ∞, using (2.4.10). This uses the metric on ℓ^{p}(X, W) associated to (2.6.3) when p ≥ 1, and the metric ∥f − g∥_{p}^{p} when p ≤ 1.

If W is complete with respect to the metric associated to ∥ · ∥_{W}, then

(2.6.13) ℓ^{p}(X, W) is complete

with respect to the metric associated to (2.6.3) when p ≥ 1, and with respect to the metric ∥f − g∥_{p}^{p} when p ≤ 1. To see this, let {f_{j}}_{j=1}^{∞} be a Cauchy sequence in ℓ^{p}(X, W) with respect to the appropriate metric. If x ∈ X, then {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in W, with respect to the metric associated to ∥·∥_{W}. This sequence converges in W, by hypothesis. This means that {f_{j}}_{j=1}^{∞} converges to a W -valued function f pointwise on X.

It is well known that a Cauchy sequence in any metric space is bounded. One can use this and (2.4.12) to get that f ∈ ℓ^{p}(X, W). Similarly, one can use the Cauchy condition for {f_{j}}_{j=1}^{∞} in ℓ^{p}(X, W) and (2.4.12) to get that the distance from f_{j} to f is small when j is sufficiently large. This means that {f_{j}}_{j=1}^{∞} converges to f with respect to the appropriate metric on ℓ^{p}(X, W).

In particular, ℓ^{p}(X, R) and ℓ^{p}(X, C) are complete with respect to their appropriate metrics.

2.7 Generalized convergence of sums

Let X be a nonempty set, and let W be a vector space over the real or complex numbers with a norm ∥ · ∥_{W}. Also let f be a function on X with values in W. We say that that the sum

∑_{x∈X} f(x) (2.7.1)

converges in the generalized sense with respect to the metric on W associated to ∥ · ∥_{W} if there is a w ∈ W with the following property: for each ϵ > 0 there is a finite set A(ϵ) ⊆ X such that for every finite set A ⊆ X with

A(ϵ) ⊆ A, (2.7.2) we have that

∥∑_{x∈A} f(x) − w∥_{W} < ϵ. (2.7.3)

One can check that such a w ∈ W is unique when it exists, in which case it is considered to be the value of the sum (2.7.1).

If X has only finitely many elements, then this is the same as treating (2.7.1) as a finite sum. If the support of f in X has only finitely many elements, then this is the same as reducing (2.7.1) to a finite sum in W, as in Section 2.3.

Of course, if A, B are finite subsets of X, then A ∪ B is a finite subset of X that contains A and B. This means that the collection of all finite subsets of X is a directed system, as a partially-ordered set with respect to inclusion. The sums

∑_{x∈A} f(x) (2.7.4)

of f over finite subsets A of X may be considered as a net in W, indexed by this directed system. The convergence of the sum (2.7.1) is the same as the convergence of this net in W, with respect to the metric associated to ∥ · ∥_{W}.

Let us say that the sum (2.7.1) satisfies the generalized Cauchy condition if for every ϵ > 0 there is a finite subset A_{0}(ϵ) of X with the following property: if A, B are finite subsets of X such that

A_{0}(ϵ) ⊆ A, B, (2.7.5)

then

∥∑_{x∈A} f(x) − ∑_{x∈B} f(x)∥_{W} < ϵ. (2.7.6)

This is the same as saying that the net of finite sums (2.7.4) is a Cauchy net with respect to the metric on W associated to ∥ · ∥_{W}. If (2.7.1) converges in the generalized sense, then it is easy to see that it satisfies the generalized Cauchy condition, with A_{0}(ϵ) = A(ϵ/2) (2.7.7)

for each ϵ > 0.

Equivalently, (2.7.1) satisfies the generalized Cauchy condition if and only if for every ϵ > 0 there is a finite subset A_{1}(ϵ) of X with the following property: if C is a finite subset of X such that

A_{1}(ϵ) ∩ C = ∅, (2.7.8)

then

∥∑_{x∈C} f(x)∥_{W} < ϵ. (2.7.9)

More precisely, the previous version implies this one, with

A_{1}(ϵ) = A_{0}(ϵ), (2.7.10) by taking A = A_{0}(ϵ) ∪ C and B = A_{0}(ϵ) in (2.7.6). Conversely, this version implies the previous one, with

A_{0}(ϵ) = A_{1}(ϵ/2) (2.7.11)

for each ϵ > 0.

If (2.7.1) satisfies the generalized Cauchy condition, then f vanishes at infinity on X with respect to ∥ · ∥_{W} on W. Indeed, (2.7.9) implies that

∥f(x)∥_{W} < ϵ (2.7.12)

when x ∈ X \ A_{1}(ϵ), by taking C = {x}. In particular, this means that the support of f has only finitely or countably many elements, as in Section 1.13.

Let {x_{j}}_{j=1}^{∞} be a sequence of distinct elements of X, and suppose for the moment that the support of f is contained in the set of x_{j}’s. If (2.7.1) converges in the generalized sense, then it is easy to see that

∑_{j=1}^{∞} f(x_{j}) (2.7.13)

converges in W, and that the value of the sum is the same as for (2.7.1).

Similarly, if (2.7.1) satisfies the generalized Cauchy condition, and (2.7.13) converges in W, then one can check that (2.7.1) converges in the generalized sense, and with the same value of the sum.

If (2.7.1) satisfies the generalized Cauchy condition, then the sequence of partial sums

∑_{j=1}^{n} f(x_{j}) (2.7.14)

of (2.7.13) is a Cauchy sequence in W with respect to the metric associated to ∥ · ∥_{W}. If W is complete with respect to this metric, then (2.7.13) converges in W. This implies that (2.7.1) converges in the generalized sense, with sum equal to (2.7.13), as in the preceding paragraph.

If W is a Banach space, and (2.7.1) satisfies the generalized Cauchy condition, then one can use the remarks in the prevous paragraphs to get that (2.7.1) converges in the generalized sense. Note that if (2.7.1) converges in the generalized sense, then every rearrangement of (2.7.13) converges, and with the same value of the sum. Conversely, if every rearrangement of (2.7.13) converges, then it is well known that (2.7.1) satisfies the generalized Cauchy condition, as in Proposition 1.c.1 on p15 of [217].

If f is any W -valued function on X, then

∥∑_{x∈C} f(x)∥_{W} ≤ ∑_{x∈C} ∥f(x)∥_{W} (2.7.15)

for every finite subset C of X, by the triangle inequality. If f ∈ ℓ^{1}(X, W), then one can use this to get that (2.7.1) satisfies the generalized Cauchy condition.

If (2.7.1) converges in the generalized sense, then we have that

∥∑_{x∈X} f(x)∥_{W} ≤ ∑_{x∈X} ∥f(x)∥_{W} (2.7.16)

in this case.

Suppose now that ∥ · ∥_{W} is the norm associated to an inner product ⟨·, ·⟩_{W} on W, and that the values of f on X are pairwise-orthogonal in W. This means that ⟨f(x), f(y)⟩_{W} = 0 (2.7.17)

for every x, y ∈ X with x ≠ y. If C is a finite subset of X, then it follows that

∥∑_{x∈C} f(x)∥_{W}^{2} = ∑_{x∈C} ∥f(x)∥_{W}^{2} . (2.7.18)

If f ∈ ℓ^{2}(X, W), then one can use this to get that (2.7.1) satisfies the generalized Cauchy condition. If (2.7.1) converges in the generalized sense, then one can verify that

∥∑_{x∈X} f(x)∥_{W}^{2} = ∑_{x∈X} ∥f(x)∥_{W}^{2} (2.7.19)

under these conditions.

2.8 More on generalized convergence

Let X be a nonempty set again, and let W be a vector space over the real or complex numbers with a norm ∥ · ∥_{W}. Consider the space

Sum(X, W) (2.8.1)

of W -valued functions f on X such that the sum (2.7.1) converges in the generalized sense. It is easy to see that this is a linear subspace of the space of all W -valued functions on X, and that

f ↦ ∑_{x∈X} f(x) (2.8.2)

is a linear mapping from Sum(X, W) into W.

Similarly, let GCC(X, W) (2.8.3)

be the space of W -valued functions f on X such that (2.7.1) satisfies the generalized Cauchy condition. This is also a linear subspace of the space of all W -valued functions on X, with

c_{00}(X, W) ⊆ Sum(X, W) ⊆ GCC(X, W) ⊆ c_{0}(X, W), (2.8.4) as in the previous section. If W is a Banach space, then

Sum(X, W) = GCC(X, W), (2.8.5)

as before.

Remember that ℓ^{1}(X, W) ⊆ GCC(X, W), (2.8.6)

so that ℓ^{1}(X, W) ⊆ Sum(X, W) (2.8.7)

when W is a Banach space. In this case, (2.8.2) is a bounded linear mapping from ℓ^{1}(X, W) into W, with operator norm equal to 1.

Let f be a nonnegative real-valued function on X. If f is summable on X, then one can check that the sum (2.7.1) converges in the generalized sense, with the same value of the sum as in Section 2.4.

Similarly, if f is a real or complex-valued function on X such that |f| is summable as a nonnegative real-valued function on X, then (2.7.1) converges in the generalized sense. This can be seen by expressing f as a linear combination of summable nonnegative real-valued functions on X, and using the remark in the preceding paragraph. Alternatively, this follows from (2.8.7), with W = R or C, as appropriate.

Let f, g be real or complex-valued functions on X such that |f|, |g| are 2summable on X. One can check that |f| |g| is summable on X. More precisely, one can use the Cauchy–Schwarz inequality for finite sums to get that

∑_{x∈X} |f(x)| |g(x)| ≤ ( ∑_{x∈X} |f(x)|^{2})^{1/2} ( ∑_{x∈X} |g(x)|^{2})^{1/2}. (2.8.8)

This could also be obtained more directly, using the same type of arguments as for finite sums.

Suppose that ∥ · ∥_{W} is the norm associated to an inner product ⟨·, ·⟩_{W} on W. If f, g are W -valued functions on X, then

|⟨f(x), g(x)⟩_{W} | ≤ ∥f(x)∥_{W} ∥g(x)∥_{W} (2.8.9)

for every x ∈ X, by the Cauchy–Schwarz inequality. If f, g ∈ ℓ^{2}(X, W), then we get that |⟨f(x), g(x)⟩_{W} | (2.8.10)

is summable as a nonnegative real-valued function on X, as in the preceding paragraph. In this case, we put

⟨f, g⟩_{ℓ^{2}(X,W)} = ∑_{x∈X}⟨f(x), g(x)⟩_{W} . (2.8.11)

where the right side converges in the generalized sense, as before. One can check that this defines an inner product on ℓ^{2}(X, W), for which the corresponding norm is the usual norm on ℓ^{2}(X, W) associated to ∥ · ∥_{W}. If W is a Hilbert space, then ℓ^{2}(X, W) is complete with respect to the metric associatived to this norm, as in Section 2.6. This means that ℓ^{2}(X, W) is a Hilbert space under these conditions.

Of course, the standard absolute value functions on R and C may be considered as norms associated to inner products. These are the same as the standard inner products on R^{n} and C^{n}, respectively, as in Section 1.9, with n = 1. This leads to inner products

⟨f, g⟩_{ℓ^{2}(X,R)} = ∑_{x∈X} f(x) g(x) (2.8.12)

on ℓ^{2}(X, R) and

⟨f, g⟩_{ℓ^{2}(X,C)} = ∑_{x∈X} f(x) \overline{g(x)} (2.8.13)

on ℓ^{2}(X, C) as in the preceding paragraph. Thus ℓ^{2}(X, R) and ℓ^{2}(X, C) are Hilbert spaces over the real and complex numbers, respectively, as before.

2.9 Bounded finite sums

Let X be a nonempty set, and let W be a vector space over the real or complex numbers with a norm ∥ · ∥_{W}. Let us say that a W -valued function f on X has bounded finite sums on X with respect to ∥ · ∥_{W} if the norms

∥∑_{x∈A} f(x)∥_{W} (2.9.1)

of the sums of f over all finite subsets A of X are bounded. It is easy to see that the space BF S(X, W) (2.9.2) of W -valued functions on X with bounded finite sums is a linear subspace of the space of all W -valued functions on X.

Let us check that

GCC(X, W) ⊆ BF S(X, W) ⊆ ℓ^{∞}(X, W). (2.9.3)

If f ∈ GCC(X, W), then there is a finite set A_{1}(1) ⊆ X such that

∥∑_{x∈C} f(x)∥_{W} < 1 (2.9.4)

for all finite sets C ⊆ X such that A_{1}(1) ∩ C = ∅, as in (2.7.9). If A is a finite subset of X, then we can take C = A \ A_{1}(1) in (2.9.4), and use the triangle inequality to get that

(2.9.5) ∥∑_{x∈A} f(x)∥_{W} ≤ ∥ ∑_{x∈A∩A_{1}(1)} f(x)∥_{W} + ∥ ∑_{x∈A\A_{1}(1)} f(x)∥_{W}
< ∑_{x∈A_{1}(1)} ∥f(x)∥_{W} + 1.

This shows that f ∈ BF S(X, W). The second inclusion in (2.9.3) corresponds to taking A to be a subset of X with only one element in (2.9.1).

If f ∈ BF S(X, W), then put

(2.9.6) ∥f∥_{BF S} = ∥f∥_{BF S(X,W)}
= sup{∥∑_{x∈A} f(x)∥_{W} : A is a finite subset of X}.

This defines a norm on BF S(X, W), with

∥f∥_{∞} ≤ ∥f∥_{BF S} (2.9.7)

for every f ∈ BF S(X, W). If f ∈ ℓ^{1}(X, W), then f ∈ BF S(X, W), with

∥f∥_{BF S} ≤ ∥f∥_{1}, (2.9.8)

by the triangle inequality. If f ∈ Sum(X, W), then f ∈ BF S(X, W), by (2.8.4) and (2.9.3), and one can check that

∥∑_{x∈X} f(x)∥_{W} ≤ ∥f∥_{BF S}. (2.9.9)

One can also verify that

(2.9.10) GCC(X, W) is the same as the closure of c_{00}(X, W) in BF S(X, W).

Let {f_{j}}_{j=1}^{∞} be a sequence of W -valued functions on X that converges pointwise to a W -valued function f on X, with respect to the metric on W associated to ∥ · ∥_{W}. If A is a finite subset of X, then it follows that

lim_{j→∞} ∑_{x∈A} f_{j}(x) = ∑_{x∈A} f(x). (2.9.11)

Suppose that f_{j} ∈ BF S(X, W) for each j, and that there is a nonnegative real number C such that ∥f_{j}∥_{BF S} ≤ C (2.9.12)

for all j. This implies that f ∈ BF S(X, W), with

∥f∥_{BF S} ≤ C, (2.9.13)

because of (2.9.11).

If W is complete with respect to the metric associated to ∥ · ∥_{W}, then

(2.9.14) BF S(X, W) is complete

with respect to the metric associated to (2.9.6). Indeed, if {f_{j}}_{j=1}^{∞} is a Cauchy sequence in BF S(X, W), then {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to the supremum metric, and in particular {f_{j}}_{j=1}^{∞} converges pointwise to a W -valued function f on X. One can check that f is an element of BF S(X, W), because Cauchy sequences are bounded. One can also verify that {f_{j}}_{j=1}^{∞} converges to f with respect to the metric associated to (2.9.6), using the fact that {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to this metric again.

Suppose for the moment that W = R, as a one-dimensional vector space over itself, with the usual absolute value function as the norm. If f ∈ BF S(X, R), then max(f, 0) and max(−f, 0) are summable on X, with

∑_{x∈X} max(f(x), 0), ∑_{x∈X} max(−f(x), 0) ≤ ∥f∥_{BF S}. (2.9.15)

This implies that f ∈ ℓ^{1}(X, R), with

∥f∥_{1} ≤ 2 ∥f∥_{BF S}. (2.9.16)

Thus BF S(X, R) = ℓ^{1}(X, R), (2.9.17)

as linear subspaces of the space of all real-valued functions on X.

Suppose now that W = C, as a one-dimensional vector space over itself, and with the usual absolute value function as the norm. If f ∈ BF S(X, C), then the real and imaginary parts of f are in BF S(X, R), with BF S norms less than or equal to the BF S norm of f. This implies that f ∈ ℓ^{1}(X, C), with

∥f∥_{1} ≤ 4 ∥f∥_{BF S}, (2.9.18)

using (2.9.16) for the real and imaginary parts of f. It follows that

BF S(X, C) = ℓ^{1}(X, C), (2.9.19)

as linear subspaces of the space of all complex-valued functions on X.

Let W be a vector space over the real or complex numbers again, and let ⟨·, ·⟩_{W} be an inner product on W, with associated norm ∥·∥_{W}. Also let f be a W - valued function on X with pairwise-orthogonal values, so that ⟨f(x), f(y)⟩_{W} = 0 for every x, y ∈ X with x ≠ y. Under these conditions,

(2.9.20) f ∈ BF S(X, W) if and only if f ∈ ℓ^{2}(X, W),

because of (2.7.18). In this case, we have that

∥f∥_{BF S} = ∥f∥_{2}. (2.9.21)

2.10 Some related linear mappings

Let X be a nonempty set, and let W, Z be vector spaces, both real or both complex. Also let T be a linear mapping from W into Z. If f is a W -valued function on X, then T (f(x)) is a Z-valued function on X. More precisely,

f ↦ T ∘ f (2.10.1)

2.10. SOME RELATED LINEAR MAPPINGS 41

is a linear mapping from the space of all W -valued functions on X into the space of all Z-valued functions on X.

Suppose that ∥ · ∥_{W}, ∥ · ∥_{Z} are norms on W, Z, respectively, and that T is bounded with respect to these norms. If f is a W -valued function on X, then ∥T (f(x))∥_{Z} ≤ ∥T ∥_{op,W Z} ∥f(x)∥_{W} (2.10.2) for every x ∈ X. If f ∈ ℓ^{p}(X, W) for some 0 < p ≤ +∞, then it follows that T ∘ f ∈ ℓ^{p}(X, Z), (2.10.3) with ∥T ∘ f∥_{ℓ^{p}(X,Z)} ≤ ∥T ∥_{op,W Z} ∥f∥_{ℓ^{p}(X,W)}. (2.10.4) Similarly, if f ∈ c_{0}(X, W), then T ∘ f ∈ c_{0}(X, Z). (2.10.5)

If f ∈ BF S(X, W), then it is easy to see that T ∘ f ∈ BF S(X, Z), (2.10.6) with ∥T ∘ f∥_{BF S(X,Z)} ≤ ∥T ∥_{op,W Z} ∥f∥_{BF S(X,W)}. (2.10.7) If f ∈ GCC(X, W), then one can check that T ∘ f ∈ GCC(X, Z). (2.10.8) If f ∈ Sum(X, W), then one can verify that T ∘ f ∈ Sum(X, Z), (2.10.9) with

∑_{x∈X} T (f(x)) = T ( ∑_{x∈X} f(x)). (2.10.10)

If ∑_{j=1}^{∞} w_{j} is a convergent series in W, then

(2.10.11)  ∑_{j=1}^{∞} T (w_{j}) is a convergent series in Z,

with

∑_{j=1}^{∞} T (w_{j}) = T ( ∑_{j=1}^{∞} w_{j}). (2.10.12)

If ∑_{j=1}^{∞} w_{j} converges absolutely with respect to ∥ · ∥_{W}, then

(2.10.13) ∑_{j=1}^{∞} T (w_{j}) converges absolutely with respect to ∥ · ∥_{Z}.

This corresponds to the fact that (2.10.1) maps ℓ^{1}(Z_{+}, W) into ℓ^{1}(Z_{+}, Z), as before.

If ∥T (w)∥_{Z} = ∥w∥_{W} (2.10.14)

for every w ∈ W, then T is said to be an isometric linear mapping from W into Z. This implies that T is an isometry from W into Z, with respect to the metrics associated to the norms. In particular, this means that T is injective.

Suppose now that ∥·∥_{W}, ∥·∥_{Z} are associated to inner products ⟨·, ·⟩_{W}, ⟨·, ·⟩_{Z} on W, Z, respectively. If

⟨T (u), T (w)⟩_{Z} = ⟨u, w⟩_{W} (2.10.15)

for every u, w ∈ W, then (2.10.14) holds for every w ∈ W, by taking u = w in (2.10.15).

Conversely, if (2.10.14) holds for every w ∈ W, then it is well known that (2.10.15) holds for every u, w ∈ W. This can be obtained from suitable polarization identities, which can be used to express an inner product in terms of the corresponding norm. If u, w ∈ W, then

(1/2) (∥u + w∥_{W}^{2} − ∥u∥_{W}^{2} − ∥w∥_{W}^{2} ) (2.10.16)

is equal to ⟨u, w⟩_{W} in the real case. In the complex case, this is equal to the real part of the inner product. To get the imaginary part, one can multiply u by −i.

If T maps W onto Z and satisfies (2.10.15) for every u, w ∈ W, then T is said to be a unitary mapping from W onto Z. In the real case, one may say that T is an orthogonal transformation.

2.11 Distances to sets

Let (X, d) be a metric space, and let A be a nonempty subset of X. If x ∈ X, then put

dist(x, A) = inf_{a∈A} d(x, a). (2.11.1)

If y ∈ X and a ∈ A, then

dist(x, A) ≤ d(x, a) ≤ d(x, y) + d(y, a). (2.11.2)

This means that dist(x, A) − d(x, y) ≤ d(y, a), (2.11.3)

so that (2.11.4) dist(x, A) − d(x, y) ≤ dist(y, A).

It follows that dist(x, A) is Lipschitz with constant 1 as a real-valued function on X, as in Section 2.1.

It is easy to see that dist(x, A) = 0 (2.11.5) if and only if x is an element of the closure \overline{A} of A in X. If A ⊆ B ⊆ X, then

(2.11.6) dist(x, B) ≤ dist(x, A)

for every x ∈ X. One can check that

(2.11.7) dist(x, A) = dist(x, \overline{A})

for every x ∈ A.

Let V be a vector space over the real or complex numbers with a norm ∥·∥_{V}, and let V_{0} be a linear subspace of V. If v ∈ V, then put

N_{V_{0}}(v) = inf_{w∈V_{0}} ∥v − w∥_{V} . (2.11.8)

Equivalently, N_{V_{0}}(v) = dist(v, V_{0}), (2.11.9)

where the right side is defined as in (2.11.1), using the metric on V associated to ∥ · ∥_{V}. Thus N_{V_{0}}(v) = 0 (2.11.10)

if and only if v is an element of the closure \overline{V_{0}} of V_{0} in V, as in the preceding paragraph. Similarly, N_{V_{0}} = N\overline{V_{0}} (2.11.11)

on V, as in (2.11.7).

One can check that N_{V_{0}}(t v) = |t| N_{V_{0}}(v) (2.11.12)

for all v ∈ V and t ∈ R or C, as appropriate. If v_{1}, v_{2} ∈ V and w_{1}, w_{2} ∈ V_{0}, then

(2.11.13)  N_{V_{0}}(v_{1} + v_{2}) ≤ ∥(v_{1} + v_{2}) − (w_{1} + w_{2})∥_{V}
≤ ∥v_{1} − w_{1}∥_{V} + ∥v_{2} − w_{2}∥_{V} .

One can use this to verify that

N_{V_{0}}(v_{1} + v_{2}) ≤ N_{V_{0}}(v_{1}) + N_{V_{0}}(v_{2}). (2.11.14)

Thus N_{V_{0}} is a seminorm on V.

Note that N_{V_{0}}(v) ≤ ∥v∥_{V} (2.11.15)

for every v ∈ V. This implies that

|N_{V_{0}}(v) − N_{V_{0}}(u)| ≤ N_{V_{0}}(u − v) ≤ ∥u − v∥_{V} (2.11.16)

for all u, v ∈ V_{0}, where the first step is as in Section 1.2. This is another way to look at the Lipschitz condition for N_{V_{0}} as a real-valued function on V, with respect to the metric associated to ∥ · ∥_{V}.

44 CHAPTER 2. LIPSCHITZ AND BOUNDED LINEAR MAPPINGS

2.12 Distances and orthogonality

Let V be a vector space over the real or complex numbers with an inner product ⟨·, ·⟩_{V} and the associated norm ∥·∥_{V}. Also let W be a linear subspace of V, and let v ∈ V be given. We shall sometimes be interested in trying to get a u ∈ W such that ⟨v − u, w⟩_{V} = 0 (2.12.1) for every w ∈ W. Let us first check that such an element of W is unique. Suppose that u′ ∈ W satisfies ⟨v − u′, w⟩_{V} = 0 (2.12.2) for every w ∈ W as well. This implies that ⟨u − u′, w⟩_{V} = 0 (2.12.3) for every w ∈ W. It follows that u = u′, (2.12.4) because we can take w = u − u′ in (2.12.3).

If u ∈ W satisfies (2.12.1), then we can take w = u to get that ∥v∥_{V}^{2} = ∥v − u∥_{V}^{2} + ∥u∥_{V}^{2} . (2.12.5) Similarly, if y ∈ W, then ∥v − y∥_{V}^{2} = ∥(v − u) + (u − y)∥_{V}^{2} = ∥v − u∥_{V}^{2} + ∥u − y∥_{V}^{2} , (2.12.6) because of (2.12.1), with w = u − y. In particular, ∥v − u∥_{V} ≤ ∥v − y∥_{V} , (2.12.7) with equality if and only if u = y. This implies that dist(v, W) = ∥v − u∥_{V} , (2.12.8) where the left side is as in the previous section, using the metric on V associated to ∥ · ∥_{V}. Note that ∥u∥_{V} ≤ ∥v∥_{V} , (2.12.9) with equality if and only if u = v, by (2.12.5).

Suppose that v_{1}, …, v_{n} are finitely many orthonormal vectors in V, so that ∥v_{j}∥_{V} = 1 (2.12.10) for each j = 1, …, n, and ⟨v_{j}, v_{l}⟩_{V} = 0 (2.12.11)

2.12. DISTANCES AND ORTHOGONALITY 45 when j ≠ l. Let W be the linear subspace of V spanned by v_{1}, …, v_{n}. If v ∈ V, then put

P_{W} (v) = ∑_{j=1}^{n}⟨v, v_{j}⟩_{V} v_{j}. (2.12.12)

This defines a linear mapping from V into W. This is known as the orthogonal projection of V onto W, because of the properties mentioned in the next two paragraphs.

It is easy to see that P_{W} (v_{l}) = v_{l} (2.12.13) for each l = 1, …, n. This implies that P_{W} (v) = v (2.12.14) for every v ∈ W, so that P_{W} maps V onto W in particular. If v ∈ V, then ⟨P_{W} (v), v_{l}⟩_{V} = ⟨v, v_{l}⟩_{V} (2.12.15) for each l = 1, …, n. This means that ⟨v − P_{W} (v), v_{l}⟩_{V} = 0 (2.12.16) for each l = 1, …, n, so that ⟨v − P_{W} (v), w⟩_{V} = 0 (2.12.17) for every w ∈ W. Note that P_{W} (v) = 0 if and only if ⟨v, w⟩_{V} = 0 for every w ∈ W, by the definition (2.12.12) of P_{W} (v) and (2.12.15).

Thus u = P_{W} (v) ∈ W satisfies (2.12.1), and is uniquely determined by these properties, as before. It follows that P_{W} (v) does not depend on the particular orthonormal vectors v_{1}, …, v_{n} spanning W. Observe that

∥P_{W} (v)∥_{V}^{2} = ∑_{j=1}^{n} |⟨v, v_{j}⟩_{V} |^{2}. (2.12.18)

Combining this with (2.12.5), we get that

∥v∥_{V}^{2} = ∥v − P_{W} (v)∥_{V}^{2} + ∑_{j=1}^{n} |⟨v, v_{j}⟩_{V} |^{2}. (2.12.19)

We also have that dist(v, W) = ∥v − P_{W} (v)∥_{V} , (2.12.20) as in (2.12.8).

2.13 Orthonormal families of vectors

Let V be a vector space over the real or complex numbers with an inner product ⟨·, ·⟩_{V} and the associated norm ∥ · ∥_{V} again. Also let A be a nonempty set, and let {v_{α}}_{α∈A} be an orthonormal family of vectors in V indexed by A. This means that ∥v_{α}∥_{V} = 1 (2.13.1)

for all α ∈ A, and that ⟨v_{α}, v_{β}⟩_{V} = 0 (2.13.2)

for all α, β ∈ A with α ≠ β.

If E is a nonempty subset of A, then

(2.13.3) let W_{E} be the linear subspace of V spanned by the v_{α}’s with α ∈ E.

This may be interpreted as being {0} when E = ∅. If E_{1} ⊆ E_{2} ⊆ A, then

W_{E_{1}} ⊆ W_{E_{2}}. (2.13.4)

If E is a nonempty finite subset of A, then let

P_{W_{E}}(v) = ∑_{α∈E}⟨v, v_{α}⟩_{V} v_{α} (2.13.5)

be the corresponding orthogonal projection of V onto W_{E}, as in the previous section. Thus

∥v∥_{V}^{2} = ∥v − P_{W_{E}}(v)∥_{V}^{2} + ∑_{α∈E} |⟨v, v_{α}⟩_{V} |^{2} (2.13.6)

and dist(v, W_{E}) = ∥v − P_{W_{E}}(v)∥_{V} (2.13.7)

for every v ∈ V, as before. If E_{1} ⊆ E_{2} are nonempty finite subsets of A, then

(2.13.8) ∥v − P_{W_{E_{2}}} (v)∥_{V} = dist(v, W_{E_{2}}) ≤ dist(v, W_{E_{1}}) = ∥v − P_{W_{E_{1}}} (v)∥_{V} ,

because of (2.13.4). This also corresponds to the fact that the sum on the right side of (2.13.6) can only increase as E gets larger.

If v ∈ V, then put f_{v}(α) = ⟨v, v_{α}⟩_{V} (2.13.9)

for every α ∈ A. This defines a real or complex-valued function on A, as appropriate. If E is a nonempty finite subset of A, then

∑_{α∈E} |f_{v}(α)|^{2} ≤ ∥v∥_{V}^{2} , (2.13.10)

by (2.13.6). This means that f_{v} ∈ ℓ^{2}(A, R) or ℓ^{2}(A, C), as appropriate, with

∥f_{v}∥_{2}^{2} = ∑_{α∈A} |f_{v}(α)|^{2} ≤ ∥v∥_{V}^{2} . (2.13.11)

Note that v ↦ f_{v} (2.13.12)

is a linear mapping from V into ℓ^{2}(A, R) or ℓ^{2}(A, C), as appropriate.

Every element of W_{A} is an element of W_{E} for some finite set E ⊆ A. Using this, one can check that

dist(v, W_{A}) = inf{dist(v, W_{E}) : E ⊆ A, E ≠ ∅, E has only

(2.13.13) finitely many elements}.

Note that

∥v∥_{V}^{2} = dist(v, W_{E})^{2} + ∑_{α∈E} |⟨v, v_{α}⟩_{V} |^{2} (2.13.14)

for every nonempty finite subset E of A, by (2.13.6) and (2.13.7). One can use (2.13.13) and (2.13.14) to get that

∥v∥_{V}^{2} = dist(v, W_{A})^{2} + ∥f_{v}∥_{2}^{2}. (2.13.15)

Let (2.13.16) W be the closure \overline{W_{A}} of W_{A} in V,

with respect to the metric associated to ∥ · ∥_{V}. It follows from (2.13.15) that v ∈ W if and only if ∥v∥_{V} = ∥f_{v}∥_{2}. (2.13.17)

Observe that

∑_{α∈A} f_{v}(α) v_{α} (2.13.18)

satisfies the generalized Cauchy condition, as in Section 2.7. If E is a nonempty finite subset of A, then

∑_{α∈E} f_{v}(α) v_{α} = P_{W_{E}}(v), (2.13.19)

by (2.13.5) and (2.13.9). If v ∈ W_{A}, then (2.13.9) is equal to 0 for all but finitely many α ∈ A, so that (2.13.18) reduces to a finite sum, which is equal to v. If v ∈ W, then one can check that (2.13.18) converges in the generalized sense, with

∑_{α∈A} f_{v}(α) v_{α} = v, (2.13.20)

using (2.13.7), (2.13.8), and (2.13.19).

Suppose for the rest of the section that V is a Hilbert space. This implies that (2.13.18) converges in V in the generalized sense for every v ∈ V, as in Section 2.7. More precisely, the sum is an element of W, and we put

P_{W} (v) = ∑_{α∈A} f_{v}(α) v_{α} (2.13.21)

for each v ∈ V. This defines a linear mapping from V into W, with

P_{W} (v) = v (2.13.22)

for every v ∈ W, as in (2.13.20). Of course, this is the same as in the previous section when A has only finitely many elements.

If v ∈ V and β ∈ A, then it is easy to see that

⟨P_{W} (v), v_{β}⟩_{V} = f_{v}(β) = ⟨v, v_{β}⟩_{V} , (2.13.23)

using a remark in Section 2.10. Equivalently,

⟨v − P_{W} (v), v_{β}⟩_{V} = 0 (2.13.24)

for every β ∈ A. This implies that

⟨v − P_{W} (v), w⟩_{V} = 0 (2.13.25)

for every w ∈ W_{A}. It follows that this holds for every w ∈ W.

Note that P_{W} (v) ∈ W is uniquely determined by (2.13.25), as in the previous section. Thus P_{W} (v) depends only on v and W, and not on the particular orthonormal family of vectors whose closed linear span is W. As before, P_{W} is known as the orthogonal projection of V onto W.

2.14 Orthonormal bases

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers, with the corresponding norm ∥ · ∥_{V}. A collection of orthonormal vectors in V is said to be an orthonormal basis for V if

(2.14.1) its linear span is dense in V,

with respect to the metric associated to ∥ · ∥_{V}.

If V has finite dimension as a vector space over R or C, then V is spanned by finitely many vectors. In this case, one can use the Gram–Schmidt process to get finitely many orthonormal vectors in V that span V.

Similarly, for any sequence of vectors in V, one can use the Gram–Schmidt process to get a finite or infinite sequence of orthonormal vectors in V with the same linear span. If the linear span of the initial sequence is dense in V, then we get an orthonormal basis for V.

Let A be a nonempty set, let {v_{α}}_{α∈A} be an orthonormal family of vectors in V indexed by A, and let W be the closure of the linear span of the v_{α}’s, α ∈ A, as in the previous section. Suppose that

W ≠ V, (2.14.2)

and let u be an element of V not in W. If P_{W} is the orthogonal projection of V onto W, as in the previous section, then

u ≠ P_{W} (u), (2.14.3) because P_{W} (u) ∈ W. Put

y = ∥u − P_{W} (u)∥_{V}^{−1} (u − P_{W} (u)), (2.14.4)

so that ∥y∥_{V} = 1 and ⟨y, w⟩_{V} = 0 (2.14.5)

for every w ∈ W, by construction. This implies that the collection of v_{α}’s, α ∈ A, together with y, is an orthonormal collection of vectors in V.

The remarks in the preceding paragraph imply that if a collection of orthonormal vectors in V is maximal with respect to inclusion, then it is an orthonormal basis for V. There are well-known arguments to get a maximal orthonormal collection of vectors using the axiom of choice, based on Zorn’s lemma or Hausdorff’s maximality principle.

Suppose now that {v_{α}}_{α∈A} is an orthonormal basis for V. If f ∈ ℓ^{2}(A, R) or ℓ^{2}(A, C), as appropriate, then

∑_{α∈A} f(α) v_{α} (2.14.6)

converges in the generalized sense in V, as in Section 2.7. Put

T (f) = ∑_{α∈A} f(α) v_{α}, (2.14.7)

which defines a linear mapping from ℓ^{2}(A, R) or ℓ^{2}(A, C), as appropriate, into V. More precisely,

(2.14.8) this defines an isometric linear mapping from ℓ^{2}(A, R) or ℓ^{2}(A, C), as appropriate, into V,

as in Section 2.7 again. This implies that T preserves the appropriate inner products, as in Section 2.10, which could also be verified more directly in this case.

In fact,

(2.14.9) T maps ℓ^{2}(A, R) or ℓ^{2}(A, C), as appropriate, onto V

under these conditions. This follows from the remarks in the previous section, because W = V in this case. Alternatively, one can observe that

(2.14.10) the image of T is a closed set in V,

because of the completeness of ℓ^{2}(A, R) or ℓ^{2}(A, C), as appropriate.

If V_{0} is a closed linear subspace of V, then V_{0} may be considered as a Hilbert space, using the restriction of ⟨·, ·⟩_{V} to V_{0}. One can use an orthonormal basis for V_{0} to get the orthogonal projection of V onto V_{0}, as in the previous section.

50 CHAPTER 2. LIPSCHITZ AND BOUNDED LINEAR MAPPINGS

2.15 Minimizing distances

Let (V, ⟨·, ·⟩_{V}) be an inner product space over the real or complex numbers, with the corresponding norm ∥ · ∥_{V}. If v, w ∈ V, then it is easy to see that ∥v + w∥_{V}^{2} + ∥v − w∥_{V}^{2} = 2 ∥v∥_{V}^{2} + 2 ∥w∥_{V}^{2} . (2.15.1) This is known as the parallelogram law.

Let E be a nonempty subset of V, and let y ∈ V be given. Also let {z_{j}}_{j=1}^{∞} be a sequence of elements of E such that

lim_{j→∞} ∥y − z_{j}∥_{V} = dist(y, E), (2.15.2)

where the distance from y to E is defined using the metric on V associated to ∥ · ∥_{V}. Using the parallelogram law, we get that ∥(y − z_{j}) + (y − z_{l})∥_{V}^{2} + ∥z_{j} − z_{l}∥_{V}^{2} = 2 ∥y − z_{j}∥_{V}^{2} + 2 ∥y − z_{l}∥_{V}^{2} (2.15.3) for all j, l ≥ 1.

Suppose now that E is convex, so that (z_{j} + z_{l})/2 ∈ E for all j, l. This implies that ∥(y − z_{j}) + (y − z_{l})∥_{V} = 2 ∥y − ((z_{j} + z_{l})/2)∥_{V} ≥ 2 dist(y, E) (2.15.4) for all j, l. It follows that {z_{j}}_{j=1}^{∞} is a Cauchy sequence in V with respect to the metric associated to ∥ · ∥_{V}.

Suppose that V is a Hilbert space, and that E is a closed set in V, with respect to the metric associated to ∥ · ∥_{V}. Under these conditions, {z_{j}}_{j=1}^{∞} converges to an element z of E, and ∥y − z∥_{V} = dist(y, E). (2.15.5)

Suppose that E is a closed linear subspace of V, so that ∥y − z + w∥_{V} ≥ dist(y, E) (2.15.6) for every w ∈ E. Equivalently, this means that ∥y − z + t w∥_{V}^{2} ≥ ∥y − z∥_{V}^{2} (2.15.7) for every w ∈ E and t ∈ R or C, as appropriate. One can use this to get that ⟨y − z, w⟩_{V} = 0 (2.15.8) for every w ∈ E.

Remember that z ∈ E is uniquely determined by (2.15.8), as in Section 2.12. Thus we get a mapping P_{E} from V into E which sends y ∈ V to the corresponding z ∈ E. It is easy to see that P_{E} is a linear mapping, because of 2.15. MINIMIZING DISTANCES 51 this characterization of z. Of course, z = y when y ∈ E. This is another way to get the orthogonal projection of V onto a closed linear subspace E.

If V is any inner product space and V_{0} is a linear subspace of V, then put (2.15.9) V_{0}^{⊥} = {u ∈ V : ⟨u, w⟩_{V} = 0 for every w ∈ V_{0}}. This is called the orthogonal complement of V_{0} in V, and it is a closed linear subspace of V. It is easy to see that V_{0} ∩ V_{0}^{⊥} = {0}. (2.15.10) We also have that (\overline{V_{0}})^{⊥} = V_{0}^{⊥}, (2.15.11) where \overline{V_{0}} is the closure of V_{0} in V with respect to the metric associated to ∥·∥_{V}. The orthogonal complement of V_{0}^{⊥} in V can be defined in the same way, and we have that V_{0} ⊆ (V_{0}^{⊥})^{⊥}. (2.15.12)

If V is a Hilbert space and V_{0} is a closed linear subspace of V, then (2.15.13) V_{0}^{⊥} is the same as the kernel

of the orthogonal projection P_{V_{0}} of V onto V_{0}.

In this case, (2.15.14) every element of V can be expressed

as a sum of elements of V_{0} and V_{0}^{⊥} in a unique way,

as before. One can use this to check that (V_{0}^{⊥})^{⊥} = V_{0}. (2.15.15) More precisely, let v ∈ (V_{0}^{⊥})^{⊥} be given, and let us express v as v = w + u, (2.15.16) with w ∈ V_{0} and u ∈ V_{0}^{⊥}. This implies that v − w = u (2.15.17) is an element of V_{0}^{⊥} and (V_{0}^{⊥})^{⊥}, and is thus equal to 0, as in (2.15.10).

Bounded linear functionals

3.1 Dual spaces

Let V a vector space over the real or complex numbers. A linear functional on V is a linear mapping from V into R or C, as appropriate, considered as a one-dimensional vector space over itself. We shall call the space V^{alg} of all linear functionals on V the algebraic dual of V. This is a linear subspace of the space of all real or complex-valued functions on V, as appropriate.

Let ∥ · ∥_{V} be a norm on V. A bounded linear functional is a linear functional on V that is bounded as a linear mapping, using the standard absolute value function on R or C as the norm, as appropriate. Let V′ be the dual space of all bounded linear functionals on V, which is a linear subspace of the algebraic dual V^{alg}.

If λ ∈ V′, then put

∥λ∥_{V′} = sup{|λ(v)| : v ∈ V, ∥v∥_{V} ≤ 1}. (3.1.1)

This is the same as the operator norm of λ, as a bounded linear mapping from V into R or C, as appropriate. Thus ∥ · ∥_{V′} defines a norm on V′, which is known as the dual norm associated to ∥ · ∥_{V} on V. Note that

(3.1.2) V′ is complete

with respect to the metric associated to the dual norm, as in Section 2.2.

Suppose now that ∥ · ∥_{V} is the norm associated to an inner product ⟨·, ·⟩_{V} on V. If w ∈ V, then λ_{w}(v) = ⟨v, w⟩_{V} (3.1.3)

defines a linear functional on V. More precisely, this is a bounded linear functional on V, because of the Cauchy–Schwarz inequality. In fact,

∥λ_{w}∥_{V′} = ∥w∥_{V} , (3.1.4)

3.2. DUALS OF ℓ^{1} SPACES 53

because λ_{w}(w) = ∥w∥_{V}^{2} . Note that

w ↦ λ_{w} (3.1.5)

is linear in the real case, and conjugate-linear in the complex case.

Suppose that V is a Hilbert space. If λ is a bounded linear functional on V, then the Riesz representation theorem states that

λ = λ_{w} (3.1.6)

for a unique w ∈ V. The uniqueness of w follows from the remarks in the preceding paragraph. To get existence, we may suppose that λ ≠ 0, since otherwise we could take w = 0. If λ ≠ 0, then the kernel of λ is a proper closed linear subspace of V. In this case, one can use the orthogonal projection from V onto the kernel of λ to get y ∈ V such that y ≠ 0 and

(3.1.7) y is in the orthogonal complement of the kernel of λ.

One can check that (3.1.6) holds with w equal to a suitable multiple of y.

3.2 Duals of ℓ^{1} spaces

Let X be a nonempty set, and let f be a real or complex-valued function on X such that |f| is summable on X. If g is a bounded real or complex-valued function on X, as appropriate, then it is easy to see that |f| |g| is summable on X, with

∑_{x∈X} |f(x)| |g(x)| ≤ ∥f∥_{1} ∥g∥_{∞}. (3.2.1)

Put

λ_{g}(f) = ∑_{x∈X} f(x) g(x), (3.2.2)

where the sum on the right converges in the generalized sense. Observe that

|λ_{g}(f)| ≤ ∥f∥_{1} ∥g∥_{∞}, (3.2.3)

by (3.2.1).

This defines a bounded linear functional on ℓ^{1}(X, R) or ℓ^{1}(X, C), as appropriate. One can check that

(3.2.4) the dual norm of λ_{g} with respect to the usual ℓ^{1} norm is equal to the supremum norm of g.

More precisely, one can verify that ∥g∥_{∞} is less than or equal to the dual norm of λ_{g}, by considering functions f on X that are equal to 1 at one point, and equal to 0 at all other points in X.

Note that g ↦ λ_{g} (3.2.5) defines a linear mapping from each of ℓ^{∞}(X, R), ℓ^{∞}(X, C) into the duals of ℓ^{1}(X, R), ℓ^{1}(X, C), respectively. We would like to show that this mapping is surjective in both cases.

If g is any real or complex-valued function on X, then (3.2.2) defines a linear functional on c_{00}(X, R) or c_{00}(X, C), as appropriate. It is easy to see that every linear functional on c_{00}(X, R) or c_{00}(X, C) corresponds to a unique real or complex-valued function g on X, as appropriate, in this way.

Let λ be a bounded linear functional on ℓ^{1}(X, R) or ℓ^{1}(X, C), with respect to the usual ℓ^{1} norm. In particular,

(3.2.6) the restriction of λ to c_{00}(X, R) or c_{00}(X, C), as appropriate, is of the form λ_{g} as in (3.2.2)

for some real or complex-valued function g on X, as appropriate, as in the preceding paragraph. One can use the boundedness of λ with respect to the ℓ^{1} norm to get that (3.2.7) g is bounded on X,

with supremum norm bounded by the dual norm of λ, as before. Thus λ_{g} also defines a bounded linear functional on ℓ^{1}(X, R) or ℓ^{1}(X, C), as appropriate, as before.

By construction, λ = λ_{g} (3.2.8)

on c_{00}(X, R) or c_{00}(X, C), as appropriate. One can use this to get that they are the same on ℓ^{1}(X, R) or ℓ^{1}(X, C), as appropriate. This uses the fact that c_{00}(X, R), c_{00}(X, C) are dense in ℓ^{1}(X, R), ℓ^{1}(X, C), respectively, with respect to the metrics associated to the ℓ^{1} norms.

Of course, if X has only finitely many elements, then c_{00}(X, R), c_{00}(X, C) are the same as ℓ^{1}(X, R), ℓ^{1}(X, C), respectively. This means that every linear functional on ℓ^{1}(X, R) or ℓ^{1}(X, C) may be expressed as in (3.2.2), with dual norm with respect to the ℓ^{1} norm corresponding to ∥g∥_{∞}, as before.

3.3 Duals of c_{0} spaces

Let X be a nonempty set, and let f be a bounded real or complex-valued function on X. If g is a real or complex-valued function on X such that |g| is summable on X, then |f| |g| is summable on X, with

∑_{x∈X} |f(x)| |g(x)| ≤ ∥f∥_{∞} ∥g∥_{1}, (3.3.1)

as in the previous section. Put

λ_{g}(f) = ∑_{x∈X} f(x) g(x), (3.3.2)

3.3. DUALS OF C_{0} SPACES 55

where the sum on the right converges in the generalized sense, as before. We also have that |λ_{g}(f)| ≤ ∥f∥_{∞} ∥g∥_{1}, (3.3.3)

by (3.3.1). This defines a bounded linear functional on ℓ^{∞}(X, R) or ℓ^{∞}(X, C), as appropriate.

It is easy to see that

(3.3.4) the dual norm of λ_{g} with respect to the supremum norm is equal to ∥g∥_{1}.

More precisely, one can do this using a bounded real or complex-valued function f on X, as appropriate, such that ∥f∥_{∞} = 1 and

f(x) g(x) = |g(x)| (3.3.5)

for every x ∈ X. Of course, g ↦ λ_{g} (3.3.6)

defines a linear mapping from each of ℓ^{1}(X, R), ℓ^{1}(X, C) into the duals of ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively.

We may also consider λ_{g} as a bounded linear functional on c_{0}(X, R) or c_{0}(X, C), as appropriate, with respect to the supremum norm. One can check that (3.3.4) holds in this case too. Indeed, if A is a nonempty finite subset of X, then one can verify that

∑_{x∈A} |g(x)| (3.3.7)

is less than or equal to the dual norm of λ_{g}, as a bounded linear functional on c_{0}(X, R) or c_{0}(X, C), as appropriate, with respect to the supremum norm. This uses a real or complex-valued function f on X supported on A such that ∥f∥_{∞} = 1 and (3.3.5) holds for every x ∈ A.

Let λ be a bounded linear functional on c_{0}(X, R) or c_{0}(X, C), with respect to the supremum norm. As in the previous section, the restriction of λ to c_{00}(X, R) or c_{00}(X, C), as appropriate, is of the form λ_{g} as in (3.3.2) for some real or complex-valued function g on X, as appropriate. If A is a nonempty finite subset of X, then (3.3.7) is less than or equal to the dual norm of λ, as in the preceding paragraph. This implies that

g ∈ ℓ^{1}(X, R) or ℓ^{1}(X, C), (3.3.8)

as appropriate, with ℓ^{1} norm less than or equal to the dual norm of λ.

Thus λ_{g} defines a bounded linear functional on c_{0}(X, R) or c_{0}(X, C), as appropriate, with respect to the supremum norm, as before. One can check that λ = λ_{g} (3.3.9)

on c_{0}(X, R) or c_{0}(X, C), as appropriate. This uses the density of c_{00}(X, R), c_{00}(X, C) in c_{0}(X, R), c_{0}(X, C), respectively, with respect to the supremum metric. If X has only finitely many elements, then c_{00}(X, R), c_{0}(X, R), and ℓ^{∞}(X, R) are all the same, as are c_{00}(X, C), c_{0}(X, C), and ℓ^{∞}(X, C).

Suppose that X has infinitely many elements. Let us say that a real or complex-valued function f on X has a limit at infinity on X if there is a real or complex number t, as appropriate, such that

(3.3.10) f(x) − t vanishes at infinity on X,

as a real or complex-valued function of x ∈ X, as appropriate. It is easy to see that t is unique when it exists, in which case t is called the limit at infinity of f on X. Under these conditions, f is bounded on X, and

|t| ≤ ∥f∥_{∞}. (3.3.11)

If X is countably infinite, and {x_{j}}_{j=1}^{∞} is a sequence of elements of X in which every element of X occurs exactly once, then f has a limit at infinity on X if and only if {f(x_{j})}_{j=1}^{∞} converges as a sequence of real or complex numbers, as appropriate.

Let c^{lai}(X, R), c^{lai}(X, C) be the spaces of real and complex-valued functions on X, respectively, with a limit at infinity. Observe that these are linear subspaces of ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively. One can check that these are also closed sets in ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively, with respect to the supremum metric. More precisely, let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued functions on X with limits at infinity such that {f_{j}}_{j=1}^{∞} converges uniformly to a real or complex-valued function f on X, as appropriate. If t_{j} is the limit of f_{j} at infinity on X for each j, then one can verify that

(3.3.12) {t_{j}}_{j=1}^{∞} is a Cauchy sequence

in R or C, as appropriate, with respect to the standard Euclidean metric. This implies that {t_{j}}_{j=1}^{∞} converges to a real or complex number t, as appropriate. One can use this to get that (3.3.10) holds.

Of course, constant functions on X have a limit at infinity. The mapping

(3.3.13) f ↦ the limit of f at infinity on X

is a bounded linear functional on each of c^{lai}(X, R) and c^{lai}(X, C), with respect to the supremum norm, and with dual norm equal to 1. If λ is any linear functional on c^{lai}(X, R) or c^{lai}(X, C) that is equal to 0 on c_{0}(X, R) or c_{0}(X, C), as appropriate, then λ is a constant multiple of (3.3.13). This constant is equal to the value of λ at the function on X equal to 1 at every x ∈ X.

3.4 Hölder’s inequality for sums

Let p, q be real numbers with 1 < p, q < ∞. Suppose that

1/p + 1/q = 1, (3.4.1) in which case p and q are said to be conjugate exponents. One may also allow p = 1, q = ∞ or p = ∞, q = 1, which were basically considered already in the previous two sections. Note that (3.4.1) implies that

p = q/(q − 1), q = p/(p − 1). (3.4.2)

Let a, b be nonnegative real numbers. It is well known that

a b ≤ a^{p}/p + b^{q}/q. (3.4.3)

Of course, this is clear when a or b is 0, and so we may as well suppose that a, b > 0. This can be obtained from the convexity of the exponential function on the real line. Alternatively, the derivative of

a^{p}/p − a b + b^{q}/q, (3.4.4)

as a function of a, is equal to a^{p−1} − b. (3.4.5)

It is easy to see that this is equal to 0 if and only if (3.4.6) holds, using (3.4.2).

If a^{p} = b^{q}, (3.4.6)

then a b = a^{p} = b^{q}, (3.4.7)

and equality holds in (3.4.3). More precisely, the strict convexity of the exponential function on R implies that equality only holds in (3.4.3) when (3.4.6) holds. This can also be obtained by considering (3.4.5).

Let X be a nonempty set, and let f, g be real or complex-valued functions on X. If |f| is p-summable on X, and |g| is q-summable on X, then |f| |g| is summable on X, with

∑_{x∈X} |f(x)| |g(x)| ≤ (1/p) ∑_{x∈X} |f(x)|^{p} + (1/q) ∑_{x∈X} |g(x)|^{q}, (3.4.8)

because of (3.4.3). Under these conditions, Hölder’s inequality states that

∑_{x∈X} |f(x)| |g(x)| ≤ ∥f∥_{p} ∥g∥_{q}. (3.4.9)

This follows from (3.4.8) when ∥f∥_{p} = ∥g∥_{q} = 1. Otherwise, one can reduce to that case, using scalar multiplication.

Let A be a nonempty subset of X, and suppose for the moment that

f(x) g(x) = |f(x)|^{p} = |g(x)|^{q} (3.4.10)

for every x ∈ A. This implies that

(3.4.11)  ∑_{x∈A} f(x) g(x) = ∑_{x∈A} |f(x)|^{p} = ∑_{x∈A} |g(x)|^{q}

= ( ∑_{x∈A} |f(x)|^{p})^{1/p} ( ∑_{x∈A} |g(x)|^{q})^{1/q}.

Of course, if A has only finitely many elements, then this works without any summability conditions on f or g.

If g ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C), then put

λ_{g}(f) = ∑_{x∈X} f(x) g(x) (3.4.12)

for all f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate, where the sum on the right converges in the generalized sense, as usual. This defines a bounded linear functional on ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate. One can check that

(3.4.13) the dual norm of λ_{g} with respect to the ℓ^{p} norm is equal to ∥g∥_{q},

using (3.4.11) with A = X. More precisely, one can choose f so that (3.4.10) holds for every x ∈ X. As before,

g ↦ λ_{g} (3.4.14)

defines a linear mapping from each of ℓ^{q}(X, R), ℓ^{q}(X, C) into the duals of ℓ^{p}(X, R), ℓ^{p}(X, C), respectively.

Let λ be a bounded linear functional on ℓ^{p}(X, R) or ℓ^{p}(X, C). As in the previous two sections, the restriction of λ to c_{00}(X, R) or c_{00}(X, C), as appropriate, is of the form λ_{g} as in (3.4.12) for some real or complex-valued function g on X, as appropriate. If A is a finite subset of X, then one can verify that

( ∑_{x∈A} |g(x)|^{q})^{1/q} (3.4.15)

is less than or equal to the dual norm of λ, using (3.4.11). Here we choose f so that (3.4.10) holds when x ∈ A, and f = 0 on X \ A.

This implies that g ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C), (3.4.16)

as appropriate, with ℓ^{q} norm less than or equal to the dual norm of λ. It follows that λ_{g} defines a bounded linear functional on ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate, as before. Of course,

λ = λ_{g} (3.4.17)

on c_{00}(X, R) or c_{00}(X, C), as appropriate, by construction. Remember that c_{00}(X, R), c_{00}(X, C) are dense in ℓ^{p}(X, R), ℓ^{p}(X, C), respectively, with respect to the metrics associated to the ℓ^{p} norms when p < ∞. One can use this to get that (3.4.17) holds on ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate, as before.

3.5 Hilbert space adjoints

Let (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) be Hilbert spaces, both real or both complex, and let ∥·∥_{V}, ∥·∥_{W} be the norms corresponding to these inner products, respectively.

3.5. HILBERT SPACE ADJOINTS 59

Also let T be a bounded linear mapping from V into W. One can check that the operator norm of T may be given equivalently by ∥T ∥_{op,V W} = sup{|⟨T (v), w⟩_{W} | : v ∈ V, w ∈ W, ∥v∥_{V} , ∥w∥_{W} ≤ 1}. (3.5.1) More precisely, right side is less than or equal to the operator norm of T, because of the Cauchy–Schwarz inequality. To get the opposite inequality, one can use the fact that ∥y∥_{W} = sup{|⟨y, w⟩_{W} | : w ∈ W, ∥w∥_{W} ≤ 1} (3.5.2) for every y ∈ W.

If w ∈ W, then μ_{w}(v) = ⟨T (v), w⟩_{W} (3.5.3) defines a bounded linear functional on V. More precisely, |μ_{w}(v)| ≤ ∥T (v)∥_{W} ∥w∥_{W} ≤ ∥T ∥_{op,V W} ∥w∥_{W} ∥v∥_{V} (3.5.4) for every v ∈ V. It follows that there is a unique element of V that we shall denote T^{∗}(w) such that ⟨T (v), w⟩_{W} = ⟨v, T^{∗}(w)⟩_{V} (3.5.5) for every v ∈ V, as in Section 3.1. Note that |⟨v, T^{∗}(w)⟩_{V} | = |⟨T (v), w⟩_{W} | ≤ ∥T ∥_{op,V W} ∥w∥_{W} ∥v∥_{V} (3.5.6) for all v ∈ V, as in (3.5.4). One can use this to get that ∥T^{∗}(w)∥_{V} ≤ ∥T ∥_{op,V W} ∥w∥_{W} , (3.5.7) as in (3.5.2).

It is easy to see that T^{∗} is a linear mapping from W into V. Using (3.5.7), we get that T^{∗} is a bounded linear mapping, with ∥T^{∗}∥_{op,W V} ≤ ∥T ∥_{op,V W} . (3.5.8) As in (3.5.1), we have that ∥T^{∗}∥_{op,W V} = sup{|⟨v, T^{∗}(w)⟩_{V} | : v ∈ V, w ∈ W, ∥v∥_{V} , ∥w∥_{W} ≤ 1}. (3.5.9) One can use this, (3.5.1), and (3.5.5) to get that ∥T^{∗}∥_{op,W V} = ∥T ∥_{op,V W} . (3.5.10) The operator T^{∗} is called the adjoint of T.

Similarly, the adjoint (T^{∗})^{∗} of T^{∗} is a bounded linear mapping from V into W. One can check that (T^{∗})^{∗} = T. (3.5.11) One can also use this to get (3.5.10) from (3.5.8) and its analogue for T^{∗}.

60 CHAPTER 3. BOUNDED LINEAR FUNCTIONALS

Note that T ↦ T^{∗} (3.5.12) is a linear mapping from BL(V, W) into BL(W, V) in the real case. In the complex case, this mapping is conjugate-linear.

Let (Z, ⟨·, ·⟩_{Z}) be another Hilbert space, which is real or complex, depending on whether V, W are real or complex. If T_{1} is a bounded linear mapping from V into W, and T_{2} is a bounded linear mapping from W into Z, then their composition T_{2} ∘ T_{1} is a bounded linear mapping from V into Z. Observe that ⟨T_{2}(T_{1}(v)), z⟩_{Z} = ⟨T_{1}(v), T_{2}^{∗}(z)⟩_{W} = ⟨v, T_{1}^{∗}(T_{2}^{∗}(z))⟩_{Z} (3.5.13) for every v ∈ V and z ∈ Z. This implies that (T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗}, (3.5.14) as bounded linear mappings from Z into V.

If T is any bounded linear mapping from V into W, then T^{∗} ∘T is a bounded linear mapping from V into itself. Note that ⟨T^{∗}(T (u)), v⟩_{V} = ⟨T (u), T (v)⟩_{W} (3.5.15) for every u, v ∈ V. It follows that T is an isometric linear mapping from V into W if and only if ⟨T^{∗}(T (u)), v⟩_{V} = ⟨u, v⟩_{V} (3.5.16) for every u, v ∈ V. One can check that this happens if and only if (3.5.17) T^{∗} ∘ T is the identity mapping I_{V} on V.

If we take u = v in (3.5.15), then we get that ∥T (v)∥_{W}^{2} = ⟨T^{∗}(T (v)), v⟩_{V} (3.5.18) for every v ∈ V. This implies that ∥T (v)∥_{V}^{2} ≤ ∥T^{∗}(T (v))∥_{V} ∥v∥_{V} ≤ ∥T^{∗} ∘ T ∥_{op,V V} ∥v∥_{V}^{2} (3.5.19) for every v ∈ V, using the Cauchy–Schwarz inequality in the first step. It follows that ∥T ∥_{op,V W}^{2} ≤ ∥T^{∗} ∘ T ∥_{op,V V} . (3.5.20) It is easy to see that the opposite inequality holds too, so that ∥T ∥_{op,V W}^{2} = ∥T^{∗} ∘ T ∥_{op,V V} . (3.5.21) This is the C^{∗} identity for the operator norm of a bounded linear mapping between Hilbert spaces.

3.6. SUBLINEAR FUNCTIONS 61

3.6 Sublinear functions

Let V be a vector space over the real numbers, and let p be a real-valued function on V. If p(v + w) ≤ p(v) + p(w) (3.6.1) for every v, w ∈ V, then p is said to be subadditive on V. If p(t v) = t p(v) (3.6.2) for every v ∈ V and nonnegative real number t, then p is said to be homogeneous of degree 1 on V. Of course, this implies that p(0) = 0, (3.6.3) by taking t = 0. If p is both subadditive and homogeneous of degree 1, then p is said to be sublinear on V.

Let us say that p is symmetric on V if p(−v) = p(v) (3.6.4) for every v ∈ V. If p is subadditive on V, then it is easy to see that p(0) ≥ 0. (3.6.5) We also have that p(0) ≤ p(v) + p(−v) (3.6.6) for every v ∈ V. If p is symmetric on V as well, then we get that 0 ≤ p(0)/2 ≤ p(v) (3.6.7) for every v ∈ V. Thus a seminorm on V is the same as a symmetric sublinear function on V.

Note that linear functionals on V are sublinear. More precisely, if p is sublinear on V, then p is linear on V if and only if p(−v) = −p(v) (3.6.8) for every v ∈ V.

Let p_{1}, p_{2} be real-valued functions on V. If p_{1}, p_{2} are subadditive on V, then it is easy to see that (3.6.9) max(p_{1}, p_{2}) is subadditive on V. If p_{1}, p_{2} are both homogeneous of degree 1, or both symmetric on V, then max(p_{1}, p_{2}) has the same property.

If p is sublinear on V, then clearly (3.6.10) p(−v) is sublinear on V.

62 CHAPTER 3. BOUNDED LINEAR FUNCTIONALS It follows that (3.6.11) max(p(v), p(−v)) is a seminorm on V. Note that (3.6.12) max(p(v), 0) is sublinear on V in this case too.

If p is subadditive on V, then p(v) ≤ p(w) + p(v − w) (3.6.13) for every v, w ∈ V. Similarly, p(w) ≤ p(v) + p(w − v) (3.6.14) for every v, w ∈ V. It follows that |p(v) − p(w)| ≤ max(p(v − w), p(w − v)) (3.6.15) for every v, w ∈ V. If p is symmetric on V, then this reduces to |p(v) − p(w)| ≤ p(v − w) (3.6.16) for every v, w ∈ V. Of course, the analogous statement for seminorms was mentioned in Section 1.2.

If p is sublinear on V, then {v ∈ V : p(v) < r} (3.6.17) is a convex set in V for every real number r. Similarly, {v ∈ V : p(v) ≤ r} (3.6.18) is a convex set in V for every r ∈ R. Note that 0 is an element of (3.6.17) when r > 0, and an element of (3.6.18) when r ≥ 0. Analogous statements for seminorms were mentioned in Section 1.2.

If E is any subset of V, then put −E = {−v : v ∈ E}. (3.6.19) If −E = E, (3.6.20) then E is said to be symmetric about 0 in V. If p is symmetric in V, then (3.6.17) and (3.6.18) are symmetric about 0 in V for every r ∈ R.

3.7. CONVEX CONES 63

3.7 Convex cones

Let V be a vector space over the real numbers again. A subset C of V is said to be a convex cone in V if it satisfies the following two conditions. First, if v ∈ C and t is a positive real number, then t v ∈ C. (3.7.1) Second, if v, w ∈ C, then v + w ∈ C. (3.7.2)

It is easy to see that (3.7.3) convex cones are convex sets. If a subset C of V satisfies (3.7.1) and is convex, then (3.7.2) holds.

Linear subspaces of V are convex cones. If a nonempty convex cone C in V is symmetric about 0 in V, then (3.7.4) C is a linear subspace of V. If C is a convex cone in V, and if C ∩ (−C) (3.7.5) is nonempty, then (3.7.5) is a linear subspace of V.

If p is a sublinear real-valued function on V, then {v ∈ V : p(v) ≤ 0} (3.7.6) is a convex cone in V. If p is a seminorm on V, then this is the same as {v ∈ V : p(v) = 0}, (3.7.7) which is a linear subspace of V.

Let C be a nonempty convex cone in V, and let p be a nonnegative realvalued function on V. If v ∈ V, then put p_{C}(v) = inf{p(v − w) : w ∈ C}. (3.7.8) If p(0) = 0, then p_{C}(v) = 0 (3.7.9) for every v ∈ C. If p is a norm on V, then p_{C} is the same as the distance to C with respect to the metric associated to C, as in Section 2.11.

If p is homogeneous of degree 1 on V, then one can check that p_{C} is homogeneous of degree 1 on V as well. More precisely, one can verify that p_{C}(t v) = t p_{C}(v) (3.7.10) for every v ∈ V and positive real number t. We also have that

p_{C}(0) = 0, (3.7.11)

so that (3.7.10) holds when t = 0 too.

Suppose that p is subadditive on V. If v_{1}, v_{2} ∈ V and w_{1}, w_{2} ∈ C, then

p_{C}(v_{1} + v_{2}) ≤ p(v_{1} + v_{2} − w_{1} − w_{2}) ≤ p(v_{1} − w_{1}) + p(v_{2} − w_{2}). (3.7.12)

One can use this to get that

p_{C}(v_{1} + v_{2}) ≤ p_{C}(v_{1}) + p_{C}(v_{2}). (3.7.13)

3.8 Minkowski functionals

Let V be a vector space over the real numbers, and let E be a subset of V. If t ∈ R, then put t E = {t v : v ∈ E}. (3.8.1)

We say that E is starlike about 0 if

t E ⊆ E (3.8.2)

for every t ∈ [0, 1]. Of course, this implies that 0 ∈ E when E ≠ ∅. If E is convex and 0 ∈ E, then E is starlike about 0.

Let us say that a subset A of V is radial at 0 if for every v ∈ V there is a positive real number r(v) such that

r v ∈ A (3.8.3)

for every r ∈ R with 0 ≤ r ≤ r(v), as on p14 of [176]. Note that this implies that 0 ∈ A. This is a type of “absorbing” property, although that term is used in slightly different ways as well. A convex set A ⊆ V may be said to be absorbing if (3.8.4) every v ∈ V is an element of t A for some t > 0,

as on p24 of [286]. This implies that 0 ∈ A, and that A is radial at 0, which would also work when A is starlike about 0 instead of convex.

Let p be a real-valued function on V that is homogeneous of degree 1. If r is a positive real number, then (3.6.17) and (3.6.18) are starlike about 0 and radial at 0.

Suppose that A ⊆ V satisfies (3.8.4). The corresponding Minkowski functional on V is defined by

ρ_{A}(v) = inf{t > 0 : t^{−1} v ∈ A} = inf{t > 0 : v ∈ t A}, (3.8.5)

as on p15 of [176], and p24 of [286]. Clearly

ρ_{A}(0) = 0, (3.8.6)

3.9. SOME ONE-STEP EXTENSIONS 65

because 0 ∈ A. It is easy to see that ρ_{A}(r v) = r ρ_{A}(v) (3.8.7) for every v ∈ V and r > 0, so that ρ_{A} is homogeneous of degree 1 on V. If A is symmetric about 0 in V, then ρ_{A} is symmetric on V.

Put B = {v ∈ V : ρ_{A}(v) < 1} (3.8.8) and C = {v ∈ V : ρ_{A}(v) ≤ 1}. (3.8.9) By construction, A ⊆ C. (3.8.10) If A is starlike about 0, then B ⊆ A. (3.8.11) If A is convex, then one can check that (3.8.12) ρ_{A} is subadditive on V, using the argument in Section 1.2.

3.9 Some one-step extensions

Let V be a vector space over the real numbers, and let p be a sublinear realvalued function on V. Also let V_{0} be a linear subspace of V, and let λ_{0} be a linear functional on V_{0}. Suppose that λ_{0}(v_{0}) ≤ p(v_{0}) (3.9.1) for every v_{0} ∈ V_{0}.

If V_{0} ≠ V, then let u_{1} be an element of V not in V_{0}. Consider V_{1} = {v_{0} + t u_{1} : v_{0} ∈ V_{0}, t ∈ R}, (3.9.2) which is the linear subspace of V spanned by V_{0} and u_{1}. Under these conditions, there is an extension λ_{1} of λ_{0} to a linear functional on V_{1} such that λ_{1}(v_{1}) ≤ p(v_{1}) (3.9.3) for every v_{1} ∈ V_{1}. This is part of the theorem of Hahn and Banach.

One can check that every element of V_{1} can be expressed in a unique way as v_{0} + t u_{1} (3.9.4) for some v_{0} ∈ V_{0} and t ∈ R, because u_{1} ∉ V_{0}. If α_{1} is a real number, then we can get an extension λ_{1} of λ_{0} to a linear functional on V_{1} by putting λ_{1}(v_{0} + t u_{1}) = λ_{0}(v_{0}) + t α_{1} (3.9.5)

66 CHAPTER 3. BOUNDED LINEAR FUNCTIONALS

for every v_{0} ∈ V_{0} and t ∈ R. In fact, every such extension λ_{1} of λ_{0} corresponds to a unique α_{1} ∈ R in this way.

Thus we would like to choose α_{1} so that λ_{0}(v_{0}) + t α_{1} ≤ p(v_{0} + t u_{1}) (3.9.6) for every v_{0} ∈ V_{0} and t ∈ R. This condition holds when t = 0, by (3.9.1). In order to get that this condition holds when t ≠ 0, one can reduce to the cases where t = ±1, because p is homogeneous of degree 1 on V, and V_{0} is a linear subspace of V. This means that it suffices to choose α_{1} so that λ_{0}(v_{0}) + α_{1} ≤ p(v_{0} + u_{1}) (3.9.7) and λ_{0}(v_{0}) − α_{1} ≤ p(v_{0} − u_{1}) (3.9.8) for every v_{0}. Equivalently, we would like to choose α_{1} so that λ_{0}(v_{0}) − p(v_{0} − u_{1}) ≤ α_{1} ≤ p(w_{0} + u_{1}) − λ_{0}(w_{0}) (3.9.9) for every v_{0}, w_{0} ∈ V_{0}.

Let us check that λ_{0}(v_{0}) − p(v_{0} − u_{1}) ≤ p(w_{0} + u_{1}) − λ_{0}(w_{0}) (3.9.10) for every v_{0}, w_{0} ∈ V_{0}. This is the same as saying that λ_{0}(v_{0}) + λ_{0}(w_{0}) ≤ p(v_{0} − u_{1}) + p(w_{0} + u_{1}) (3.9.11) for all v_{0}, w_{0} ∈ V_{0}. If v_{0}, w_{0} ∈ V_{0}, then

(3.9.12) λ_{0}(v_{0}) + λ_{0}(w_{0}) = λ_{0}(v_{0} + w_{0}) ≤ p(v_{0} + w_{0})
≤ p(v_{0} − u_{1}) + p(w_{0} + u_{1}).

This uses (3.9.1) in the second step, and the subadditivity of p on V in the third step. This shows that (3.9.10) holds, which implies that there is an α_{1} ∈ R such that (3.9.9) holds.

More precisely, sup{λ_{0}(v_{0}) − p(v_{0} − u_{1}) : v_{0} ∈ V_{0}} (3.9.13)

≤ inf{p(w_{0} + u_{1}) − λ_{0}(w_{0}) : w_{0} ∈ V_{0}},

because of (3.9.10). In order to get (3.9.9), one can take any α_{1} ∈ R such that sup{λ_{0}(v_{0}) − p(v_{0} − u_{1}) : v_{0} ∈ V_{0}} (3.9.14)

≤ α_{1} ≤ inf{p(w_{0} + u_{1}) − λ_{0}(w_{0}) : w_{0} ∈ V_{0}}.

Of course, (3.9.1) is the same as saying that −λ_{0}(v_{0}) = λ_{0}(−v_{0}) ≤ p(−v_{0}) (3.9.15)

3.10. THE HAHN–BANACH THEOREM 67

for every v_{0} ∈ V_{0}. This means that −p(−v_{0}) ≤ λ_{0}(v_{0}) (3.9.16) for every v_{0} ∈ V_{0}. Thus (3.9.1) implies that |λ_{0}(v_{0})| ≤ max(p(v_{0}), p(−v_{0})) (3.9.17) for every v_{0} ∈ V_{0}.

Let ∥ · ∥_{V} be a norm on V, and suppose that p(v) ≤ C ∥v∥_{V} (3.9.18) for some nonnegatve real number C and every v ∈ V. This implies that p(−v) ≤ C ∥ − v∥_{V} = C ∥v∥_{V} (3.9.19) for every v ∈ V as well. It follows that |p(v) − p(w)| ≤ max(p(v − w), p(w − v)) ≤ C ∥v − w∥_{V} (3.9.20) for every v, w ∈ V, using (3.6.15) in the first step. In particular, this means that p is continuous as a real-valued function on V, with respect to the metric associated to ∥ · ∥_{V}.

Similarly, |λ_{0}(v_{0})| ≤ C ∥v_{0}∥_{V} (3.9.21) for every v_{0} ∈ V_{0}, by (3.9.17). Under these conditions, (3.9.22) λ_{0} has a unique extension to a bounded linear functional

on the closure \overline{V_{0}} of V_{0} in V,

with respect to the metric associated to ∥ · ∥_{V}, as in Section 2.2. This extension satisfies the analogue of (3.9.1) on \overline{V_{0}}, because p is continuous on V, as in the preceding paragraph.

3.10 The Hahn–Banach theorem

Let us continue with the same notation and hypotheses as at the beginning of the previous section. Under these conditions, the Hahn–Banach theorem states that there is an extension λ of λ_{0} to a linear functional on V that satisfies λ(v) ≤ p(v) (3.10.1) for every v ∈ V.

If V is spanned by V_{0} and finitely many additional vectors, then λ can be obtained by repeating the argument in the previous section finitely many times. If V is spanned by V_{0} and a sequence of additional vectors, then one can continue to repeat the argument to get λ.

Otherwise, there is an argument based on the axiom of choice using Zorn’s lemma or Hausdorff’s maximality principle to get a maximal extension of λ_{0} to a linear subspace of V on which the extension is less than or equal to p. The argument in the previous section implies that such a maximal extension is defined on all of V.

Let ∥ · ∥_{V} be a norm on V again, and suppose that p is less than or equal to a nonnegative real number C times ∥ · ∥_{V} on V, as in (3.9.18). In this case, it suffices to find an extension of λ_{0} to a dense linear subspace of V, with respect to the metric associated to ∥ · ∥_{V}, on which the extension is less than or equal to p, as in the previous section. If the linear span of V_{0} and a sequence of additional vectors is dense in V, then such an extension of λ_{0} can be obtained by repeating the argument in the previous section, as before.

If p is a seminorm on V, then (3.9.1) is the same as saying that

|λ_{0}(v_{0})| ≤ p(v_{0}) (3.10.2)

for every v_{0} ∈ V_{0}, as in the previous section. Similarly, (3.10.1) is the same as saying that |λ(v)| ≤ p(v) (3.10.3)

for every v ∈ V. This corresponds to Theorem 3.3 on p57 of [286] in the real case.

In particular, we can take

p(v) = C ∥v∥_{V} (3.10.4)

on V for some C ≥ 0. Using this, we get that the Hahn–Banach theorem implies that if λ_{0} is a bounded linear functional on V_{0} with respect to the restriction of ∥ · ∥_{V} to V_{0}, then λ_{0} has an extension to a bounded linear functional on V, with the same dual norm with respect to ∥ · ∥_{V}.

Let u_{0} ∈ V with u_{0} ≠ 0 be given, and let V_{0} be the linear span of u_{0} in V. Consider the linear functional λ_{0} defined on V_{0} by

λ_{0}(t u_{0}) = t ∥u_{0}∥_{V} (3.10.5)

for every t ∈ R. The Hahn–Banach theorem implies that λ_{0} has an extension to a bounded linear functional on V with dual norm equal to 1. This corresponds to the corollary on p58 of [286] in the real case.

Let W_{0} be a closed linear subspace of V, with respect to the metric associated to ∥ · ∥_{V}, and suppose that u_{0} ∈ V \ W_{0}. Thus there is a positive real number r such that ∥u_{0} − w_{0}∥_{V} ≥ r (3.10.6)

for every w_{0} ∈ W_{0}. Let V_{0} be the linear subspace of V spanned by W_{0} and u_{0}, so that every element of V_{0} can be expressed in a unique way as

w_{0} + t u_{0} (3.10.7)

for some w_{0} ∈ W_{0} and t ∈ R. This uses the fact that t u_{0} ∈ W_{0} only when t = 0.

Consider the linear functional λ_{0} defined on V_{0} by

λ_{0}(w_{0} + t u_{0}) = t (3.10.8)

for every w_{0} ∈ W_{0} and t ∈ R. One can check that

|λ_{0}(w_{0} + t u_{0})| = |t| ≤ r^{−1} ∥w_{0} + t u_{0}∥_{V} (3.10.9)

for every w_{0} ∈ W_{0} and t ∈ R, using (3.10.6) in the second step. The Hahn– Banach theorem implies that λ_{0} can be extended to a bounded linear functional λ on V, with dual norm less than or equal to 1/r. Note that λ(u_{0}) = 1, and that λ(w_{0}) = 0 for every w_{0} ∈ W_{0}. This corresponds to a simplification of Theorem 3.5 on p59 of [286] in the real case.

3.11 Some separation results

Let V be a vector space over the real numbers, and let ∥ · ∥_{V} be a norm on V. Also let A be a nonempty convex open subset of V, with respect to the metric associated to ∥ · ∥_{V}, and let v be an element of V not in A. Under these conditions, there is a bounded linear functional λ on V such that

λ(a) < λ(v) (3.11.1)

for every a ∈ A. This corresponds to a simplification of part (a) of Theorem 3.4 on p58 of [286] in the real case.

To see this, we start by reducing to the case where 0 ∈ A, which we can do using a translation on V. It is easy to see that A is radial at 0, or equivalently that A has the absorbing property mentioned in Section 3.8, because A is an open set in V. Let ρ_{A} be the Minkowski functional on V associated to A, as in Section 3.8. Remember that ρ_{A} is sublinear on V under these conditions.

Let V_{0} be the linear subspace of V spanned by v, and let λ_{0} be the linear functional on V_{0} defined by

λ_{0}(t v) = t ρ_{A}(v) (3.11.2)

for every t ∈ R. Observe that

λ_{0}(t v) ≤ ρ_{A}(t v) (3.11.3)

for every t ∈ R, with equality when t ≥ 0. More precisely, if t < 0, then the right side is greater than or equal to 0, and the left side is less than or equal to 0. The Hahn–Banach theorem implies that there is an extension λ of λ_{0} to a linear functional on V that satisifes

λ(w) ≤ ρ_{A}(w) (3.11.4)

for every w ∈ V.

It is easy to see that ρ_{A}(v) ≥ 1, (3.11.5)

because v ∉ A, by hypothesis. If a ∈ A, then one can check that

ρ_{A}(a) < 1, (3.11.6)

because A is an open set in V. This implies (3.11.1).

Because A is an open set in V that contains 0, there is a positive real number r such that a ∈ A when a ∈ V and ∥a∥_{V} < r. Using this, one can verify that

ρ_{A}(w) ≤ r^{−1} ∥w∥_{V} (3.11.7)

for every w ∈ V. It follows from this and (3.11.4) that λ is a bounded linear functional on V, as in Section 3.9.

If E_{1}, E_{2} ⊆ V, then put

E_{1} + E_{2} = {w_{1} + w_{2} : w_{1} ∈ E_{1}, w_{2} ∈ E_{2}}. (3.11.8)

It is easy to see that this is an open set in V, with respect to the metric associated to ∥ · ∥_{V}, when E_{1} or E_{2} is an open set. If E_{1} is an open set in V, for instance, then E_{1} + E_{2} is the same as the union of the translates of E_{1} by elements of E_{2}, and each of these translates of E_{1} is an open set in V. If E_{1} and E_{2} are both convex sets in V, then E_{1} + E_{2} is convex in V as well.

Now let B be a nonempty closed convex set in V, with respect to the metric associated to ∥ · ∥_{V}, and suppose that v ∉ B. Thus there is a positive real number r_{1} such that ∥v − b∥_{V} ≥ r_{1} (3.11.9)

for every b ∈ B. Put A_{1} = B + B(0, r_{1}), (3.11.10)

where B(0, r_{1}) is the usual open ball in V centered at 0 with radius r_{1} with respect to the metric associated to ∥ · ∥_{V}. This is a convex open subset of V, as in the preceding paragraph. We also have that v ∉ A_{1}, by construction.

It follows that there is a bounded linear functional λ on V such that

λ(a_{1}) < λ(v) (3.11.11)

for every a_{1} ∈ A_{1}, as before. Equivalently, this means that

λ(b) + λ(w) = λ(b + w) < λ(v) (3.11.12)

for every b ∈ B and w ∈ V with ∥w∥_{V} < r_{1}. Of course, λ ≢ 0 on V, so that

λ(w) > 0 (3.11.13)

for some w ∈ V with ∥w∥_{V} < r_{1}. Using this, we get that

sup_{b∈B} λ(b) ≤ λ(v) − λ(w) < λ(v). (3.11.14)

This corresponds to a simplification of part (b) of Theorem 3.4 on p58 of [286] in the real case.

3.12 Complex linear functionals

Let V be a vector space over the complex numbers, and let V_{R} be V considered as a vector space over the real numbers, as before. If λ is a linear functional on V, then μ = Re λ (3.12.1)

is a linear functional on V_{R}. It is easy to see that λ is uniquely determined by μ. More precisely, λ(v) = μ(v) − i μ(i v) (3.12.2)

for every v ∈ V, as in (1) on p56 of [286]. Conversely, if μ is a linear functional on V_{R}, then one can check that (3.12.2) defines a linear functional on V, as in [286].

Let ∥ · ∥_{V} be a norm on V, which may also be considered as a norm on V_{R}. It is easy to see that λ is bounded as a linear functional on V with respect to ∥ · ∥_{V} if and only if μ is bounded as a linear functional on V_{R} with respect to ∥ · ∥_{V}. In fact, the corresponding dual norms of λ and μ are the same, which is to say that ∥λ∥_{V′} = ∥μ∥_{V′_{R}}. (3.12.3)

Indeed, if v ∈ V, then

(3.12.4)  |λ(v)| = sup{Re(tλ(v)) : t ∈ C, |t| = 1}
= sup{μ(t v) : t ∈ C, |t| = 1}.

Let N be a seminorm on V, which may be considered as a seminorm on V_{R} too. Also let V_{0} be a linear subspace of V, and let λ_{0} be a linear functional on V_{0} such that |λ_{0}(v_{0})| ≤ N(v_{0}) (3.12.5)

for every v_{0} ∈ V_{0}. Thus μ_{0} = Re λ_{0} is a linear functional on V_{0}, as a linear subspace of V_{R}, with μ_{0}(v_{0}) ≤ N(v_{0}) (3.12.6)

for every v_{0} ∈ V_{0}. It follows that μ_{0} can be extended to a linear functional μ on V_{R} such that μ(v) ≤ N(v) (3.12.7)

for every v ∈ V_{R}, as in Section 3.10. Let λ be the linear functional on V whose real part is equal to μ, as before. Observe that

|λ(v)| ≤ N(v) (3.12.8)

for every v ∈ V, because of (3.12.4). We also have that λ = λ_{0} on V_{0}, because their real parts are the same on V_{0}, by construction. This corresponds to Theorem 3.3 on p57 of [286] in the complex case.

If u_{0} ∈ V and u_{0} ≠ 0, then there is a bounded linear functional λ on V such that λ(u_{0}) = ∥u_{0}∥_{V} and ∥λ∥_{V′} = 1. (3.12.9)

This can be obtained from the remarks in the preceding paragraph in the same way as in Section 3.10. This corresponds to the corollary on p58 of [286] in the complex case.

Similarly, let W_{0} be a closed linear subapace of V, with respect to the metric associated to ∥·∥_{V}, and let u_{0} be an element of V \W_{0}. Under these conditions, there is a bounded linear function λ on V such that

(3.12.10) λ(u_{0}) = 1 and λ(w_{0}) = 0 for every w_{0} ∈ W_{0},

as in Section 3.10 again. This corresponds to a simplification of Theorem 3.5 on p59 of [286] in the complex case.

3.13 Dual linear mappings

Let V, W be vector spaces, both real or both complex. Also let T be a linear mapping from V into W. If λ is a linear functional on W, then

λ ∘ T (3.13.1)

is a linear functional on V. Put

T^{alg}(λ) = λ ∘ T, (3.13.2)

which defines a linear mapping from W^{alg} into V^{alg}. This is the dual linear mapping associated to T between the corresponding algebraic dual spaces.

Observe that T ↦ T^{alg} (3.13.3)

defines a linear mapping from L(V, W) into L(W^{alg}, V^{alg}). Let Z be another vector space over the real or complex numbers, depending on whether V, W are real or complex. If T_{1} is a linear mapping from V into W, and T_{2} is a linear mapping from W into Z, then one can check that

(T_{2} ∘ T_{1})^{alg} = T_{1}^{alg} ∘ T_{2}^{alg}, (3.13.4)

as linear mappings from Z^{alg} into V^{alg}. More precisely, if μ ∈ Z^{alg}, then

(3.13.5) ((T_{2} ∘ T_{1})^{alg})(μ) = μ ∘ (T_{2} ∘ T_{1}) = (μ ∘ T_{2}) ∘ T_{1} = T_{1}^{alg}(T_{2}^{alg}(μ)).

Let ∥ · ∥_{V}, ∥ · ∥_{W} be norms on V, W, respectively. If T is a bounded linear mapping from V into W, and λ is a bounded linear functional on W, then (3.13.1) is a bounded linear functional on V. Put

T′(λ) = λ ∘ T, (3.13.6)

which defines a linear mapping from W′ into V′. Of course, this is the same as the restriction of T^{alg} to W′. This is the dual linear mapping of T between the dual spaces W′, V′ of W, V associated to the norms ∥ · ∥_{W}, ∥ · ∥_{V}, respectively.

3.13. DUAL LINEAR MAPPINGS 73

Note that ∥T′(λ)∥_{V′} ≤ ∥T ∥_{op,V W} ∥λ∥_{W′} (3.13.7) for every λ ∈ W′. This implies that T′ is a bounded linear mapping from W′ into V′, with respect to the dual norms corresponding to ∥ · ∥_{W}, ∥ · ∥_{V}, with ∥T′∥_{op,W′V′} ≤ ∥T ∥_{op,V W} . (3.13.8) We also have that T ↦ T′ (3.13.9) is a linear mapping from BL(V, W) into BL(W′, V′).

If v ∈ V and λ ∈ W′, then

(3.13.10)  |λ(T (v))| = |(T′(λ))(v)| ≤ ∥T′(λ)∥_{V′} ∥v∥_{V}
≤ ∥T′∥_{op.W′V′} ∥λ∥_{W′} ∥v∥_{V} .

This implies that ∥T (v)∥_{W} ≤ ∥T′∥_{op,W′V′} ∥v∥_{V} , (3.13.11) because of (3.12.9) and its analogue in the real case. It follows that ∥T ∥_{op,V W} ≤ ∥T′∥_{op,W′V′}. (3.13.12) This means that ∥T′∥_{op,W′V′} = ∥T ∥_{op,V W} , (3.13.13) because of (3.13.8).

Suppose that T is an isometric linear mapping from V into W, as in Section 2.10. In particular, this implies that the operator norm of T is equal to 1, at least if V ≠ {0}. If V is a linear subspace of W, for instance, and ∥ · ∥_{V} is the same as the restriction of ∥ · ∥_{W} to V, then one can take T to be the obvious inclusion mapping from W into V. This means that for each v ∈ V, T (v) = v, (3.13.14) considered as an element of W.

In this case, T′ is the obvious restriction mapping from W′ into V′, which sends λ ∈ W′ to its restriction to V, as an element of V′. Note that T′(W′) = V′, (3.13.15) by the Hahn–Banach theorem. If T is any isometric linear mapping into W, then it is easy to see that T′ is surjective, for essentially the same reason.

3.14 Second dual spaces

Let V be a vector space over the real or complex numbers. Consider the algebraic dual (V^{alg})^{alg} (3.14.1)

of the algebraic dual V^{alg} of V. If v ∈ V and λ ∈ V^{alg}, then put

L̂_{v}(λ) = λ(v). (3.14.2)

This defines a linear functional on V^{alg}, and thus an element of (V^{alg})^{alg}. In fact, v ↦ L̂_{v} (3.14.3)

is a linear mapping from V into (V^{alg})^{alg}.

If v ≠ 0, then it is well known that there is a λ ∈ V^{alg} such that λ(v) ≠ 0. This means that L̂_{v} ≠ 0, so that (3.14.3) is one-to-one. If V has finite dimension, then V^{alg} has the same dimension, by standard arguments. This implies that (V^{alg})^{alg} has the same dimension in this case, using the same argument. It follows that (3.14.3) maps V onto (V^{alg})^{alg} under these conditions.

Let ∥ · ∥_{V} be a norm on V, so that the corresponding dual norm ∥ · ∥_{V′} can be defined on the dual space V′ of bounded linear functionals on V in the usual way. Thus the dual space V′′ = (V′)′ (3.14.4)

of bounded linear functionals on V′ can be defined in the usual way too, with the dual norm ∥ · ∥_{V′′} = ∥ · ∥_{(V′)′} (3.14.5)

associated to ∥ · ∥_{V′}.

If v ∈ V and λ ∈ V′, then put

L_{v}(λ) = λ(v). (3.14.6)

Equivalently, L_{v} is the same as the restriction of L̂_{v} to V′. Of course,

|L_{v}(λ)| = |λ(v)| ≤ ∥v∥_{V} ∥λ∥_{V′}. (3.14.7)

This implies that L_{v} is a bounded linear functional on V′, with

∥L_{v}∥_{V′′} ≤ ∥v∥_{V} . (3.14.8)

More precisely, ∥L_{v}∥_{V′′} = ∥v∥_{V} , (3.14.9)

because of (3.12.9) and its analogue in the real case.

Note that v ↦ L_{v} (3.14.10)

is a linear mapping from V into V′′, as before. This mapping is one-to-one, because of (3.14.9).

If (3.14.10) maps V onto V′′, then V is said to be reflexive. This can only happen when V is a Banach space, because V′′ is automatically a Banach space.

Of course, {L_{v} : v ∈ V } (3.14.11)

is a linear subspace of V′′. If V is a Banach space, then (3.14.11) is a closed set in V′′, with respect to the metric associated to ∥ · ∥_{V′′}. This follows from (3.14.9), and a remark about isometries in Section 1.15.

3.15 Second duals of linear mappings

Let V, W be vector spaces, both real or both complex, and let T be a linear mapping from V into W. Remember that T^{alg} is the dual mapping from W^{alg} into V^{alg} corresponding to T, as in Section 3.13. Using this, we get the corresponding second dual mapping

(T^{alg})^{alg} (3.15.1)

from (V^{alg})^{alg} into (W^{alg})^{alg}. Note that

T ↦ (T^{alg})^{alg} (3.15.2)

is a linear mapping from L(V, W) into L((V^{alg})^{alg}, (W^{alg})^{alg}), because of the analogous statement in Section 3.13.

If v ∈ V, then let L̂_{v}^{V} = L̂_{v} ∈ (V^{alg})^{alg} be as in (3.14.2). Similarly, if w ∈ W and μ ∈ W^{alg}, then L̂_{w}^{W} (μ) = μ(w) (3.15.3)

defines an element of (W^{alg})^{alg}. Remember that v ↦ L̂_{v}^{V} and w ↦ L̂_{w}^{W} define linear mappings from V and W into (V^{alg})^{alg} and (W^{alg})^{alg}, respectively, as in the previous section.

Let us check that (T^{alg})^{alg}(L̂_{v}^{V}) = L̂_{T (v)}^{W}. (3.15.4)

Of course, (3.15.5) (T^{alg})^{alg}(L̂_{v}^{V}) = L̂_{v}^{V} ∘ T^{alg},

by construction. If μ ∈ W^{alg}, then

(L̂_{v}^{V} ∘ T^{alg})(μ) = L̂_{v}^{V} (T^{alg}(μ)) = L̂_{v}^{V} (μ ∘ T) = μ(T (v)) = L̂_{T (v)}^{W}(μ). (3.15.6)

This means that L̂_{v}^{V} ∘ T^{alg} = L̂_{T (v)}^{W}, (3.15.7)

so that (3.15.4) holds.

Let ∥ · ∥_{V}, ∥ · ∥_{W} be norms on V, W, respectively, and suppose now that T is a bounded linear mapping from V into W. Using the dual mapping T′ from W′ into V′, we get the second dual mapping T′′ = (T′)′ from V′′ into W′′. We also have that

∥T′′∥_{op,V′′W′′} = ∥T′∥_{op,W′V′} = ∥T ∥_{op,V W} , (3.15.8)

by (3.13.13). As before, T ↦ T′′ (3.15.9)

is a linear mapping from BL(V, W) into BL(V′′, W′′), because of the analogous statement in Section 3.13.

If v ∈ V, then let L_{v}^{V} = L_{v} ∈ V′′ be as in (3.14.6). Similarly, if w ∈ W and μ ∈ W′, then L_{w}^{W} (μ) = μ(w) (3.15.10)

defines an element of W′′, which is the same as the restriction of L̂_{w}^{W} to W′. Of course, v ↦ L_{v}^{V} and w ↦ L_{w}^{W} are linear mappings from V and W into V′′ and W′′, respectively, as before. Note that

L_{v}^{V} ∘ T′ = L_{T (v)}^{W} (3.15.11)

on W′, as in (3.15.7). This implies that

T′′(L_{v}^{V}) = L_{v}^{V} ∘ T′ = L_{T (v)}^{W}, (3.15.12)

as in (3.15.4).

Uniform boundedness and related topics

4.1 The Baire category theorem

Let X be a metric space, or a topological space, and let E be a subset of X. As usual, we say that E is dense in X if every element of X is an element of E, a limit point of E, or both. Equivalently, this means that

(4.1.1) the closure \overline{E} of E in X is equal to X.

It is easy to see that E is dense in X if and only if for every nonempty open subset V of X, we have that E ∩ V ≠ ∅. (4.1.2)

Sometimes this is used as the definition of a dense set, particularly in arbitrary topological spaces.

If E is a dense set in X and U is a dense open set in X, then one can use this characterization of density to check that

(4.1.3) E ∩ U is dense in X.

If U_{1}, …, U_{n} are finitely many dense open sets in X, then one can use the previous statement to get that

(4.1.4)  ⋂_{j=1}^{n} U_{j} is a dense open set in X

too. If U_{1}, U_{2}, U_{3}, … is a sequence of dense open sets in X, then the Baire category theorem gives conditions under which

(4.1.5)  ⋂_{j=1}^{∞} U_{j} is dense in X.

78 CHAPTER 4. UNIFORM BOUNDEDNESS AND RELATED TOPICS

More precisely, this holds when X is a complete metric space, and also when X is a locally compact Hausdorff topological space.

It is easy to see that A ⊆ X has empty interior if and only if

(4.1.6) X \ A is dense in X.

Thus A is a closed set in X with empty interior if and only if

(4.1.7) X \ A is a dense open set in X.

If A_{1}, …, A_{n} are finitely many closed sets in X with empty interior, then it follows that

(4.1.8)  ⋃_{j=1}^{n} A_{j} is a closed set in X with empty interior

as well. The conclusion of the Baire category theorem is equivalent to saying that if A_{1}, A_{2}, A_{3}, … is a sequence of closed sets in X with empty interior, then

(4.1.9)  ⋃_{j=1}^{∞} A_{j} has empty interior in X.

A subset A of X is said to be nowhere dense in X if

(4.1.10) the closure \overline{A} of A in X has empty interior.

If A_{1}, …, A_{n} are finitely many nowhere dense sets in X, then

(4.1.11)  ⋃_{j=1}^{n} A_{j} is nowhere dense in X.

This uses the well-known fact that

\overline{( ⋃_{j=1}^{n} A_{j})} = ⋃_{j=1}^{n} \overline{A_{j}}. (4.1.12)

A subset E of X is said to be of first category or meager in X if E can be expressed as

E = ⋃_{j=1}^{∞} A_{j}, (4.1.13)

where A_{1}, A_{2}, A_{3}, … is a sequence of nowehere dense sets in X. Otherwise, E is said to be of second category or non-meager in X.

The conclusion of the Baire category theorem is the same as saying that if E is of first category in X, then

(4.1.14) the interior of E is empty.

This means that a subset of X with nonempty interior is of second category.

4.2 Pointwise and uniform boundedness

Let X be a nonempty set, let (Y, d_{Y}) be a nonempty metric space, and let E be a nonempty collection of mappings from X into Y. If x ∈ X, then put

E(x) = {f(x) : f ∈ E}. (4.2.1)

We say that E is pointwise bounded on a subset A of X if for each x ∈ A,

(4.2.2) E(x) is a bounded subset of Y,

with respect to d_{Y}.

Similarly, put E(A) = {f(x) : x ∈ A, f ∈ E} (4.2.3)

for each A ⊆ X. If this is a bounded set in Y, then we say that E is uniformly bounded on A. Of course, this implies that E is pointwise bounded on A.

Suppose for the moment that Y is the real line, equipped with the standard Euclidean metric. If n is a positive integer, then put

(4.2.4) E_{n} = {x ∈ X : |f(x)| ≤ n for every f ∈ E}.

Thus

(4.2.5)  ⋃_{n=1}^{∞} E_{n} = {x ∈ X : E(x) is bounded in R}.

Similarly, E is uniformly bounded on A ⊆ X if and only if A ⊆ E_{n} for some n.

Now let (X, d_{X}) be a nonempty metric space, or even a topological space. Suppose that each f ∈ E is a continuous real-valued function on X. This implies that (4.2.6) E_{n} is a closed set in X

for each n. If (4.2.5) is of second category in X, then

(4.2.7) E_{n} has nonempty interior

for some n. This means that

(4.2.8) E is uniformly bounded on a nonempty open subset of X.

Let Y be any nonempty metric space again, and let y_{0} be an element of Y. Remember that d_{Y} (y, y_{0}) is Lipschitz with constant 1 as a function of y ∈ Y, as in Section 2.1. If f is a continuous mapping from X into Y, then it follows that F (x) = d_{Y} (f(x), y_{0}) (4.2.9)

is a continuous real-valued function on X.

Let E_{0} be the collection of real-valued functions on X of the form (4.2.9), with f ∈ E. If x ∈ X, then it is easy to see that E(x) is a bounded subset of Y if and only if (4.2.10) E_{0}(x) is a bounded set in R.

Similarly, if A ⊆ X, then

(4.2.11) E(A) is bounded in Y

if and only if (4.2.12) E_{0}(A) is bounded in R.

If the set of x ∈ X such that E_{0}(x) is bounded in R is of second category in X, then

(4.2.13) E_{0} is uniformly bounded on a nonempty open subset of X,

as before. This means that if the set of x ∈ X such that E(x) is bounded in Y is of second category in X, then E is uniformly bounded on a nonempty open subset of X.

4.3 The Banach–Steinhaus theorem

Let V, W be vector spaces, both real or both complex, and with norms ∥ · ∥_{V}, ∥ · ∥_{W}, respectively. Also let E be a collection of bounded linear mappings from V into W. If v ∈ V, then put

E(v) = {T (v) : T ∈ E}, (4.3.1)

as in the previous section.

Suppose that (4.3.2) {v ∈ V : E(v) is bounded in W }

is of second category in V, using the metrics on V, W associated to the norms. Note that this holds when V is a Banach space, and (4.3.2) is equal to V. This implies that E is uniformly bounded on a nonempty open subset of V, as before.

Equivalently, this means that E is uniformly bounded on an open ball in V of positive radius. One can reduce to the case where the ball is centered at 0, using linearity of the elements of E. Similarly, one can reduce to the case where the radius of the ball is one, using linearity.

It follows that the operator norms of the elements of E are bounded, so that

∥T ∥_{op,V W} ≤ C (4.3.3)

for some nonnegative real number C and every T ∈ E. This is the Banach– Steinhaus theorem, which is also known as the uniform boundedness principle. Note that this condition implies that E is pointwise bounded on V, and uniformly bounded on bounded subsets of V.

One may also consider (4.3.3) as an equicontinuity property of E. It is the same as saying that each T ∈ E is Lipschitz with constant C as a mapping from V into W, with respect to the metrics associated to their norms.

4.4 Pointwise convergence

Let V, W be vector spaces, both real or both complex, with norms ∥·∥_{V}, ∥·∥_{W}, respectively. Also let {T_{j}}_{j=1}^{∞} be a sequence of bounded linear mappings from V into W. Consider

(4.4.1) {v ∈ V : {T_{j}(v)}_{j=1}^{∞} is a bounded sequence in W }.

It is easy to see that this is a linear subspace of V. If (4.4.1) is of second category in V, with respect to the metric associated to ∥ · ∥_{V}, then it follows that {T_{j}}_{j=1}^{∞} is bounded with respect to the operator norm, by the Banach– Steinhaus theorem.

Similarly, consider

(4.4.2) {v ∈ V : {T_{j}(v)}_{j=1}^{∞} is a Cauchy sequence in W }.

If W is a Banach space, then this is the same as

(4.4.3) {v ∈ V : {T_{j}(v)}_{j=1}^{∞} converges in W }.

It is well known and easy to see that Cauchy sequences in metric spaces are bounded, so that (4.4.2) is contained in (4.4.1). Convergent sequences in metric spaces are Cauchy sequences, so that (4.4.3) is contained in (4.4.2). Note that (4.4.2) and (4.4.3) are linear subspaces of V.

If v ∈ V is an element of (4.4.3), then put

T (v) = lim_{j→∞} T_{j}(v). (4.4.4)

This defines a linear mapping from (4.4.3) into W.

Suppose that {T_{j}}_{j=1}^{∞} is bounded with respect to the operator norm, so that

∥T_{j}∥_{op,V W} ≤ C (4.4.5)

for some nonnegative real number C and each j ≥ 1. In this case, one can check that (4.4.2) is a closed set in V, with respect to the metric associated to ∥ · ∥_{V}. In particular, if (4.4.2) is dense in V, then it follows that (4.4.2) is equal to V.

Suppose for the moment that T is any linear mapping from V into W. Consider (4.4.6) {v ∈ V : {T_{j}(v)}_{j=1}^{∞} converges to T (v) in W }.

This is another linear subspace of V. If {T_{j}}_{j=1}^{∞} is bounded with respect to the operator norm, and T is a bounded linear mapping from V into W, then one can verify that (4.4.6) is a closed set in V, with respect to the metric associated to ∥ · ∥_{V}.

Suppose again that {T_{j}}_{j=1}^{∞} is bounded with respect to the operator norm, so that (4.4.5) holds for some C ≥ 0 and each j. Equivalently, this means that

∥T_{j}(v)∥_{W} ≤ C ∥v∥_{V} (4.4.7) for each v ∈ V and j ≥ 1. Suppose also that (4.4.3) is equal to V, so that (4.4.4) defines a linear mapping from V into W. Under these conditions, we have that

∥T (v)∥_{W} ≤ C ∥v∥_{V} (4.4.8)

for every v ∈ V. This is the same as saying that T is bounded as a linear mapping from V into W, with ∥T ∥_{op,V W} ≤ C.

Of course, if {T_{j}}_{j=1}^{∞} converges to a bounded linear mapping T from V into W with respect to the metric associated to the operator norm, then {T_{j}}_{j=1}^{∞} is bounded with respect to the operator norm, and {T_{j}}_{j=1}^{∞} converges to T pointwise on V. Pointwise convergence of {T_{j}}_{j=1}^{∞} to a bounded linear mapping T from V into W is equivalent to the convergence of {T_{j}}_{j=1}^{∞} to T with respect to the strong operator topology on the space BL(V, W) of bounded linear mappings from V into W. We shall not discuss this topology in detail for the moment.

4.5 A sequential compactness theorem

Let V be a vector space over the real or complex numbers with a norm ∥·∥_{V}, and let V′ be the dual of V with respect to ∥ · ∥_{V}, as usual. Pointwise convergence of a sequence of bounded linear functionals {λ_{j}}_{j=1}^{∞} on V to another bounded linear functional λ on V is equivalent to the convergence of {λ_{j}}_{j=1}^{∞} to λ with respect to the weak^{∗} topology on V′. This may be described as the weakest topology on V′ such that λ ↦ λ(v) (4.5.1)

is continuous for every v ∈ V. This corresponds to taking W = R or C, as appropriate, in the previous section, using the standard absolute value function as the norm.

This terminology is related to the common use of V^{∗} for the dual of V with respect to ∥ · ∥_{V}, instead of V′. The notation V′ is being used here to be consistent with the notation T′ for the dual of a bounded linear mapping T, as compared to the notation T^{∗} for the adjoint of a bounded linear mapping T between Hilbert spaces.

Let {λ_{j}}_{j=1}^{∞} be a bounded sequence in V′ with respect to the dual norm ∥ · ∥_{V′}. If v ∈ V, then {λ_{j}(v)}_{j=1}^{∞} is a bounded sequence of real or complex numbers, as appropriate, and there is a subsequence of {λ_{j}(v)}_{j=1}^{∞} that converges, by a well-known result. Let E be a subset of V with only finitely or countably many elements. Under these conditions, there is a subsequence {λ_{j_{l}}}_{l=1}^{∞} of {λ_{j}}_{j=1}^{∞} such that

(4.5.2) {λ_{j_{l}}(v)}_{l=1}^{∞} converges in R or C,

as appropriate, for every v ∈ E. This can be obtained from the previous statement and standard arguments.

It follows that (4.5.2) holds for every element v of the linear span of E in V, by linearity. In fact, (4.5.2) holds for every element v of the closure of the linear span of E in V, with respect to the metric associated to ∥ · ∥_{V}, as in the previous section. Suppose now that the linear span of E is dense in V, so that (4.5.2) holds for every v ∈ V. This means that

(4.5.3) {λ_{j_{l}}}_{l=1}^{∞} converges pointwise to a bounded linear functional on V,

as before. Remember that there is a subset E of V with these properties exactly when V is separable with respect to the metric associated to ∥·∥_{V}, as in Section 1.14.

The Banach–Alaoglu theorem states that

(4.5.4) the closed unit ball in V′ is compact with respect to the weak^{∗} topology.

The remarks in the previous paragraphs correspond to a variant of this, namely, the sequential compactness of the closed unit ball in V′ with respect to the weak^{∗} topology when V is separable. Of course, the compactness or sequential compactness of the closed unit ball in V′ with respect to the weak^{∗} topology implies the analogous property of any closed ball in V′. If V is separable, then it is not too difficult to show that the topology induced on the closed unit ball in V′ by the weak^{∗} topology is determined by a metric. In this case, compactness and sequential compactness of the closed unit ball in V′ with respect to the weak^{∗} topology are the same.

4.6 A criterion for density

Let W be a vector space over the real or complex numbers with a norm ∥ · ∥_{W}, and let W_{0} be a linear subspace of W. Suppose that there is a real number a, 0 ≤ a < 1, such that for every w ∈ W there is a w_{0} ∈ W_{0} with

∥w − w_{0}∥_{W} ≤ a∥w∥_{W} . (4.6.1)

It is not too difficult to show that W_{0} is dense in W with respect to the metric associated to ∥ · ∥_{W}. More precisely, one can use the same condition with w replaced by w − w_{0} to get a w_{1} ∈ W_{0} such that

∥w − w_{0} − w_{1}∥_{W} ≤ a ∥w − w_{0}∥_{W} ≤ a^{2} ∥w∥_{W} . (4.6.2)

Continuing in this way, we get that w is in the closure of W_{0} in W.

Let 0 ≤ a < 1 be given, and let E_{a} be a subset of W such that for every u ∈ W with ∥u∥_{W} = 1 there is a u_{0} ∈ E_{a} with

∥u − u_{0}∥_{W} ≤ a. (4.6.3)

Let W_{0} be the linear span of E_{a} in W, which may be interpreted as being {0} when W = {0}. It is easy to see that W_{0} satisfies the condition mentioned in the preceding paragraph. This implies that W_{0} is dense in W, as before.

If W_{0} is a finite-dimensional linear subspace of W, then W_{0} is complete with respect to the metric associated to the restriction of ∥ · ∥_{W} to W_{0}, as in Section 1.11. This implies that W_{0} is a closed set in W, with respect to the metric associated to ∥ · ∥_{W}, as in Section 1.6. If W_{0} is the linear span of a subset E_{a} of W as in the preceding paragraph, and if E_{a} has only finitely many elements, then W_{0} has finite dimension, and it follows that W_{0} = W. In particular, this means that (4.6.4) W has finite dimension

under these conditions.

A subset A of a metric space X is said to be totally bounded if for every r > 0, A can be covered by finitely many balls of radius r. It is easy to see that compact subsets of X are totally bounded. If the closed unit sphere in W is totally bounded with respect to the metric associated to ∥ · ∥_{W}, then (4.6.4) holds, as before.

4.7 Pointwise convergence in ℓ^{q} spaces

Let X be a nonempty set, and let q be a positive extended real number. Suppose that {g_{j}}_{j=1}^{∞} is a bounded sequence in ℓ^{q}(X, R) or ℓ^{q}(X, C), so that

∥g_{j}∥_{q} ≤ C (4.7.1)

for some nonnegative real number C and each j ≥ 1. Suppose also that {g_{j}}_{j=1}^{∞} converges pointwise to a real or complex-valued function g on X, as appropriate. Under these conditions, we have that g ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C), as appropriate, with ∥g∥_{q} ≤ C. (4.7.2)

More precisely, if q = +∞, then (4.7.1) says that

|g_{j}(x)| ≤ C (4.7.3)

for each x ∈ X and j ≥ 1. This implies that

|g(x)| ≤ C (4.7.4)

for every x ∈ X, so that (4.7.2) holds. If q < ∞, then (4.7.1) is the same as saying that

∑_{x∈X} |g_{j}(x)|^{q} ≤ C^{q} (4.7.5)

for each j. It follows that

∑_{x∈X} |g(x)|^{q} ≤ C^{q}, (4.7.6)

as in Section 2.4.

Let {g_{j}}_{j=1}^{∞} be a bounded sequence in ℓ^{q}(X, R) or ℓ^{q}(X, C) again. If x ∈ X, then {g_{j}(x)}_{j=1}^{∞} is a bounded sequence of real or complex numbers. If A ⊆ X 4.8. WEAK^{∗} CONVERGENCE IN ℓ^{Q} SPACES 85

has only finitely or countably many elements, then it follows that there is a subsequence {g_{j_{l}}}_{l=1}^{∞} of {g_{j}}_{j=1}^{∞} such that

(4.7.7) {g_{j_{l}}(x)}_{l=1}^{∞} converges in R or C,

as appropriate, for every x ∈ A. This uses the same type of argument as for (4.5.2). Of course, if X has only finitely or countably many elements, then one can simply take A = X.

If q < ∞, then the support of g_{j} has only finitely or countably many elements for each j. This implies that the union of the supports of the g_{j}’s has only finitely or countably many elements as well. If we take

A = ⋃_{j=1}^{∞} supp g_{j}, (4.7.8)

then we get a subsequence {g_{j_{l}}}_{l=1}^{∞} of {g_{j}}_{j=1}^{∞} such that (4.7.7) holds for every x ∈ A, as before. In this case, it follows that (4.7.7) holds for every x ∈ X.

The space of all real or complex-valued functions on X is the same as the Cartesian product of a family of copies of R or C, as appropriate, indexed by X. Let us consider the corresponding product topology on this space, using the standard topology on each factor of R or C. Closed balls in ℓ^{∞}(X, R) or ℓ^{∞}(X, C) are compact sets with respect to the product topology, by Tychonoff’s theorem. If q < ∞, then one can check that closed balls in ℓ^{q}(X, R) or ℓ^{q}(X, C) are closed sets with respect to the product topology. This implies that they are compact sets with respect to the product topology as well, because closed sets contained in compact sets are compact.

4.8 Weak^{∗} convergence in ℓ^{q} spaces

Let X be a nonempty set, and let 1 ≤ p, q ≤ ∞ be conjugate exponents, so that 1/p + 1/q = 1. Also let {g_{j}}_{j=1}^{∞} be a bounded sequence in ℓ^{q}(X, R) or ℓ^{q}(X, C), so that (4.7.1) holds for some C ≥ 0 and all j ≥ 1. Note that

λ_{g_{j}}(f) = ∑_{x∈X} f(x) g_{j}(x) (4.8.1)

defines a bounded linear functional on ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate, for each j, with dual norm equal to ∥g_{j}∥_{q}, as in Sections 3.2, 3.3, and 3.4.

Suppose that {g_{j}}_{j=1}^{∞} converges pointwise to a real or complex-valued function g on X, as appropriate. Thus g ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C), as appropriate, as in the previous section, so that

λ_{g}(f) = ∑_{x∈X} f(x) g(x) (4.8.2)

defines a bounded linear functional on ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate. If f has finite support in X, then it is easy to see that

lim_{j→∞} λ_{g_{j}}(f) = λ_{g}(f). (4.8.3)

The set of f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate, such that (4.8.3) holds is a closed set with respect to the metric associated to the ℓ^{p} norm, as in Section 4.4. If q > 1, so that p < ∞, then it follows that (4.8.3) holds for all f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate. This uses the fact that c_{00}(X, R), c_{00}(X, C) are dense in ℓ^{p}(X, R), ℓ^{p}(X, C), respectively, when p < ∞, as in Section 2.6.

If q = 1, then (4.8.1) and (4.8.2) may also be considered as bounded linear functionals on c_{0}(X, R) or c_{0}(X, C), as appropriate, with respect to the supremum norm, as in Section 3.3. In this case, (4.8.3) holds for all real or complex-valued functions f on X, as appropriate, that vanish at infinity. This is because c_{00}(X, R), c_{00}(X, C) are dense in c_{0}(X, R), c_{0}(X, C), respectively, with respect to the supremum metric, as in Section 1.13.

4.9 Weak convergence

Let V be a vector space over the real or complex numbers with a norm ∥ · ∥_{V}, and let V′ be the corresponding dual space. A sequence {v_{j}}_{j=1}^{∞} of elements of V is said to converge weakly to v ∈ V if

lim_{j→∞} λ(v_{j}) = λ(v) (4.9.1)

for every λ ∈ V′. This is equivalent to the convergence of {v_{j}}_{j=1}^{∞} to v with respect to the weak topology on V. This may be described as the weakest topology on V with respect to which every λ ∈ V′ is continuous.

If {v_{j}}_{j=1}^{∞} converges to v with respect to the metric associated to the norm, then {v_{j}}_{j=1}^{∞} converges weakly to v. Similarly, the topology determined on V by the metric associated to the norm is at least as strong as the weak topology on V, because every λ ∈ V′ is continuous on V with respect to the metric associated to the norm.

Remember that V′ separates points in V, as in Section 3.10. If a sequence in V has a weak limit, then one can use this to get that the limit is unique. Similarly, one can use this to get that the weak topology on V is Hausdorff.

Let V′′ = (V′)′ be the dual of V′ with respect to the dual norm ∥ · ∥_{V′}, as in Section 3.14. If v ∈ V, then L_{v}(λ) = λ(v) defines an element of V′′, as before. Using this notation, (4.9.1) is the same as saying that

lim_{j→∞} L_{v_{j}}(λ) = L_{v}(λ) (4.9.2)

for every λ ∈ V′. This means that {v_{j}}_{j=1}^{∞} converges to v weakly in V if and only if {L_{v_{j}}}_{j=1}^{∞} converges to L_{v} in the weak^{∗} sense in V′′, as the dual of V′.

Let E be a subset of V, and put

E_{E} = {L_{v} : v ∈ E}, (4.9.3)

which is a subset of V′′. If λ ∈ V′, then put

E_{E}(λ) = {L_{v}(λ) : v ∈ E}, (4.9.4) as in Section 4.3. Equivalently, E_{E}(λ) = λ(E).

Let us say that E is weakly bounded in V if λ(E) is a bounded set in R or C, as appropriate, for each λ ∈ V′. This is the same as saying that E_{E} is pointwise bounded on V′. Remember that V′ is complete with respect to the metric associated to ∥ · ∥_{V′}, as in Sections 2.2 and 3.1. Let ∥ · ∥_{V′′} be the dual norm on V′′ associated to ∥ · ∥_{V′} on V′, as in Section 3.14. If E_{E} is pointwise bounded on V′, then it follows that E_{E} is bounded with respect to ∥ · ∥_{V′′}, because of the Baire category theorem and the Banach–Steinhaus theorem.

Remember that ∥L_{v}∥_{V′′} = ∥v∥_{V} for every v ∈ V, as in Section 3.14. If E is weakly bounded in V, then we get that E is bounded with respect to ∥ · ∥_{V}, using the remarks in the preceding paragraph.

If {v_{j}}_{j=1}^{∞} is a sequence of elements of V that converges weakly in V, then it is easy to see that the set of v_{j}’s, j ≥ 1, is weakly bounded in V. This implies that {v_{j}}_{j=1}^{∞} is bounded with respect to ∥ · ∥_{V}, as before.

Suppose that ∥v_{j}∥_{V} ≤ C (4.9.5)

for some nonnegative real number C and each j ≥ 1. This implies that

|λ(v_{j})| ≤ C ∥λ∥_{V′} (4.9.6)

for each λ ∈ V′ and j ≥ 1. If {v_{j}}_{j=1}^{∞} converges weakly to v ∈ V, then it follows that |λ(v)| ≤ C ∥λ∥_{V′} (4.9.7)

for every λ ∈ V′. Using this, we get that

∥v∥_{V} ≤ C, (4.9.8)

as in Sections 3.10 and 3.12.

4.10 Some multiplication operators

Let X be a nonempty set, and let p, q, and r be positive extended real numbers. Suppose that 1/p + 1/q = 1/r, (4.10.1)

with suitable interpretations when any of p, q, or r is +∞. Let f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C) and g ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C) be given. Under these conditions, f g ∈ ℓ^{r}(X, R) or ℓ^{r}(X, C), as appropriate, with

∥f g∥_{r} ≤ ∥f∥_{p} ∥g∥_{q}. (4.10.2)

This can be verified directly when p or q is +∞.

Suppose that p, q < +∞, which implies that r < +∞. In this case, p, q > r, so that p/r, q/r > 1, and (4.10.1) is the same as saying that

r/p + r/q = 1. (4.10.3)

Observe that |f|^{r} is (p/r)-summable on X, and that |g|^{r} is (q/r)-summable on X. Hölder’s inequality implies that |f|^{r} |g|^{r} is summable on X, with

∑_{x∈X} |f(x)|^{r} |g(x)|^{r} ≤ ∥(|f|^{r})∥_{p/r} ∥(|g|^{r})∥_{q/r}. (4.10.4)

This is the same as saying that |f| |g| is r-summable on X, and that (4.10.2) holds.

Let a be a real or complex-valued function on X. If f is another real or complex-valued function on X, as appropriate, then

M_{a}(f) = a f (4.10.5)

defines real or complex-valued function on X too. This defines a linear mapping from the space of real or complex-valued functions on X, as appropriate, into itself, which is the multiplication operator associated to a. Note that if f has finite support in X, then M_{a}(f) has finite support as well. Thus the restriction of M_{a} to c_{00}(X, R) or c_{00}(X, C), as appropriate, defines a linear mapping from that space into itself.

Suppose for the moment that a ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C) for some q > 0. If p, r > 0 satisfy (4.10.1), then

(4.10.6) M_{a} maps ℓ^{p}(X, R) or ℓ^{p}(X, C) into ℓ^{r}(X, R) or ℓ^{r}(X, C),

as appropriate, as before. We also have that

∥M_{a}(f)∥_{r} ≤ ∥a∥_{q} ∥f∥_{p} (4.10.7)

for every f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate, by (4.10.2). If q = +∞, then it is easy to see that

(4.10.8) M_{a} maps c_{0}(X, R) or c_{0}(X, C) into itself

too, as appropriate.

Suppose now that a is a real or complex-valued function on X such that (4.10.6) holds, with ∥M_{a}(f)∥_{r} ≤ C ∥f∥_{p} (4.10.9)

for some C ≥ 0 and all f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate. More precisely, if this holds for all real or complex-valued functions f on X with finite support, as appropriate, then it is easy to see that this holds for all f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate. We would like to check that a ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C), as appropriate, with ∥a∥_{q} ≤ C. (4.10.10)

If p = +∞, then q = r, and this can be verified directly, by taking f ≡ 1 on X. One can also consider functions f that are equal to 1 on a finite subset of X and to 0 elsewhere, so that f has finite support in X. If q = +∞, then p = r, and this can be verified directly again, by considering functions f equal to 1 at one point in X, and to 0 elsewhere.

Suppose that p, q < +∞, so that r < +∞ and p, q > r, as before. Let A be a nonempty finite subset of X, and let f_{A} be a real or complex-valued function on X, as appropriate, such that

|f_{A}(x)| = |a(x)|^{(q−r)/r} (4.10.11)

for each x ∈ A, and f_{A}(x) = 0 when x ∈ X \ A. Thus

|f_{A}(x)|^{r} |a(x)|^{r} = |a(x)|^{q} (4.10.12)

for every x ∈ A. This implies that

∥M_{a}(f_{A})∥_{r} = ∥a f_{A}∥_{r} = ( ∑_{x∈A} |a(x)|^{q})^{1/r}. (4.10.13)

Observe that r/p = 1 − r/q = (q − r)/q, (4.10.14)

so that p/r = q/(q − r). It follows that

|f_{A}(x)|^{p} = |a(x)|^{((q−r) p)/r} = |a(x)|^{q} (4.10.15)

for every x ∈ A. This means that

∥f_{A}∥_{p} = ( ∑_{x∈A} |a(x)|^{q})^{1/p}. (4.10.16)

Combining this with (4.10.13), we get that

( ∑_{x∈A} |a(x)|^{q})^{1/q} ≤ C. (4.10.17)

This implies that a ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C), as appropriate, and that (4.10.10) holds.

If r ≥ 1, then p, q ≥ 1, and it follows that M_{a} defines a bounded linear mapping from ℓ^{p}(X, R) or ℓ^{p}(X, C) into ℓ^{r}(X, R) or ℓ^{r}(X, C), as appropriate, with operator norm equal to ∥a∥_{q}. There is an analogous statement when r < 1, using suitable versions of some of our previous definitions, and related facts.

4.11 Convergence of multiplication operators

Let X be a nonempty set, and remember that ℓ^{p}(X, R), ℓ^{p}(X, C) may be considered as metric spaces when 0 < p ≤ +∞. This uses the metric associated to the ℓ^{p} norm when p ≥ 1, and the analogous metric mentioned in Section 2.6 when p < 1.

Let {a_{j}}_{j=1}^{∞} be a sequence of real or complex-valued functions on X that converges pointwise to a real or complex-valued function a on X, as appropriate. If f is any real or complex-valued function on X, as appropriate, then

(4.11.1) {M_{a_{j}}(f)}_{j=1}^{∞} converges to M_{a}(f) pointwise on X.

Let p, q, r be positive extended real numbers satisfying (4.10.1) again. Suppose that a_{j} ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C) for each j, as appropriate, with

∥a_{j}∥_{q} ≤ C (4.11.2)

for some C ≥ 0 and each j. This implies that a ∈ ℓ^{q}(X, R) or ℓ^{q}(X, C), as appropriate, with ∥a∥_{q} ≤ C, (4.11.3)

as in Section 4.7. If f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate, then we would like to have conditions under which

(4.11.4) {M_{a_{j}}(f)}_{j=1}^{∞} converges to M_{a}(f) in ℓ^{r}(X, R) or ℓ^{r}(X, C),

as appropriate. It is easy to see that this holds when

(4.11.5) {a_{j}}_{j=1}^{∞} converges to a in ℓ^{q}(X, R) or ℓ^{q}(X, C),

as appropriate, because of (4.10.2).

Of course, the set of f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C) such that (4.11.4) holds is a linear subspace of ℓ^{p}(X, R) or ℓ^{p}(X, R), as appropriate. This is also a closed set in ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate, because of (4.11.2) and (4.11.3), as in Section 4.4.

Note that (4.11.4) holds when f has finite support in X. If p < +∞, then it follows that (4.11.4) holds for every f ∈ ℓ^{p}(X, R) or ℓ^{p}(X, C), as appropriate. This uses the fact that c_{00}(X, R), c_{00}(X, C) are dense in ℓ^{p}(X, R), ℓ^{p}(X, C), respectively, when p < +∞, as in Section 2.6.

Suppose that p = +∞, so that q = r. If f is equal to 1 at every point in X, then (4.11.4) implies (4.11.5).

If f vanishes at infinity on X, then (4.11.4) holds, with q = r. This is because c_{00}(X, R), c_{00}(X, C) are dense in c_{0}(X, R), c_{0}(X, C), respectively, with respect to the supremum metric, as in Section 1.13.

4.12 L^{p} Spaces

Let (X, A, μ) be a (nonempty) measure space, so that X is a nonempty set, A is a σ-algebra of measurable subsets of X, and μ is a countably additive (nonnegative) measure on the measurable space (X, A). If p is a positive real number, then let L^{p}(X, R) and L^{p}(X, C) be the corresponding spaces of real and complex-valued measurable functions f on X, as appropriate, such that |f|^{p} is integrable with respect to μ. More precisely, these spaces consist of equivalence 4.12. L^{P} SPACES 91

classes of functions that are equal almost everywhere on X with respect to μ. If f ∈ L^{p}(X, R) or L^{p}(X, C), then we put

∥f∥_{p} = ( ∫_{X} |f(x)|^{p} dμ(x))^{1/p}, (4.12.1)

as usual.

Similarly, let L^{∞}(X, R) and L^{∞}(X, C) be the spaces of equivalence classes of real and complex-valued measurable functions f on X that are essentially bounded on X with respect to μ. In this case, ∥f∥_{∞} is the essential supremum norm of f with respect to μ.

It is well known that L^{p}(X, R) and L^{p}(X, C) are vector spaces over the real and complex numbers, respectively, for each p > 0. If p ≥ 1, then ∥f∥_{p} defines a norm on each of L^{p}(X, R) and L^{p}(X, C). In this case, the triangle inequality for ∥ · ∥_{p} is known as Minkowski’s inequality for integrals.

If 0 < p ≤ 1, then ∥f∥_{p} satisfies the usual homogeneity property of a norm, and ∥f + g∥_{p}^{p} ≤ ∥f∥_{p}^{p} + ∥g∥_{p}^{p} (4.12.2)

for all f, g ∈ L^{p}(X, R) or L^{p}(X, C). This means that

∥f − g∥_{p}^{p} (4.12.3)

defines a metric on each of L^{p}(X, R) and L^{p}(X, C) when p ≤ 1.

It is well known that L^{p}(X, R) and L^{p}(X, C) are complete with respect to the appropriate metric for each p > 0. In particular, L^{p}(X, R) and L^{p}(X, C) are Banach spaces when p ≥ 1.

Of course, ℓ^{p}(X, R) and ℓ^{p}(X, C) are the same as L^{p}(X, R) and L^{p}(X, C), respectively, when μ is counting measure on X. In this case, one can take all subsets of X to be measurable.

If f, g ∈ L^{2}(X, R) or L^{2}(X, C), then it is well known that f g ∈ L^{1}(X, R) or L^{1}(X, C), as appropriate, with

∥f g∥_{1} ≤ ∥f∥_{2} ∥g∥_{2}. (4.12.4)

This is the integral version of the Cauchy–Schwarz inequality. Put

⟨f, g⟩ = ∫_{X} f(x) g(x) dμ(x) (4.12.5)

in the real case, and

⟨f, g⟩ = ∫_{X} f(x) \overline{g(x)} dμ(x) (4.12.6)

in the complex case. These define inner products on L^{2}(X, R) and L^{2}(X, C), respectively, for which the corresponding norm is ∥ · ∥_{2}. Thus L^{2}(X, R) and L^{2}(X, C) are Hilbert spaces over the real and complex numbers, respectively.

Suppose that 1 ≤ p, q ≤ ∞ are conjugate exponents, so that 1/p+1/q = 1. If f ∈ L^{p}(X, R) or L^{p}(X, C) and g ∈ L^{q}(X, R) or L^{q}(X, C), then f g ∈ L^{1}(X, R) or L^{1}(X, C), as appropriate, with

∥f g∥_{1} ≤ ∥f∥_{p} ∥g∥_{q}. (4.12.7)

This is the integral version of Hölder’s inequality.

If g ∈ L^{q}(X, R) or L^{q}(X, C), then put

λ_{g}(f) = ∫_{X} f(x) g(x) dμ(x) (4.12.8)

for all f ∈ L^{p}(X, R) or L^{p}(X, C), as appropriate. Of course,

|λ_{g}(f)| ≤ ∥f∥_{p} ∥g∥_{q} (4.12.9)

for all f ∈ L^{p}(X, R) or L^{p}(X, C), as appropriate, by Hölder’s inequality. Thus λ_{g} is a bounded linear functional on L^{p}(X, R) or L^{p}(X, C), as appropriate, with dual norm less than or equal to ∥g∥_{q}. Note that

g ↦ λ_{g} (4.12.10)

defines a linear mapping from each of L^{q}(X, R), L^{q}(X, C) into the duals of L^{p}(X, R), L^{p}(X, C), respectively.

If g ∈ L^{q}(X, R) or L^{q}(X, C) and 1 ≤ q < ∞, then one can check directly that

(4.12.11) the dual norm of λ_{g} with respect to the L^{p} norm is equal to ∥g∥_{q}.

This also works when q = ∞, under the additional condition that

(4.12.12) every measurable subset of X of positive measure contain a measurable set of positive finite measure.

Note that this condition holds when X is σ-finite with respect to μ. This condition also holds when μ is counting measure on X, and all subsets of X are measurable.

More precisely, suppose that g ∈ L^{∞}(X, R) or L^{∞}(X, C), with ∥g∥_{∞} > 0, and let r be a nonnegative real number such that ∥g∥_{∞} > r. This implies that the set where |g| > r has positive measure with respect to μ, by the definition of the essential supremum norm of g. If (4.12.12) holds, then there is a measurable subset of X with positive finite measure on which |g| > r. One can use this to get that (4.12.11) holds with q = ∞.

If 1 < p < ∞, then it is well known that every bounded linear functional on L^{p}(X, R) or L^{p}(X, C) is of this form. This follows from the analogous statement for Hilbert spaces when p = 2. If p = 1, then the analogous statement holds when X is σ-finite with respect to μ.

If μ(X) is finite and 0 < r < p ≤ ∞, then it is well known that L^{p}(X) is contained in L^{r}(X). A more precise version of this will be mentioned in Section 4.15.

4.13 A measure-theoretic lemma

Let (X, A, μ) be a measure space, and let E be a measurable subset of X. Suppose that every measurable subset of E of positive measure contains a measurable set of positive finite measure. We would like to check that

(4.13.1) μ(E) = sup{μ(A) : A ⊆ E, A measurable, μ(A) < ∞}.

Of course, the supremum on the right is automatically less than or equal to μ(E). If μ(E) < ∞, then (4.13.1) holds trivially.

Let {A_{j}}_{j=1}^{∞} be a sequence of measurable subsets of E with finite measure such that {μ(A_{j})}_{j=1}^{∞} converges to the supremum on the right side of (4.13.1). Put

B_{l} = ⋃_{j=1}^{l} A_{j} (4.13.2)

for each l ≥ 1, which is a measurable subset of E with finite measure. Note that μ(B_{l}) is less than or equal to the right side of (4.13.1) for each l. We also have that μ(A_{l}) ≤ μ(B_{l}) (4.13.3)

for each l, by construction. It follows that {μ(B_{l})}_{l=1}^{∞} converges to the supremum on the right side of (4.13.1) as well.

Put

B = ⋃_{l=1}^{∞} B_{l}, (4.13.4)

which is a measurable subset of E. Of course, B_{l} ⊆ B_{l+1} for each l, by construction. Thus μ(B_{l}) → μ(B) as l → ∞, (4.13.5)

by a standard argument. This means that μ(B) is equal to the right side of (4.13.1).

If the supremum on the right side of (4.13.1) is +∞, then equality holds trivially. Thus we may suppose that the supremum is finite, which means that μ(B) < +∞. In this case, we would like to check that

μ(E \ B) = 0. (4.13.6)

This will imply that μ(E) = μ(B), (4.13.7)

so that (4.13.1) holds.

Suppose for the same of a contradiction that

μ(E \ B) > 0. (4.13.8)

This implies that there is a measurable set C ⊆ E \B such that μ(C) is positive and finite, by hypothesis. It follows that B ∪C is a measurable subset of E such that μ(B ∪ C) = μ(B) + μ(C) > μ(B). (4.13.9)

This contradicts the fact that μ(B) is equal to the right side of (4.13.1), because μ(B ∪ C) < ∞.

4.14 Some more multiplication operators

Let (X, A, μ) be a nonempty measure space, and let p, q, and r be positive extended real numbers such that 1/p + 1/q = 1/r. If f ∈ L^{p}(X, R) or L^{p}(X, C) and g ∈ L^{q}(X, R) or L^{q}(X, C), then f g ∈ L^{r}(X, R) or L^{r}(X, C), as appropriate, with ∥f g∥_{r} ≤ ∥f∥_{p} ∥g∥_{q}. (4.14.1)

This can be obtained from the integral version of Hölder’s inequality, as in Section 4.10.

Let a be a real or complex-valued measurable function on X. If f is another real or complex-valued measurable function on X, as appropriate, then

M_{a}(f) = a f (4.14.2)

defines a real or complex-valued measurable function on X as well. This defines a linear mapping from the space of real or complex-valued measurable functions on X, as appropriate, into itself, which is the multiplication operator associated to a. We shall use the same notation for the induced linear mapping on the corresponding space of equivalence classes of functions that are equal almost everywhere on X with respect to μ. Note that this induced linear mapping only depends on the analogous equivalence class that contains a.

Suppose for the moment then a ∈ L^{q}(X, R) or L^{q}(X, C). In this case,

(4.14.3) M_{a} maps L^{p}(X, R) or L^{p}(X, C) into L^{r}(X, R) or L^{r}(X, C),

as appropriate, with ∥M_{a}(f)∥_{r} ≤ ∥a∥_{q} ∥f∥_{p} (4.14.4)

for every f ∈ L^{p}(X, R) or L^{p}(X, C), as appropriate, by (4.14.1).

Suppose now that a is a real or complex-valued measurable function on X such that (4.14.3) holds, with

∥M_{a}(f)∥_{r} ≤ C ∥f∥_{p} (4.14.5)

for some C ≥ 0 and all f ∈ L^{p}(X, R) or L^{p}(X, C), as appropriate. Under suitable conditions, we would like to show that a ∈ L^{q}(X, R) or L^{q}(X, C), as appropriate, with ∥a∥_{q} ≤ C. (4.14.6)

If p = +∞, then q = r, and this follows by taking f ≡ 1 on X.

Suppose from now on in this section that p < +∞, and that every measurable subset of {x ∈ X : a(x) ≠ 0} (4.14.7) of positive measure contains a measurable set of positive finite measure. If q = +∞, then p = r, and one can check directly that a is essentially bounded on X, with L^{∞} norm less than or equal to C, as in (4.14.6).

Suppose now that q < +∞ too, so that p, q > r. Let ϵ > 0 be given, and let us show that μ({x ∈ X : |a(x)| ≥ ϵ}) < ∞. (4.14.8)

Note that every measurable subset of

{x ∈ X : |a(x)| ≥ ϵ} (4.14.9)

of positive measure contains a measurable set of positive finite measure, by hypothesis.

If A ⊆ X, then let 1_{A} be the corresponding indicator function on X, which is equal to 1 on A and to 0 on X \ A. Suppose that A is measurable, so that 1_{A} is measurable on X. If μ(A) < ∞, then 1_{A} ∈ L^{p}(X, R), and we get that M_{a}(1_{A}) = a 1_{A} ∈ L^{r}(X, R) or L^{r}(X, C), as appropriate. We also get that

∫_{A} |a(x)|^{r} dμ(x) ≤ C^{r} ∥1_{A}∥_{p}^{r} = C^{r} μ(A)^{r/p}, (4.14.10)

by (4.14.5).

If A is contained in (4.14.9), then it follows that

ϵ^{r} μ(A) ≤ C^{r} μ(A)^{r/p}. (4.14.11)

This implies that μ(A)^{(p−r)/p} ≤ (C/ϵ)^{r}, (4.14.12)

so that μ(A) ≤ (C/ϵ)^{p r/(p−r)}. (4.14.13)

This means that

μ({x ∈ X : |a(x)| ≥ ϵ}) ≤ (C/ϵ)^{pr/(p−r)}, (4.14.14)

as in the previous section. Note that (p − r)/(pr) = 1/r − 1/p = 1/q, so that pr/(p − r) = q.

Let b be a real or complex-valued measurable function on X such that

|b| ≤ |a| (4.14.15)

almost everywhere on X with respect to μ. This implies that b satisfies the analogues of (4.14.3) and (4.14.5) in place of a. If b ∈ L^{q}(X, R) or L^{q}(X, C), as appropriate, then one can verify that

∥b∥_{q} ≤ C. (4.14.16)

This can be obtained by considering f such that

|f(x)| = |b(x)|^{(q−r)/r} (4.14.17) on X, so that |f(x)|^{r} |b(x)|^{r} = |b(x)|^{q} (4.14.18)

and |f(x)|^{p} = |b(x)|^{q} (4.14.19)

on X, as in Section 4.10.

Let us check that |a|^{q} is integrable on (4.14.9) for each ϵ > 0, with

( ∫_{{x∈X:|a(x)|≥ϵ}} |a(x)|^{q} dμ(x))^{1/q} ≤ C. (4.14.20)

If b is a bounded real or complex-valued measurable function on X that is equal to 0 when |a| < ϵ, then b ∈ L^{q}(X, R) or L^{q}(X, C), as appropriate, because of (4.14.8). If b also satisfies (4.14.15), then (4.14.16) holds, as before. One can approximate a on (4.14.9) by such functions b, to get that |a|^{q} is integrable on (4.14.9), and that (4.14.20) holds.

Using this, it is easy to see that |a|^{q} is integrable on X, and that (4.14.6) holds.

If r ≥ 1, then p, q ≥ 1, and we get that M_{a} is a bounded liinear mapping from L^{p}(X, R) or L^{p}(X, C) into L^{r}(X, R) or L^{r}(X, C), as appropriate, with operator norm equal to ∥a∥_{q} under the conditions mentioned earlier. There is an analogous statement when r < 1, using suitable versions of some of our previous definitions, as before.

4.15 Convergence in measure

Let (X, A, μ) be a nonempty measure space, and let p be a positive extended real number. Remember that L^{p}(X, R) and L^{p}(X, C) may be considered as metric spaces, as in Section 4.12.

Let 0 < r < p be given, and suppose that q > 0 satisfies 1/q = 1/r − 1/p. If f ∈ L^{p}(X, R) or L^{p}(X, C), A ⊆ X is measurable, and μ(A) < ∞, then f 1_{A} ∈ L^{r}(X, R) or L^{r}(X, C), as appropriate, with

∥f 1_{A}∥_{r} ≤ ∥f∥_{p} ∥1_{A}∥_{q} = ∥f∥_{p} μ(A)^{1/q}, (4.15.1)

as in (4.14.1).

Let {f_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X, and let f be another real or complex-valued measurable function on X, as appropriate. This sequence is said to converge to f in measure on a measurable set A ⊆ X with respect to μ if for every ϵ > 0,

lim_{j→∞} μ({x ∈ A : |f_{j}(x) − f(x)| ≥ ϵ}) = 0. (4.15.2)

If μ(A) < ∞, and {f_{j}}_{j=1}^{∞} converges to f pointwise almost everywhere on A with respect to μ, then it is well known that

(4.15.3) {f_{j}}_{j=1}^{∞} converges to f in measure on A.

4.15. CONVERGENCE IN MEASURE 97

If {f_{j}}_{j=1}^{∞} converges to f in measure on X, then it is well known that (4.15.4) there is a subsequence of {f_{j}}_{j=1}^{∞} that converges to f

pointwise almost everywhere on X

with respect to μ.

Suppose for the moment that f_{j} ∈ L^{p}(X, R) or L^{p}(X, C) for some p > 0 and each j, and that f ∈ L^{p}(X, R) or L^{p}(X, C), as appropriate, too. If (4.15.5) {f_{j}}_{j=1}^{∞} converges to f with respect to the L^{p} metric, then it is easy to see that (4.15.6) {f_{j}}_{j=1}^{∞} converges to f in measure on X with respect to μ.

Suppose that {f_{j}}_{j=1}^{∞} converges to f in measure on X, and that ∥f_{j}∥_{p} ≤ C (4.15.7) for some C ≥ 0 and each j. One can use this to get that ∥f∥_{p} ≤ C, (4.15.8) by passing to a subsequence that converges to f pointwise almost everywhere with respect to μ and using Fatou’s lemma when p < ∞. If p = ∞, then (4.15.7) can also be obtained more directly from convergence in measure and the definition of the assential supremum norm.

Suppose that 0 < r < p, and that A is a measurable subset of X with finite measure. Observe that ∥(f_{j} − f) 1_{A}∥_{r} ≤ ∥f_{j} − f∥_{p} μ(A)^{1/q} (4.15.9) for each j, as in (4.15.1). It is easy to see that ∥f_{j} − f∥_{p} is uniformly bounded in this case, because of (4.15.7) and (4.15.8). If μ(X) < ∞, then one can use (4.15.9) to get that (4.15.10) {f_{j}}_{j=1}^{∞} converges to f in L^{r}(X, R) or L^{r}(X, C), as appropriate.

Let {a_{j}}_{j=1}^{∞} be a sequence of real or complex-valued measurable functions on X, and let a, f be real or complex-valued measurable functions on X as well. If (4.15.11) {a_{j}}_{j=1}^{∞} converges to a pointwise almost everywhere on X with respect to μ, then (4.15.12) {M_{a_{j}}(f)}_{j=1}^{∞} converges to M_{a}(f)

pointwise almost everywhere on X with respect to μ too.

Let A be a measurable subset of X with finite measure again. It is well known that

lim_{n→∞} μ({x ∈ A : |f(x)| > n}) = 0, (4.15.13)

by a standard argument. If

(4.15.14) {a_{j}}_{j=1}^{∞} converges to a on A in measure

with respect to μ, then one can use this to check that

(4.15.15) {M_{a_{j}}(f)}_{j=1}^{∞} converges to M_{a}(f) on A in measure

with respect to μ.

Suppose now that a_{j} ∈ L^{q}(X, R) or L^{q}(X, C), as appropriate, for some q > 0 and each j, with ∥a_{j}∥_{q} ≤ C (4.15.16)

for some C ≥ 0 and each j. Similarly, suppose that a ∈ L^{q}(X, R) or L^{q}(X, C), as appropriate. If f ∈ L^{p}(X, R) or L^{p}(X, C), as appropriate, for some p > 0, and if r > 0 satisfies 1/p + 1/q = 1/r, then we would like to have conditions under which

(4.15.17) {M_{a_{j}}(f)}_{j=1}^{∞} converges to M_{a}(f) in L^{r}(X, R) or L^{r}(X, C),

as appropriate. Of course, this holds when

(4.15.18) {a_{j}}_{j=1}^{∞} converges to a in L^{q}(X, R) or L^{q}(X, C),

as appropriate, by (4.14.1).

The set of f ∈ L^{p}(X, R) or L^{p}(X, C) such that (4.15.17) holds is a linear subspace of L^{p}(X, R) or L^{p}(X, C), as appropriate. This is a closed set in L^{p}(X, R) or L^{p}(X, C) too, as appropriate, because of (4.15.16), as in Section 4.4.

Suppose for the moment that p = +∞, so that q = r. If f ≡ 1 on X, then (4.15.17) implies (4.15.18).

Suppose now that p < ∞, so that q > r. Let A be a measurable subset of X with finite measure. In this case, (4.15.17) holds with f = 1_{A} if and only if

(4.15.19) {a_{j}}_{j=1}^{∞} converges to a on A in measure with respect to μ,

as before.

If this holds for every such A, then (4.15.17) holds for all simple functions in L^{p}(X, R) or L^{p}(X, C), as appropriate. This implies that (4.15.17) holds for all f ∈ L^{p}(X, R) or L^{p}(X, C), as appropriate, because simple functions are dense in these spaces.

Some more spaces and mappings

5.1 Compact support and local compactness

Let X be a nonempty metric space, or topological space, and let f be a real or complex-valued function on X. The support of f in X is defined to be the closure of the set where f ≠ 0,

supp f = \overline{\{x ∈ X : f(x) ≠ 0\}}. (5.1.1)

The support of a function on a set was previously defined to simply be the set where the function is nonzero. That corresponds to taking the set to be equipped with the discrete metric or topology here.

We shall often be interested in functions f on X whose support is a compact set. Note that this happens when supp f is contained in a compact subset of X, because closed sets that are contained in compact sets are compact as well. If X is a metric space, or a Hausdorff topological space, then it is well known that compact sets in X are closed sets. In this case, f has compact support in X when the set where f ≠ 0 is contained in a compact subset of X.

Let C_{com}(X, R), C_{com}(X, C) be the spaces of continuous real and complexvalued functions on X with compact support, respectively. One can check that these are linear subspaces of the spaces C(X, R), C(X, C) of continuous real and complex-valued functions on X, respectively. This uses the fact that the union of two compact subsets of X is compact as well. If X is compact, then C_{com}(X, R), C_{com}(X, C) are the same as the spaces C(X, R), C(X, C) of all continuous real or complex-valued functions on X, respectively.

Suppose for the moment that X is equipped with the discrete metric or topology. It is easy to see that only the finite subsets of X are compact in this case. This means that C_{com}(X, R), C_{com}(X, C) are the same as the spaces c_{00}(X, R), c_{00}(X, C), respectively, defined in Section 1.12.

Let us say that X is locally compact at a point x ∈ X if there are an open set U ⊆ X and a compact set K ⊆ X such that x ∈ U and U ⊆ K. If X is a metric space or a Hausdorff topological space, then K is a closed set in X, so that \overline{U} ⊆ K. This implies that \overline{U} is compact, and local compactness is sometimes defined in this way. If X is locally compact at every x ∈ X, then X is simply said to be locally compact as a metric or topological space.

Suppose that f is a continuous real or complex-valued function on X with compact support. If x ∈ X and f(x) ≠ 0, then it is easy to see that X is locally compact at x.

Suppose now that X is a locally compact Hausdorff space. Let K ⊆ X be a compact set, and let U ⊆ X be an open set, with K ⊆ U. Under these conditions, it is well known that there is a continuous real-valued function f on X with compact support contained in U such that f = 1 on K, and 0 ≤ f ≤ 1 on X. This is a version of Urysohn’s lemma, which can be shown using analogous arguments, or obtained from the usual formulation for normal topological spaces. If X is a locally compact metric space, then this can be verified more directly.

5.2 Vanishing at infinity and compactness

Let X be a nonempty metric space or topological space again, and let f be a real or complex-valued function on X. Let us say that f vanishes at infinity on X if for each ϵ > 0, {x ∈ X : |f(x)| ≥ ϵ} (5.2.1)

is contained in a compact subset of X. Of course, this holds trivially when X is compact. If X is equipped with the discrete metric or topology, then this reduces to the analogous definition in Section 1.13.

If f is continuous on X, then (5.2.1) is a closed set in X for each ϵ > 0. If (5.2.1) is contained in a compact subset of X for some ϵ > 0, then it follows that (5.2.1) is compact. Note that f is bounded on X in this case, because f is bounded on compact subsets of X.

Suppose that f is continuous and vanishes at infinity on X. If x ∈ X and f(x) ≠ 0, then one can check that X is locally compact at x.

Let ϕ be a mapping from R or C into itself, as appropriate, such that ϕ(0) = 0 and ϕ is continuous at 0. If f vanishes at infinity on X, then it is easy to see that (5.2.2) ϕ ∘ f vanishes at infinity on X

as well. If ϕ ≡ 0 on a neighborhood of 0 in R or C, as appropriate, then f ≡ 0 on the complement of a compact subset of X.

Let C_{0}(X, R), C_{0}(X, C) be the spaces of continuous real and complex-valued functions on X that vanish at infinity. It is easy to see that these are linear subspaces of the spaces C_{b}(X, R), C_{b}(X, C) of bounded continuous real and complex-valued functions on X, respectively. One can verify that C_{0}(X, R), C_{0}(X, C) are also closed sets in C_{b}(X, R), C_{b}(X, C), respectively, with respect to the supremum metric. If X is compact, then C_{0}(X, R), C_{0}(X, C) are the same as the spaces C(X, R), C(X, C) of all continuous real and complex-valued functions on X, respectively. If X is equipped with the discrete metric or topology, then C_{0}(X, R), C_{0}(X, C) are the same as the spaces c_{0}(X, R), c_{0}(X, C), respectively, defined in Section 1.13.

Of course, any real or complex-valued function on X with compact support vanishes at infinity. If f is any continuous real or complex-valued function on X that vanishes at infinity, then f can be approximated uniformly by continuous functions with compact support in X. This can be seen using suitable compositions with continuous functions on R or C, as appropriate, as in (5.2.2). If X is a locally compact Hausdorff topological space, then one can also use the version of Urysohn’s lemma mentioned in the previous section. This means that C_{com}(X, R), C_{com}(X, C) are dense in C_{0}(X, R), C_{0}(X, C), respectively, with respect to the supremum metric.

Let X be a locally compact Hausdorff topological space, and let λ be a bounded linear functional on C_{0}(X, R) or C_{0}(X, C), with respect to the supremum norm. It is well known that λ can be represented in a unique way in terms of integration with respect to a real or complex Borel measure on X, with suitable regularity properties. If there is a base for the topology of X with only finitely or countably many elements, then these additional regularity properties hold automatically.

5.3 Uniform convergence on compact sets

Let X be a metric or topological space, and let (Y, d_{Y}) be a metric spaces. Also let {f_{j}}_{j=1}^{∞} be a sequence of mappings from X into Y, and let f be another mapping from X into Y. Suppose that

(5.3.1) {f_{j}}_{j=1}^{∞} converges to f uniformly on compact subsets of X,

so that for each compact set K ⊆ X, {f_{j}}_{j=1}^{∞} converges to f uniformly on K. If f_{j} is continuous on X for each j, then it follows that

(5.3.2) f is continuous on K for every compact K ⊆ X.

More precisely, this uses the topology induced on K by the topology on X, or the restriction of the metric on X to K when X is a metric space.

Of course, the same conclusion holds when f_{j} is only asked to be continuous on compact subsets of X for each j. Let us now consider conditions under which (5.3.2) implies that f is continuous on X.

If X is locally compact at a point x ∈ K, then (5.3.2) implies that f is continuous at x. Thus (5.3.2) implies that f is continuous on X when X is locally compact.

We say that f is sequentially continuous at a point x ∈ X if for every sequence {x_{l}}_{l=1}^{∞} of elements of X that converges to x,

(5.3.3) {f(x_{l})}_{l=1}^{∞} converges to f(x) in Y.

If f is continuous at x, then f is sequentially continuous at x, by a standard argument. If there is a local base for the topology of X with only finitely or countably many elements, and if f is sequentially continuous at x, then one can check that f is continuous at x. In particular, metric spaces have this property.

Let {x_{l}}_{l=1}^{∞} be a sequence of elements of X that converges to an element x of X. One can check that

{x_{l} : l ∈ Z_{+}} ∪ {x} (5.3.4)

is a compact subset of X. If the restriction of f to this compact set is continuous at x, then it follows that (5.3.3) holds.

If f satisfies (5.3.2), then we get that f is sequentially continuous on X, which is to say that f is sequentially continuous at every point in X.

If for every x ∈ X there is a local base for the topology of X at x with only finitely or countably many elements, then X is said to satisfy the first countability condition. Metric spaces have this property, as before. In this case, (5.3.2) implies that f is continuous on X.

5.4 Some related multiplication operators

Let X be a nonempty metric or topological space, and let a be a continuous real or complex-valued function on X. If f is another continuous real or complexvalued function on X, as appropriate, then

M_{a}(f) = a f (5.4.1)

defines a continuous real or complex-valued function on X as well. This defines a linear mapping from C(X, R) or C(X, C) into itself, as appropriate, which is the multiplication operator associated to a.

If f has compact support in X, then M_{a}(f) has compact support too. This means that the restriction of M_{a} to C_{com}(X, R) or C_{com}(X, C), as appropriate, defines a linear mapping from that space into itself.

If a is bounded on X, then

(5.4.2) M_{a} maps C_{b}(X, R) or C_{b}(X, C) into itself,

as appropriate. Similarly, one can check that

(5.4.3) M_{a} maps C_{0}(X, R) or C_{0}(X, C) into itself,

as appropriate, in this case.

Let ∥ · ∥_{sup} be the usual supremum norm on C_{b}(X, R) or C_{b}(X, C). This notation is often preferable, to avoid confusion with L^{∞} norms. If a is bounded on X, then ∥M_{a}(f)∥_{sup} ≤ ∥a∥_{sup} ∥f∥_{sup} (5.4.4)

for every f ∈ C_{b}(X, R) or C_{b}(X, C), as appropriate.

Suppose that (5.4.2) holds, and that

∥M_{a}(f)∥_{sup} ≤ C ∥f∥_{sup} (5.4.5)

for some C ≥ 0 and all f ∈ C_{b}(X, R) or C_{b}(X, C), as appropriate. This implies that a is bounded on X, with

∥a∥_{sup} ≤ C, (5.4.6)

by taking f ≡ 1 on X. This means that the operator norm of M_{a} on C_{b}(X, R) or C_{b}(X, C) with respect to the supremum norm is equal to ∥a∥_{sup}, because of (5.4.4).

Suppose now that (5.4.5) holds for some C ≥ 0 and all f ∈ C_{com}(X, R) or C_{com}(X, C), as appropriate. If X is a locally compact Hausdorff topological space, then one can use the version of Urysohn’s lemma mentioned in Section 5.1 to get that a is bounded on X, and that (5.4.6) holds.

Let {a_{j}}_{j=1}^{∞} be a sequence of continuous real or complex-valued functions on X, as appropriate. Suppose that

(5.4.7) {a_{j}}_{j=1}^{∞} converges to a uniformly on compact subsets of X.

If f is a continuous real or complex-valued function on X, then one can check that

(5.4.8) {M_{a_{j}}(f)}_{j=1}^{∞} converges to M_{a}(f) uniformly on compact subsets of X.

This uses the fact that f is bounded on compact subsets of X.

Note that (5.4.9) supp M_{a}(f) ⊆ supp f,

and similarly (5.4.10) supp M_{a_{j}}(f) ⊆ supp f

for each j. If f has compact support in X, then it follows that

(5.4.11) {M_{a_{j}}(f)}_{j=1}^{∞} converges to M_{a}(f) uniformly on X.

Of course, (5.4.11) holds with f ≡ 1 on X if and only if

(5.4.12) {a_{j}}_{j=1}^{∞} converges to a uniformly on X.

In this case, (5.4.11) holds when f is bounded on X.

Suppose that the a_{j}’s are uniformly bounded on X, so that

∥a_{j}∥_{sup} ≤ C (5.4.13)

for some C ≥ 0 and each j. If {a_{j}}_{j=1}^{∞} converges to a pointwise on X, then a is bounded on X, and (5.4.6) holds. If (5.4.7) holds, then (5.4.11) holds when f vanishes at infinity on X. This uses the fact that C_{com}(X, R), C_{com}(X, C) are dense in C_{0}(X, R), C_{0}(X, C), respectively, with respect to the supremum metric, as in Section 5.2.

5.5 Some separation conditions

Let X be a nonempty topological space. If continuous real-valued functions on X separate points in X, then X is said to be a Urysohn space. More precisely, this means that if x, y are distinct elements of X, then there is a continuous real-valued function f on X such that

f(x) ≠ f(y). (5.5.1)

In this case, one can choose f so that

f(x) = 0, f(y) = 1, and 0 ≤ f ≤ 1 on X. (5.5.2)

Alterntively, one might choose f so that

f(x) = −1, f(y) = 1, and |f| ≤ 1 on X. (5.5.3)

It is easy to see that Urysohn spaces are Hausdorff. If the topology on X is determined by a metric, then one can check directly that X is a Urysohn space. Locally compact Hausdorff topological spaces are Urysohn spaces, by the version of Urysohn’s lemma mentioned in Section 5.1.

Let us say that X is completely regular in the strict sense if for every x ∈ X and closed set E ⊆ X with x ∉ E there is a continuous real-valued function f on X such that f ≡ 0 on E and f(x) ≠ 0. (5.5.4)

One can choose f so that

f(x) = 1 and 0 ≤ f ≤ 1 on X. (5.5.5)

If X also satisfies the first or even zeroth separation condition, then we say that X is completely regular in the strong sense. This implies that X is a Urysohn space, and that X is Hausdorff in particular.

Sometimes one says that X is completely regular when X is completely regular in the strict sense. Similarly, if X is completely regular in the strong sense, then one may say that X satisfies separation condition number three and a half, or equivalently that X is a T_{3 \frac{1}{2}} -space. However, the opposite convention is sometimes used too. Sometimes each of these names is used when X is completely regular in the strong sense, and one may refer to complete regularity in the strict sense in some other way.

If X is a metric space, then one can check directly that X is completely regular in the strong sense. If X is a locally compact Hausdorff topological space, then the version of Urysohn’s lemma mentioned in Section 5.1 implies that X is completely regular in the strong sense.

Let us say that X is normal in the strict sense if for every pair A, B of disjoint closed subsets of X there are disjoint open sets U, V ⊆ X such that

A ⊆ U, B ⊆ V. (5.5.6)

If X satisfies the first separation condition too, then we say that X is normal in the strong sense. Note that this implies that X is Hausdorff. It is well known that metric spaces are normal in the strong sense.

Sometimes one says that X is normal when X is normal in the strict sense, and that X satisfies the fourth separation condition, or equivalently that X is a T_{4} space, when X is normal in the strong sense. The opposite convention is sometimes used as well. Sometimes each of these names is used when X is normal in the strong sense, and normality in the strict sense may be described another way.

Suppose that X is normal in the strict sense, and that A, B are disjoint closed subsets of X. Under these conditions, Urysohn’s lemma states that there is a continuous real-valued function f on X such that

(5.5.7) f ≡ 0 on A, f ≡ 1 on B, and 0 ≤ f ≤ 1 on X.

If X is normal in the strong sense, then it follows that f is completely regular in the strong sense. Note that the conclusion of Urysohn’s lemma can be obtained more directly for metric spaces, using distances to sets, as in Section 2.11.

Let us say that X is regular in the strict sense if for every x ∈ X and closed set E ⊆ X with x ∉ E there are disjoint open sets U, V ⊆ X such that

x ∈ U and E ⊆ V. (5.5.8)

If X also satisfies the first or zeroth separation condition, then we say that X is regular in the strong sense. If X is completely regular in the strict or strong sense, then X is regular in the strict or strong sense, respectively. If X is regular in the strong sense, then X is Hausdorff.

As before, one sometimes says that X is regular when X is regular in the strict sense, and that X satisfies the third separation condition, or equivalently that X is a T_{3} space, when X is regular in the strong sense. The opposite convention is sometimes used too, and sometimes each of these names is used when X is regular in the strong sense, and regularity in the strict sense may be described in some other way.

One can check that X is regular in the strict sense if and only if for every x ∈ X and open set W ⊆ X with x ∈ W, there is an open set U ⊆ X such that

x ∈ U and \overline{U} ⊆ W, (5.5.9)

where \overline{U} is the closure of U in X, as usual. If X is a metric space, or a locally compact Hausdorff space, then it is somewhat easier to verify directly that X is regular in the strong sense than complete regularity. Similarly, if X is normal in the strong sense, then it is easy to see directly that X is regular in the strong sense.

We say that X is completely Hausdorff if for every x, y ∈ X with x ≠ y there are open sets U, V ⊆ X such that

x ∈ U, y ∈ V, and \overline{U} ∩ \overline{V} = ∅. (5.5.10)

Note that completely Hausdorff spaces are Hausdorff in particular. One can verify that X is completely Hausdorff when X is a Urysohn space, and when X is regular in the strong sense.

Suppose that X satisfies the second countability condition. If X is also normal in the strong sense, then Urysohn’s metrization theorem states that there is a metric on X that determines the same topology. Tychonoff showed that normality in the strong sense can be replaced with regularity in the strong sense.

5.6 Point evaluations

Let X be a nonempty metric or topological space. If x ∈ X, then put

δ_{x}(f) = f(x) (5.6.1)

for every continuous real or complex-valued function f on X. This defines a bounded linear functional on each of C_{b}(X, R) and C_{b}(X, C), with respect to the supremum norm. It is easy to see that the dual norm of this linear functional is equal to 1.

The restriction of δ_{x} to each of C_{0}(X, R) and C_{0}(X, C) is a bounded linear functional with dual norm less than or equal to 1, with respect to the supremum norm. If X is a locally compact Hausdorff topological space, then one can use the version of Urysohn’s lemma mentioned in Section 5.1 to get that the dual norm of δ_{x} on these spaces is also equal to 1.

If x, y ∈ X, then δ_{x} − δ_{y} defines a bounded linear functional on each of C_{b}(X, R) and C_{b}(X, C), with dual norm with respect to the supremum norm less than or equal to 2. If X is a Urysohn space, then the dual norm of δ_{x} − δ_{y} on each of these spaces if equal to 2, because of (5.5.3). Similarly, if X is a locally compact Hausdorff topological space, then one can check that the dual norm of δ_{x} − δ_{y} on each of C_{0}(X, R) and C_{0}(X, C) is equal to 2, with respect to the supremum norm.

Suppose that {x_{j}}_{j=1}^{∞} is a sequence of elements of X that converges to x ∈ X. Observe that (5.6.2) {δ_{x_{j}}}_{j=1}^{∞} converges to δ_{x}

with respect to the weak^{∗} topology on the dual of each of C_{b}(X, R), C_{b}(X, C).

Consider the mapping x ↦ δ_{x} (5.6.3)

from X into the dual of C_{b}(X, R) or C_{b}(X, C). This mapping is continuous with respect to the weak^{∗} topology on the dual space. If X is a Urysohn space, then this mapping is one-to-one. If X is completely regular in the strong sense, then this mapping is a homeomorphism onto its image, with respect to the topology induced on its iimage by the weak^{∗} topology on the dual space.

Now consider (5.6.3) as a mapping from X into the dual of C_{0}(X, R) or C_{0}(X, C). This mapping is continuous with respect to the weak^{∗} topology on the dual space, as before. If X is a locally compact Hausdorff space, then this mapping is a homeomorphism onto its image, with respect to the topology induced on its image by the weak^{∗} topology on the dual space.

Suppose that X is a locally compact Hausdorff topological space that is not compact. Let us say that a sequence {x_{l}}_{l=1}^{∞} of elements of X tends to infinity if for every compact set K ⊆ X, we have that

x_{l} ∈ X \ K (5.6.4)

for all sufficiently large l. This implies that

(5.6.5) {δ_{x_{l}}}_{l=1}^{∞} converges to 0

with respect to the weak^{∗} topology on the dual of each of C_{0}(X, R), C_{0}(X, C). It is easy to see that the converse holds as well, using the version of Urysohn’s lemma mentioned in Section 5.1.

5.7 Some remarks about Borel measures

Let X be a nonempty metric or topological space. A subset of X is called an F_{σ} set if it can be expressed as the union of a sequence of closed sets in X. Similarly, a subset of X is said to be a G_{δ} set if it can be expressed as the intersection of a sequence of open sets. Thus a G_{δ} set is the same as the complement of an F_{σ} set in X.

Suppose for the moment that the topology on X is determined by a metric d. If A ⊆ X and r is a positive real number, then put

A_{r} = ⋃_{x∈A} B(x, r), (5.7.1)

where B(x, r) is the open ball in X centered at x ∈ X of radius r. This is an open set in X that contains A, and one can check that

\overline{A} = ⋂_{l=1}^{∞} A_{1/l}. (5.7.2)

It follows that (5.7.3) every closed set in X is a G_{δ} set.

Equivalently, this means that

(5.7.4) every open set in X is an F_{σ} set.

Let X be any nonempty metric or topological space again. The collection of Borel sets in X is the smallest σ-algebra of subsets of X that contains the open sets, or equivalently the closed sets. Thus F_{σ} sets and G_{δ} sets in X are Borel sets.

Let μ be a nonnegative Borel measure on X, which is to say a nonnegative countably-additive measure on the σ-algebra of Borel sets in X. Suppose that (5.7.3) holds, which is the same as saying that (5.7.4) holds. Supose also that

μ(X) < +∞, (5.7.5)

for the moment. If E ⊆ X is a Borel set, then it is well known that there are closed sets A ⊆ X and open sets U ⊆ X such that

A ⊆ E ⊆ U (5.7.6)

and (5.7.7) μ(U \ A) is as small as we like.

This basically corresponds to parts of Proposition D.1 on p419 of [26], Theorem 2.2.2 on p60 of [99], and Theorem 1.10 on p11 of [232].

To see this, observe that open and closed sets in X have these properties, because of (5.7.3), (5.7.4). Thus it suffices to check that the collection C of Borel sets E ⊆ X with these properties is a σ-algebra. It is easy to see that C is closed under taking complements. It follows that the closure of C under countable unions and intersections are equivalent to each other. It is not too difficult to verify either of these conditions.

Equivalently, (5.7.7) means that

(5.7.8) μ(U \ E) is as small as we like

and (5.7.9) μ(E \ A) is as small as we like.

Suppose that μ(E) < +∞, (5.7.10)

in place of (5.7.5). Note that

μ_{E}(B) = μ(B ∩ E) (5.7.11)

is a nonnegative Borel measure on X such that μ_{E}(X) = μ(E) < +∞. If (5.7.3) or equivalently (5.7.4) holds, then the previous argument implies that there are closed sets A in X such that A ⊆ E (5.7.12)

and μ_{E}(E \ A) is as small as we like. Of course, the latter is the same as saying that (5.7.9) holds.

Suppose now that W_{1}, W_{2}, W_{3}, … is a sequence of open subsets of X such that μ(W_{j}) < +∞ (5.7.13)

for each j, and

E ⊆ ⋃_{j=1}^{∞} W_{j}. (5.7.14)

As before, μ_{j}(B) = μ(B ∩ W_{j}) (5.7.15)

is a nonnegative Borel measure on X for each j, with μ_{j}(X) = μ(W_{j}) < +∞. If (5.7.3) or equivalently (5.7.4) holds, then the earlier argument implies that for each j there is an open set U_{j} ⊆ X such that

E ∩ W_{j} ⊆ U_{j} (5.7.16)

and (5.7.17) μ_{j}(U_{j} \ (E ∩ W_{j})) is as small as we like.

We may as well take U_{j} so that

U_{j} ⊆ W_{j} (5.7.18)

for each j, by replacing U_{j} with its intersection with W_{j}, if necessary. In this case, (5.7.17) is the same as saying that

(5.7.19) μ(U_{j} \ (E ∩ W_{j})) is as small as we like

for each j.

If we put

U = ⋃_{j=1}^{∞} U_{j}, (5.7.20)

then U is an open set in X such that

E ⊆ U. (5.7.21)

We can also use (5.7.14) and (5.7.19) to get that (5.7.8) holds. These two variants of the earlier argument correspond to parts of Theorem 2.2.2 on p60 of [99], and Theorem 1.10 on p11 of [232].

5.8 Approximation by continuous functions

Let X be a nonempty metric or topological space, and let μ be a nonnegative Borel measure on X. Also let A ⊆ X be a closed set and U ⊆ X be an open set such that A ⊆ U. (5.8.1)

If X is normal in the strict sense, then there is a continuous real-valued function ϕ on X such that

(5.8.2) ϕ ≡ 1 on A, ϕ ≡ 0 on X \ U, and 0 ≤ ϕ ≤ 1 on X,

by Urysohn’s lemma. This implies that

μ(A) ≤ ∫_{X} ϕ dμ ≤ μ(U). (5.8.3)

If E ⊆ X is a Borel set such that

A ⊆ E ⊆ U, (5.8.4)

then μ(A) ≤ μ(E) ≤ μ(U). (5.8.5)

If μ(U) < ∞, then one can use this and (5.8.3) to get that

|μ(E) − ∫_{X} ϕ dμ| ≤ μ(U) − μ(A) = μ(U \ A). (5.8.6)

If μ(X) < +∞, and X satisfies (5.7.3) or equivalently (5.7.4), then we can find A, U such that μ(U \ A) is as small as we like, as in the previous section. Note that {x ∈ X : 1_{E}(x) ≠ ϕ(x)} ⊆ U \ A. (5.8.7)

Let us say that a topological space X is perfectly normal in the strict sense if X is normal in the strict sense, and (5.7.3) or equivalently (5.7.4) holds. If X also satisfies the first separation condition, then we say that X is perfectly normal in the strong sense. As usual, one may say that X is perfectly normal when X is perfectly normal in the strict sense, and perfectly T_{4} when X is perfectly normal in the strict sense, but the opposite convention may be used as well. Alternatively, both of these terms may be used when X is perfectly normal in the strong sense, and perfectly normal spaces in the strict sense may be described in other ways.

Let X be a nonempty topological space, and let μ_{1}, μ_{2} be nonnegative finite Borel measures on X. If f is a bounded continuous real-valued function on X, then put

λ_{j}(f) = λ_{μ_{j}}(f) = ∫_{X} f dμ_{j} (5.8.8)

for j = 1, 2. These define bounded linear functionals on C_{b}(X, R) with respect to the supremum norm, with dual norms equal to

μ_{j}(X), (5.8.9)

j = 1, 2.

Suppose from now on in this section that X is perfectly normal in the strict sense. If λ_{1} = λ_{2} (5.8.10)

on C_{b}(X, R), then μ_{1} = μ_{2} (5.8.11)

on X. Of course, this means that

μ_{1}(E) = μ_{2}(E) (5.8.12)

for all Borel sets E ⊆ X. To see this, one can try to approximate μ_{1}(E) and μ_{2}(E) by λ_{1}(ϕ) = λ_{2}(ϕ) (5.8.13) for suitable ϕ ∈ C_{b}(X, R).

More precisely, μ = μ_{1} + μ_{2} (5.8.14)

is another nonnegative finite Borel measure on X. Thus there are closed sets A ⊆ X and open sets U ⊆ X such that (5.8.4) holds and μ(U \ A) is as small as we like, as in the previous section. If ϕ is as in (5.8.2), then

|μ_{j}(E) − ∫_{X} ϕ dμ_{j}| ≤ μ_{j}(U \ A), (5.8.15)

j = 1, 2, as in (5.8.6).

Alternatively, one can look at μ_{1} − μ_{2} as a real-valued or signed measure on X. Before doing this, we shall review real and complex-valued measures in the next section.

5.9 Real and complex measures

Let (X, A) be a (nonempty) measurable space, so that X is a nonempty set, and A is a σ-algebra of measurable subsets of X. A real or complex-valued function μ on A is said to be a measure on (X, A) if

μ( ⋃_{j=1}^{∞} A_{j}) = ∑_{j=1}^{∞} μ(A_{j}) (5.9.1)

for every sequence A_{1}, A_{2}, A_{3}, … of pairwise-disjoint measurable subsets of X. In the real case, μ may also be called a signed measure on X. More precisely, the convergence of the series on the right side of (5.9.1) is considered to be part of the definition of a measure. It follows that the series should converge absolutely, because any rearrangement of the series converges too.

If A ⊆ X is measurable, then put

|μ|(A) = sup { ∑_{j=1}^{∞} |μ(A_{j})| : A_{1}, A_{2}, A_{3}, … is a pairwise-disjoint sequence

(5.9.2) of measurable subsets of X such that ⋃_{j=1}^{∞} A_{j} = A}.

It is well known that this defines a nonnegative finite measure on (X, A), which is the total variation measure associated to μ. Note that

|μ(A)| ≤ |μ|(A) (5.9.3)

for every measurable set A ⊆ X. The total variation measure is the smallest nonnegative measure on (X, A) with this property.

Let M(X, R), M(X, C) be the spaces of real and complex measures on (X, A), respectively. It is easy to see that these are vector spaces over the real and complex numbers, respectively, with respect to pointwise addition and scalar multiplication on A. One can check that

∥μ∥ = |μ|(X) (5.9.4)

defines a norm on each of M(X, R) and M(X, C). One can show that M(X, R) and M(X, C) are Banach spaces over the real and complex numbers, respectively, with respect to this norm.

If μ is a real measure on X, then

μ^{+} = (1/2) (|μ| + μ), μ^{−} = (1/2) (|μ| − μ) (5.9.5)

are nonnegative finite measures on (X, A). These are the positive and negative variation measures associated to μ on X, respectively. Of course,

μ = μ^{+} − μ^{−} (5.9.6)

and |μ| = μ^{+} + μ^{−}, (5.9.7)

by construction. The former is known as the Jordan decomposition of μ.

The Hahn decomposition theorem implies that there is a measurable set B ⊆ X such that μ(A ∩ B) ≥ 0 (5.9.8)

and μ(A ∩ (X \ B)) ≤ 0 (5.9.9)

for all measurable sets A ⊆ X. Using this, one can check that

|μ|(A) = μ(A ∩ B) − μ(A ∩ (X \ B)) (5.9.10)

for all measurable sets A ⊆ X. Similarly,

μ^{+}(A) = μ(A ∩ B) (5.9.11)

and μ^{−}(A) = −μ(A ∩ (X \ B)) (5.9.12)

for all measurable sets A ⊆ X.

If μ is a complex measure on X, then it is well known that there is a measurable complex-valued function h on X such that

|h| = 1 on X (5.9.13)

and μ(A) = ∫_{A} h d|μ| (5.9.14)

for all measurable sets A ⊆ X. This reduces to the Hahn decomposition theorem in the real case.

If f is a measurable complex-valued function on X that is integrable with respect to |μ|, then the integral of f with respect to μ on X can be defined by

∫_{X} f dμ = ∫_{X} f h d|μ|. (5.9.15)

In this case, we get that

|∫_{X} f dμ| ≤ ∫_{X} |f| d|μ|. (5.9.16)

Note that f is integrable with respect to |μ| when f is bounded on X, because |μ|(X) < ∞.

5.10 Real and complex Borel measures

Let X be a topological space, so that X may be considered as a measurable space too, using the σ-algebra of Borel sets in X. Also let μ be a real or complex Borel measure on X, which is to say a real or complex measure on X with respect to the Borel sets. If f is a bounded continuous real or complex-valued function on X, then put

λ_{μ}(f) = ∫_{X} f dμ. (5.10.1)

Observe that |λ_{μ}(f)| ≤ ∫_{X} |f| d|μ| ≤ ∥f∥_{sup} |μ|(X), (5.10.2)

by (5.9.16). Thus λ_{μ} defines a bounded linear functional on C_{b}(X, R) or C_{b}(X, C), as appropriate, with respect to the supremum norm, and with dual norm less than or equal to |μ|(X).

If X is perfectly normal in the strict sense, then

(5.10.3) the dual norm of λ_{μ} with respect to the supremum norm is equal to |μ|(X).

In particular, this implies that λ_{μ} = 0 only when μ = 0 on X.

Suppose that μ is a real Borel measure on X, and let B ⊆ X be a Borel set as in the Hahn decomposition of μ. In this case,

λ_{μ}(f) = ∫_{B} f d|μ| − ∫_{X\B} f d|μ| (5.10.4)

for all f ∈ C_{b}(X, R). Let C ⊆ X be a closed set and V ⊆ X an open set such that C ⊆ B ⊆ V. (5.10.5)

Using Urysohn’s lemma, we can get a continuous real-valued function ψ on X such that (5.10.6) ψ ≡ 1 on C, ψ ≡ −1 on X \ V, and |ψ| ≤ 1 on X.

Observe that

(5.10.7) λ_{μ}(ψ) − |μ|(X) = ∫_{B}(ψ − 1) d|μ| − ∫_{X\B}(ψ + 1) d|μ|
= ∫_{B\C}(ψ − 1) d|μ| − ∫_{V \B}(ψ + 1) d|μ|.

This implies that

|λ_{μ}(ψ) − |μ|(X)| ≤ ∫_{B\C} |ψ − 1| d|μ| + ∫_{V \B} |ψ + 1| d|μ|
(5.10.8)  ≤ 2 |μ|(B \ C) + 2 |μ|(V \ B) = 2 |μ|(V \ C).

The right side can be as small as we want, as in Section 5.7. One can use this to get (5.10.3) in the real case.

In the complex case, there is a complex-valued Borel measurable function h on X that satisfies (5.9.13) such that

λ_{μ}(f) = ∫_{X} f h d|μ| (5.10.9)

for all f ∈ C_{b}(X, C), as in (5.9.15). Note that

|λ_{μ}(f) − |μ|(X)| = |∫_{X}(f h − 1) d|μ|| ≤ ∫_{X} |f h − 1| d|μ|. (5.10.10)

To get (5.10.3), one can find f so that |f| ≤ 1 on X, and

∫_{X} |f h − 1| d|μ| = ∫_{X} |f − \overline{h}| d|μ| (5.10.11)

is as small as we like. One way to do this is to approximate \overline{h} by Borel measurable simple functions, and to approximate those simple functions by bounded continuous functions on X. The condition |f| ≤ 1 on X can be obtained by replacing f with f/|f| on the set where |f| ≥ 1.

5.11 Some remarks about product spaces

If X and Y are any two sets, then their Cartesian product X ×Y is the set of all ordered pairs (x, y), with x ∈ X and y ∈ Y. If X and Y are topological spaces, then the corresponding product topology on X × Y can be defined in a standard way. More precisely, if U ⊆ X and V ⊆ Y are open sets, then U × V is an open set in X × Y, and the open subsets of X × Y of this type form a base for the product topology.

Suppose now that (X, d_{X}), (Y, d_{Y}) are metric spaces. There are various ways that one might try to define a metric on X × Y, using d_{X}, d_{Y}. If 1 ≤ p < ∞, then put

(5.11.1)  d_{p}((x, y), (x′, y′)) = d_{X×Y,p}((x, y), (x′, y′))
= (d_{X}(x, x′)^{p} + d_{Y} (y, y′)^{p})^{1/p}

for all x, x′ ∈ X and y, y′ ∈ Y. One can check that this defines a metric on X ×Y, using Minkowski’s inequality for finite sums, as in Section 1.3. Of course, this is much easier when p = 1.

Similarly, put

(5.11.2)  d_{∞}((x, y), (x′, y′)) = d_{X×Y,∞}((x, y), (x′, y′))
= max(d_{X}(x, x′), d_{Y} (y, y′))

for all x, x′ ∈ X and y, y′ ∈ Y. One can verify that this defines a metric on X × Y as well. If 1 ≤ p < ∞, then

(5.11.3)  d_{∞}((x, y), (x′, y′)) ≤ d_{p}((x, y), (x′, y′))
≤ 2^{1/p} d_{∞}((x, y), (x′, y′))

for all x, x′ ∈ X and y, y′ ∈ Y.

If x ∈ X and y ∈ Y, then let B_{X}(x, r), B_{Y} (y, r) be the open balls in X, Y centered at x, y, respectively, with radius r > 0, and let \overline{B}_{X}(x, r), \overline{B}_{Y} (y, r) be the corresponding closed balls of radius r ≥ 0. Similarly, if 1 ≤ p ≤ ∞, then let B_{p}((x, y), r) be the open ball in X ×Y with respect to d_{p} centered at (x, y) with radius r > 0, and let \overline{B}_{p}((x, y), r) be the corresponding closed ball of radius r ≥ 0. It is easy to see that

B_{∞}((x, y), r) = B_{X}(x, r) × B_{Y} (y, r) (5.11.4)

for every r > 0, and that

\overline{B}_{∞}((x, y), r) = \overline{B}_{X}(x, r) × \overline{B}_{Y} (y, r) (5.11.5)

for every r ≥ 0.

One can use (5.11.4) to check that the topology determined on X × Y by d_{∞} is the same as the product topology associated to the topologies determined on X, Y by d_{X}, d_{Y}, respectively. This is the same as the topology determined on X × Y by d_{p} for any p ≥ 1, because of (5.11.3).

If 1 ≤ p ≤ ∞, then a sequence {(x_{j}, y_{j})}_{j=1}^{∞} of elements of X × Y is a Cauchy sequence with respect to d_{p} if and only if {x_{j}}_{j=1}^{∞} and {y_{j}}_{j=1}^{∞} are Cauchy sequences in X and Y, respectively. If X, Y are complete with respect to d_{X}, d_{Y}, respectively, then it follows that X × Y is complete with respect to d_{p}.

We may sometimes be interested in uniform continuity properties of a mapping from X × Y into another metric space Z. Uniform continuity with respect to any of the metrics d_{p} implies uniform continuity with respect to the other metrics of this type, because of (5.11.3).

Similarly, boundedness of a subset of X × Y with respect to any of the metrics d_{p} implies boundedness with respect to the other metrics of this type. We may also be interested in uniform continuity properties of mappings from X × Y into Z on bounded sets in X × Y, for which the choice of metric d_{p} may not matter too much.

5.12 Some remarks about direct sums

Let V, W be vector spaces, both real or both complex. Their Cartesian product V × W may be considered as a vector space over the real or complex numbers too, as appropriate, with respect to coordinatewise addition and scalar multiplication. This is the direct sum of V and W, as vector spaces. One may wish to identify V, W with the linear subspaces V × {0}, {0} × W of V × W in the obvious way.

Let ∥ · ∥_{V}, ∥ · ∥_{W} be norms on V, W, respectively. If 1 ≤ p < ∞, then put

∥(v, w)∥_{V ×W,p} = (∥v∥_{V}^{p} + ∥w∥_{W}^{p})^{1/p} (5.12.1)

for every (v, w) ∈ V × W. It is easy to see that this is a norm on V × W when p = 1. If p > 1, then one can verify that this is a norm on V × W using Minkowski’s inequality for finite sums, as in Section 1.3.

Similarly, put

∥(v, w)∥_{V ×W,∞} = max(∥v∥_{V} .∥w∥_{W}) (5.12.2)

for every (v, w) ∈ V × W. One can check that this defines a norm on V × W as well. We also have that

∥(v, w)∥_{V ×W,∞} ≤ ∥(v, w)∥_{V ×W,p} ≤ 2^{1/p} ∥(v, w)∥_{V ×W,∞} (5.12.3)

for every (v, w) ∈ V × W and 1 ≤ p < ∞.

If 1 ≤ p ≤ ∞, then

(5.12.4)  d_{V ×W,p}((v, w), (v′, w′)) = ∥(v, w) − (v′, w′)∥_{V ×W,p}
= ∥(v − v′, w − w′)∥_{V ×W,p}

defines a metric on V ×W, as usual. This is the same as the metric corresponding to p and the metrics on V, W associated to their norms as in the previous section. If V, W are Banach spaces, then V × W is complete with respect to this metric, as before. This means that V × W is a Banach space with respect to ∥ · ∥_{V ×W,p}.

Suppose that ⟨·, ·⟩_{V}, ⟨·, ·⟩_{W} are inner products on V, W, respectively. It is easy to see that

⟨(v, w), (v′, w′)⟩_{V ×W} = ⟨v, v′⟩_{V} + ⟨w, w′⟩_{W} (5.12.5)

defines an inner product on V × W. If ∥ · ∥_{V}, ∥ · ∥_{W} are the norms on V, W associated to their inner products, respectively, then the norm on V × W associated to this inner product is the same as (5.12.1) with p = 2. If V, W are Hilbert spaces, then it follows that V × W is a Hilbert space with respect to this inner product, as in the preceding paragraph.

Let Z be another vector space over the real or complex numbers, as appropriate. If T_{1}, T_{2} are linear mappings from V, W into Z, respectively, then

T ((v, w)) = T_{1}(v) + T_{2}(w) (5.12.6) defines a linear mapping from V ×W into Z. Note that T is uniquely determined by the conditions that T ((v, 0)) = T_{1}(v) (5.12.7)

for every v ∈ V, and T ((0, w)) = T_{2}(w) (5.12.8)

for every w ∈ W. Conversely, if T is any linear mapping from V × W into Z, then (5.12.7) and (5.12.8) define linear mappings T_{1}, T_{2} from V, W into Z, respectively.

Let 1 ≤ p ≤ ∞ be given, and suppose that T is a bounded linear mapping from V × W into Z, with respect to ∥ · ∥_{V ×W,p} on V × W. Let ∥T ∥_{op,p} be the corresponding operator norm of T. It is easy to see that the associated linear mappings T_{1}, T_{2} from V, W into Z, respectively, are bounded, with

∥T_{1}∥_{op,V Z}, ∥T_{2}∥_{op,W Z} ≤ ∥T ∥_{op,p}. (5.12.9)

Suppose now that T_{1}, T_{2} are bounded linear mappings from V, W into Z, respectively. If v ∈ V and w ∈ W, then

(5.12.10) ∥T ((v, w))∥_{Z} ≤ ∥T_{1}(v)∥_{Z} + ∥T_{2}(w)∥_{Z}
≤ ∥T_{1}∥_{op,V Z} ∥v∥_{V} + ∥T_{2}∥_{op,W Z} ∥w∥_{W} .

This implies that T is bounded as a linear mapping from V × W into Z, with respect to ∥ · ∥_{V ×W,p} on V × W. More precisely, if p = 1, then we get that

∥T ((v, w))∥_{Z} ≤ max(∥T_{1}∥_{op,V Z}, ∥T_{2}∥_{op,W Z}) ∥(v, w)∥_{V ×W,1}, (5.12.11)

so that ∥T ∥_{op,1} ≤ max(∥T_{1}∥_{op,V Z}, ∥T_{2}∥_{op,W Z}). (5.12.12)

If p > 1, then let 1 ≤ q < ∞ be the exponent conjugate to p. In this case,

∥T ((v, w))∥_{Z} ≤ (∥T_{1}∥_{op,V Z}^{q} + ∥T_{2}∥_{op,W Z}^{q})^{1/q} ∥(v, w)∥_{V ×W,p} (5.12.13)

for every v ∈ V and w ∈ W. This follows easily from (5.12.10) when p = ∞, so that q = 1. If 1 < p < ∞, then one can use Hölder’s inequality for functions on a set with two elements. Thus

∥T ∥_{op,p} ≤ (∥T_{1}∥_{op,V Z}^{q} + ∥T_{2}∥_{op,W Z}^{q})^{1/q} (5.12.14)

when 1 < p ≤ ∞.

5.13 Bilinear mappings

Let V, W, and Z be vector spaces, all real or all complex. Also let B be a mapping from V × W into Z, so that B(v, w) is defined as an element of Z for each v ∈ V and w ∈ W. We say that B is bilinear on V × W if B(v, w) is linear as a function of v ∈ V for each w ∈ W, and linear as a function of w ∈ W for each v ∈ V. Of course, if B is a bilinear mapping from V × W into Z, then

B̃(w, v) = B(v, w) (5.13.1)

defines a bilinear mapping from W × V into Z.

If w ∈ W, then put B_{1,w}(v) = B(v, w) (5.13.2)

for every v ∈ V. It is easy to see that B is bilinear on V × W if and only if B_{1,w} is a linear mapping from V into Z for each w ∈ W, and

w ↦ B_{1,w} (5.13.3)

is a linear mapping from W into the space L(V, Z) of all linear mappings from V into Z. Similarly, if v ∈ V, then put

B_{2,v}(w) = B(v, w) (5.13.4)

for every w ∈ W. As before, B is bilinear on V × W if and only if B_{2,v} is a linear mapping from W into Z for each v ∈ V, and

v ↦ B_{2,v} (5.13.5)

is a linear mapping from V into L(W, Z).

Suppose that B is bilinear, and let ∥ · ∥_{V}, ∥ · ∥_{W}, and ∥ · ∥_{Z} be norms on V, W, and Z, respectively. We say that B is bounded on V × W if there is a nonnegative real number C such that

∥B(v, w)∥_{Z} ≤ C ∥v∥_{V} ∥w∥_{W} (5.13.6)

for every v ∈ V and w ∈ W. This is the same as saying that B_{1,w} is a bounded linear mapping from V into Z for each w ∈ W, with

∥B_{1,w}∥_{op,V Z} ≤ C ∥w∥_{W} . (5.13.7)

This means that (5.13.3) is a bounded linear mapping from W into the space BL(V, Z) of bounded linear mappings from V into Z, with operator norm less than or equal to C. Similarly, (5.13.6) is the same as saying that B_{2,v} is a bounded linear mapping from W into Z for each v ∈ V, with

∥B_{2,v}∥_{op,W Z} ≤ C ∥v∥_{V} . (5.13.8)

Equivalently, this means that (5.13.5) is a bounded linear mapping from V into BL(W, Z), with operator norm less than or equal to C. Of course, (5.13.6) holds with C = 0 if and only if B ≡ 0 on V × W.

Remember that V × W may be considered as a metric space, using a metric obtained from the metrics associated to the norms on V and W as in Section 5.11. If B is bounded on V × W, then it is easy to see that B is continuous at (0, 0), as a mapping from V × W into Z.

If v, v′ ∈ V and w, w′ ∈ W, then

B(v, w) − B(v′, w′) = B(v − v′, w) + B(v′, w − w′). (5.13.9)

If (5.13.6) holds for some C ≥ 0, then we get that

∥B(v, w) − B(v′, w′)∥_{Z} ≤ ∥B(v − v′, w)∥_{Z} + ∥B(v′, w − w′)∥_{Z}
(5.13.10)  ≤ C ∥v − v′∥_{V} ∥w∥_{W} + ∥v′∥_{V} ∥w − w′∥_{W} .

It follows from (5.13.10) that B is uniformly continuous on bounded subsets of V × W, with respect to the metric on Z associated to the norm. In particular, this implies that B is continuous on V × W.

If B is continuous at (0, 0) on V × W, then one can check that (5.13.6) holds for some C ≥ 0. More precisely, it suffices to ask that B be bounded on a product of balls in V, W with positive radii centered at 0.

Let V_{0}, W_{0} be dense linear subspaces of V, W, respectively, with respect to the metrics associated to their norms. Suppose that B_{0} is a bounded bilinear mapping from V_{0} × W_{0} into Z, using the restriction of the norms on V, W to V_{0}, W_{0}, respectively. If Z is complete with respect to the metric associated to its norm, then there is a unique extension of B_{0} to a bounded bilinear mapping from V × W into Z. This can be obtained from the analogous statement for bounded linear mappings in Section 2.2, one variable at a time. This could also be obtained from the uniform continuity properties of B_{0} on bounded subsets of V_{0} × W_{0}, as in the preceding paragraph, and the extension result for uniformly continuous mappings mentioned in Section 1.15.

5.14 Separate continuity

Let V, W and Z be metric or topological spaces, and let B be a mapping from V × W into Z. We say that B is separately continuous on V × W if B(v, w) is continuous as a function of v ∈ V for each w ∈ W, and continuous as a function of w ∈ W for each v ∈ V. If B is continuous with respect to the product topology on V × W, then B is sometimes said to be jointly continuous on V × W. It is easy to see that this implies that B is separately continuous on V × W.

Suppose now that V, W, and Z are vector spaces, all real or all complex, and equipped with norms ∥ · ∥_{V}, ∥ · ∥_{W}, and ∥ · ∥_{Z}, respectively. Suppose also that B is a bilinear mapping from V × W into Z, and let B_{1,w}, B_{2,v} be as in (5.13.2), (5.13.4), respectively, for each v ∈ V, w ∈ W. In this case, separate continuity of B on V × W means that

(5.14.1) B_{1,w} is a bounded linear mapping from V into Z

for every w ∈ W, and

(5.14.2) B_{2,v} is a bounded linear mapping from W into Z for every v ∈ V.

Suppose that B satisfies these two conditions, and let E be a nonempty subset of W. Thus E_{1,E} = {B_{1,w} : w ∈ E} (5.14.3)

is a nonempty collection of bounded linear mappings from V into Z. If v ∈ V, then put E_{1,E}(v) = {B_{1,w}(v) : w ∈ E}, (5.14.4)

as in Section 4.3. Equivalently,

E_{1,E}(v) = B_{2,v}(E). (5.14.5)

If E is a bounded subset of W, then we get that (5.14.5) is a bounded subset of Z, because of (5.14.2).

If V is a Banach space, then the Banach–Steinhaus theorem implies that the operator norms of B_{1,w}, w ∈ E, are uniformly bounded. If we take E to be the closed unit ball in W, then we get that B is bounded as a bilinear mapping from V × W into Z. This corresponds to a simplification of Theorem 2.17 on p51 of [286]. Of course, there is an analogous statement when W is a Banach space.

5.15 Bilinear and sesquilinear forms

Let V, W be vector spaces, both real or both complex. A bilinear mapping b from V × W into R or C, as appropriate, may be called a bilinear functional on V × W, as on p106, 137 of [176]. The term bilinear form is used for this on p88 of [291], although this term is often used for the case where V = W.

If w ∈ W, then put b_{1,w}(v) = b(v, w), as in Section 5.13. This is a linear functional on V, and w ↦ b_{1,w} (5.15.1)

defines a linear mapping from W into the algebraic dual V^{alg} of V, as before. Similarly, if v ∈ V, then b_{2,v}(w) = b(v, w) is a linear functional on W, and

v ↦ b_{2,v} (5.15.2)

defines a linear mapping from V into W^{alg}. Conversely, any linear mapping from W into V^{alg} or from V into W^{alg} corresponds to a bilinear functional on V × W in this way.

Now let ∥ · ∥_{V}, ∥ · ∥_{W} be norms on V, W, respectively. If b is bounded as a bilinear mapping on V × W, with the standard absolute value function as the norm on R or C, as appropriate, then b_{1,w} is a bounded linear functional on V for each w ∈ W, and (5.15.1) defines a bounded linear mapping from W into the dual V′ of V with respect to ∥·∥_{V}. Similarly, b_{2,v} is a bounded linear functional on W for each v ∈ V, and (5.15.2) defines a bounded linear mapping from V into W′. Conversely, any bounded linear mapping from W into V′ or from V into W′ corresponds to a bounded bilinear functional on V × W in this way.

Let us take V = W for the rest of the section. Suppose for the moment that V is a vector space over R. Let ⟨·, ·⟩_{V} be an inner product on V, with associated norm ∥ · ∥_{V}. If A is a linear mapping from V into itself, then

b_{A}(v, w) = ⟨A(v), w⟩_{V} (5.15.3)

defines a bilinear form on V. One can check that this is bounded as a bilinear mapping from V × V into R if and only if A is a bounded linear mapping from V into itself.

Conversely, if V is a Hilbert space, then it is well known that any bounded bilinear form b on V is of this form, for a unique bounded linear mapping A on V. More precisely, suppose that b is a bilinear form on V such that b_{2,v}(w) = b(v, w) is a bounded linear functional on V, as a function of w, for each v ∈ V. This implies that for each v ∈ V there is a unique A(v) ∈ V such that

b_{2,v}(w) = ⟨A(v), w⟩_{V} (5.15.4)

for every w ∈ V, as in Section 3.1. It is easy to see that A is a linear mapping from V into itself, because of uniqueness. If b is bounded as a bilinear form on V, then A is bounded as a linear mapping on V, as in the preceding paragraph.

Suppose that V is a vector space over C for the rest of the section. A mapping b from V × V into C is said to be a sesquilinear form on V if b(v, w) is complex-linear in v for each w ∈ V, and conjugate-linear in w for each v ∈ V. In this case, if v ∈ V, then

b̃_{2,v}(w) = \overline{b(v, w)} (5.15.5)

is complex-linear in w. Note that b may be considered as a real-bilinear mapping from V × V into C, which is to say that b is bilinear over R, or equivalently that b is bilinear when V and C are considered as vector spaces over R.

Let ⟨·, ·⟩_{V} be an inner product on V, as a complex vector space, and let ∥·∥_{V} be the associated norm. In particular, ⟨·, ·⟩_{V} is sesquilinear on V. A sesquilinear form b on V is said to be bounded with respect to ∥ · ∥_{V} if there is a nonnegative real number C such that

|b(v, w)| ≤ C ∥v∥_{V} ∥w∥_{V} (5.15.6)

for every v, w ∈ V. This is the same as saying that b is bounded as a realbilinear mapping from V × V into C, where ∥ · ∥_{V} is considered as a norm on V as a vector space over R, and the standard absolute value function on C is considered as a norm on C as a vector space over R.

If A is a linear mapping from V into itself, then (5.15.3) defines a sesquilinear form on V. One can check that this is a bounded sesquilinear form on V with respect to ∥ · ∥_{V} if and only if A is a bounded linear mapping from V into itself, as before. If V is a complex Hilbert space, then it is well known that every bounded sesquilinear form b on V is of this form, for a unique bounded linear mapping A on V.

Indeed, suppose that b is a sesquilinear form on V such that (5.15.5) is a bounded linear functional on V, as a function of w, for each v ∈ V. This implies that for each v ∈ V there is a unique A(v) ∈ V such that

b̃_{2,v}(w) = ⟨w, A(v)⟩_{V} (5.15.7)

for every w ∈ V, as in Section 3.1 again. Equivalently, this means that

b(v, w) = \overline{⟨w, A(v)⟩_{V}} = ⟨A(v), w⟩_{V} (5.15.8)

for every w ∈ V. One can use uniqueness to get that A is a linear mapping from V into itself, as before. If b is bounded as a sesquilinear form on V, then A is bounded as a linear mapping on V, as in the previous paragraph.

Part II

Algebras, norms, and operators Algebras and norms

6.1 Algebras in the strict sense

Let A be a vector space over the real or complex numbers. If A is also equipped with a bilinear mapping from A × A into A, then we say that A is an algebra in the strict sense. This bilinear mapping may be expressed as

(a, b) ↦ a b, (6.1.1)

although one may use other notation, depending on the situation.

If (a b) c = a (b c) (6.1.2)

for every a, b, c ∈ A, then A is said to be an associative algebra. If

a b = b a (6.1.3)

for every a, b ∈ A, then A is said to be commutative. An element e = e_{A} of A is said to be the multiplicative identity element in A if

e_{A} a = a e_{A} = a (6.1.4)

for every a ∈ A. It is easy to see that the multiplicative identity element in A is unique when it exists.

If X is a nonempty set, then the spaces of all real and complex-valued functions on X are commutative associative algebras over R and C, respectively, with respect to pointwise multiplication of functions. The function 1_{X} equal to 1 at every point in X is the multiplicative identity element of each of these algebras.

Let V be a vector space over the real or complex numbers, and let L(V) = L(V, V) be the space of linear mappings from V into itself. This is an associative algebra over the real or complex numbers, as appropriate, with respect to composition of linear mappings. The identity mapping I = I_{V} on V is the multiplicative identity element in L(V).

Let A be an algebra in the strict sense over R or C. Also let A_{0} be a linear subspace of A such that a b ∈ A_{0} (6.1.5)

for every a, b ∈ A_{0}. Under these conditions, A_{0} is an algebra in the strict sense over R or C, as appropriate, with respect to the restriction of multiplication on A to A_{0}. We say that A_{0} is a subalgebra of A in this case. Of course, if A is associative or commutative, then A_{0} has the same property.

Let X be a nonempty metric or topological space. The spaces C(X, R), C(X, C) of continuous real and complex-valued functions on X are subalgebras of the algebras of all real or complex-valued functions on X, respectively. If X is equipped with the discrete metric or topology, then every function on X is continuous.

If U is a nonempty open subset of the complex plane, then the space H(U) of holomorphic functions on U is a subalgebra of C(U, C). Similarly, the space A(U) of continuous complex-valued functions on the closure \overline{U} of U in C that are holomorphic on U is a subalgebra of C(\overline{U}, C). There are analogous statements for holomorphic functions of several complex variables.

6.2 Norms on algebras

Let A be an algebra in the strict sense over the real or complex numbers, and let ∥·∥_{A} be a norm on A, as a vector space over R or C. To say that multiplication on A is bounded as a bilinear mapping with respect to ∥ · ∥_{A} means that

∥a b∥_{A} ≤ C ∥a∥_{A} ∥b∥_{A} (6.2.1)

for some nonnegative real number C and all a, b ∈ A, as in Section 5.13. This happens exactly when multiplication on A is continuous as a mapping from A × A into A, with respect to the metric on A associated to the norm, and the corresponding product topology on A × A, as before.

If (6.2.1) holds with C = 1, then ∥ · ∥_{A} is said to be submultiplicative on A. If A has finite dimension as a vector space over R or C, then it is easy to see that (6.2.1) holds for some C ≥ 0, using the remarks in Section 1.11.

If A has a multiplicative identity element e_{A}, then (6.2.1)

∥e_{A}∥_{A} = ∥e_{A} e_{A}∥_{A} ≤ C ∥e_{A}∥_{A}^{2}. (6.2.2)

If e_{A} ≠ 0, so that ∥e_{A}∥_{A} > 0, then it follows that

1 ≤ C ∥e_{A}∥_{A}. (6.2.3)

Note that A = {0} when e_{A} = 0. Sometimes

∥e_{A}∥_{A} = 1 (6.2.4)

is included as a condition on a norm on A.

If X is a nonempty set, then the spaces ℓ^{∞}(X, R), ℓ^{∞}(X, C) of bounded real and complex-valued functions on X are subalgebras of the spaces of all real and complex-valued functions on X, respectively. If f and g are bounded real or complex-valued functions on X, then it is easy to see that

∥f g∥_{∞} ≤ ∥f∥_{∞} ∥g∥_{∞}. (6.2.5)

Note that ∥1_{X}∥_{∞} = 1.

If X is a nonempty metric or topological space, then the spaces C_{b}(X, R), C_{b}(X, C) of bounded continuous real and complex-valued functions are subalgebras of C(X, R), C(X, C) and of ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively. If U is a nonempty open subset of C, then the space H^{∞}(U) of bounded holomorphic functions on U is a subalgebra of H(U) and C_{b}(U, C). Similarly, the space A_{b}(U) of bounded continuous complex-valued functions on \overline{U} that are holomorphic on U is a subalgebra of A(U) and C_{b}(\overline{U}, C).

Let V be a vector space over the real or complex numbers with a norm ∥·∥_{V}. The space BL(V) = BL(V, V) of all bounded linear mappings from V into itself with respect to ∥·∥_{V} is a subalgebra of L(V). The corresponding operator norm ∥ · ∥_{op} = ∥ · ∥_{op,V V} is submultiplicative on BL(V), as in Section 2.2. Note that I_{V} is automatically bounded on V, with

∥I_{V} ∥_{op} = 1 (6.2.6)

when V ≠ {0}.

Let A be an algebra in the strict sense over R or C with a norm ∥·∥_{A} again. If A is not complete with respect to the metric associated to the norm, then one can pass to a completion to get a Banach space, as in Section 1.15. If (6.2.1) holds for some C ≥ 0, then multiplication on A has a unique extension to the completion that satisfies the analogous conditions, as in Section 5.13.

Suppose that A is an associative algebra over R or C. If a ∈ A and j is a positive integer, then a^{j} can be defined in A in the usual way. If ∥ · ∥_{A} is a submultiplicative norm on A, then

∥a^{j}∥_{A} ≤ ∥a∥_{A}^{j}. (6.2.7)

If ∥a∥_{A} < 1, then it follows that

∥a^{j}∥_{A} → 0 as j → ∞. (6.2.8)

6.3 Algebra homomorphisms

Let A, B be algebras in the strict sense, both real or both complex. Also let ϕ be a linear mapping from A into B, as vector spaces over R or C. If

ϕ(x y) = ϕ(x) ϕ(y) (6.3.1) for all x, y ∈ A, then ϕ is said to be an algebra homomorphism from A into B. If A, B also have multiplicative identity elements e_{A}, e_{B}, respectively, then one may wish to ask that ϕ(e_{A}) = e_{B} (6.3.2)

in addition. If A has a multiplicative identity element e_{A} and ϕ maps A onto B, then (6.3.1) implies that ϕ(e_{A}) is the multiplicative identity element in B.

Let C be another real or complex algebra in the strict sense, as appropriate. If ϕ is a homomorphism from A into B, and ψ is a homomorphism from B into C, then their composition ψ ∘ ϕ is a homomorphism from A into C.

A one-to-one homomorphism ϕ from A onto B is called an algebra isomorphism, as usual. In this case, ϕ^{−1} is an isomorphism from B onto A. If ψ is an isomorphism from B onto C, then it follows that ψ ∘ ϕ is an isomorphism from A onto C.

Let A be an algebra in the strict sense over R or C again. If a, x ∈ A, then put L_{a}(x) = a x. (6.3.3)

This defines L_{a} as a linear mapping from A into itself, as a vector space over R or C, which is the left multiplication operator associated to a. Note that

a ↦ L_{a} (6.3.4)

is linear as a mapping from A into the space L(A) of linear mappings from A into itself.

Suppose that e_{A} is a multiplicative identity element in A. Thus

L_{e_{A}} = I_{A}, (6.3.5)

the identity mapping on A. We also have that

L_{a}(e_{A}) = a (6.3.6)

for every a ∈ A. In particular, this implies that (6.3.4) is injective.

Suppose that A is an associative algebra. If a, b, x ∈ A, then

L_{a}(L_{b}(x)) = a (b x) = (a b) x = L_{a b}(x). (6.3.7)

Equivalently, this means that

L_{a} ∘ L_{b} = L_{a b}. (6.3.8)

Thus (6.3.4) is an algebra homomorphism from A into L(A) in this case.

Let ∥ · ∥_{A} be a norm on A, and suppose that (6.2.1) holds for some C ≥ 0. If a ∈ A, then we get that

∥L_{a}(x)∥_{A} ≤ C ∥a∥_{A} ∥x∥_{A} (6.3.9)

for every x ∈ A. This means that L_{a} is a bounded linear mapping from A into itself, with ∥L_{a}∥_{op} ≤ C ∥a∥_{A}. (6.3.10)

It follows that (6.3.4) is a bounded linear mapping from A into the space BL(A) of bounded linear mappings from A into itself.

If e_{A} is a multiplicative identity element in A, then

∥a∥_{A} ≤ ∥L_{a}∥_{op} ∥e_{A}∥_{A} (6.3.11)

for each a ∈ A, by (6.3.6). If C = 1 and (6.2.4) holds, then we get that (6.3.4) is an isometric linear mapping from A into BL(A).

6.4 Opposite algebra homomorphisms

Let A, B be algebras in the strict sense, both real or both complex, and let ϕ be a linear mapping from A into B, as vector spaces over R or C. If

ϕ(x y) = ϕ(y) ϕ(x) (6.4.1)

for all x, y ∈ A, then ϕ is said to be an opposite algebra homomorphism from A into B. If A, B have multiplicative identity elements e_{A}, e_{B}, then one may wish to ask that ϕ(e_{A}) = e_{B} too. If A has a multiplicative identity element e_{A} and ϕ maps A onto B, then (6.4.1) implies that ϕ(e_{A}) is the multiplicative identity element in B, as before.

A one-to-one opposite algebra homomorphism ϕ from A onto B is called an opposite algebra isomorphism. This implies that ϕ^{−1} is an opposite algebra isomorphism from B onto A.

Let ϕ be an opposite algebra isomorphism from A onto itself. If

ϕ ∘ ϕ = I_{A}, (6.4.2)

then ϕ is called an algebra involution on A.

If A is an algebra in the strict sense over C, then A may also be considered as an algebra in the strict sense over R. In this case, one may be interested in conjugate-linear algebra involutions on A. These are algebra involutions on A as an algebra in the strict sense over R that are also conjugate-linear.

Let (V, ⟨·, ·⟩_{V}) be a real or complex Hilbert space, with the associated norm ∥ · ∥_{V}. If T is a bounded linear mapping from V into itself, then its adjoint T^{∗} is a bounded linear mapping on V as well, as in Section 3.5. In fact, T ↦ T^{∗} is an algebra involution on BL(V), which is conjugate-linear in the complex case.

Let A be any algebra in the strict sense over the real or complex numbers again. If a, x ∈ A, then put R_{a}(x) = x a. (6.4.3)

This defines a linear mapping from A into itself, which is the right multiplication operator associated to a. Clearly

a ↦ R_{a} (6.4.4)

is a linear mapping from A into L(A). If A has a multiplicative identity element e_{A}, then R_{e_{A}} = I_{A} (6.4.5) and R_{a}(e_{A}) = a (6.4.6)

for every a ∈ A.

If A is an associative algebra, then

R_{b}(R_{a}(x)) = (x a) b = x (a b) = R_{a b}(x) (6.4.7)

for every a, b, x ∈ A. This means that

R_{b} ∘ R_{a} = R_{a b}, (6.4.8)

so that (6.4.4) is an opposite algebra homomorphism from A into L(A).

Let ∥ · ∥_{A} be a norm on A that satisfies (6.2.1) for some C ≥ 0. If a ∈ A, then ∥R_{a}(x)∥_{A} ≤ C ∥a∥_{A} ∥x∥_{A} (6.4.9)

for every x ∈ A, so that R_{a} is a bounded linear mapping on A, with

∥R_{a}∥_{op} ≤ C ∥a∥_{A}. (6.4.10)

If e_{A} is a multiplicative identity element in A, then

∥a∥_{A} ≤ ∥R_{a}∥_{op} ∥e_{A}∥_{A} (6.4.11)

for every a ∈ A. In particular, (6.4.4) is an isometric linear mapping from A into BL(A) when C = 1 and (6.2.4) holds.

6.5 Banach algebras

Let A be an associative algebra over the real or complex numbers, with a submultiplicative norm ∥·∥_{A}. If A is complete with respect to the metric associated to ∥ · ∥_{A}, then A is said to be a Banach algebra with respect to ∥ · ∥_{A}. Sometimes one may also ask that A have a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1.

Let A be an associative algebra over R or C with a multiplicative identity element e_{A}. An element x of A is said to be invertible in A if there is an element x^{−1} of A such that x x^{−1} = x^{−1} x = e_{A}. (6.5.1)

One can check that x^{−1} is unique when it exists, in which case it is called the multiplicative inverse of x in A.

If a ∈ A and n is a nonnegative integer, then

(e_{A} − a) ∑_{j=0}^{n} a^{j} = ( ∑_{j=0}^{n} a^{j}) (e_{A} − a) = e_{A} − a^{n+1}, (6.5.2)

by a standard argument. Here a^{j} is interpreted as being equal to e_{A} when j = 0. More precisely, the left side and middle part of (6.5.2) are equal to

∑_{j=0}^{n} a^{j} − ∑_{j=0}^{n} a^{j+1} = ∑_{j=0}^{n} a^{j} − ∑_{j=1}^{n+1} a^{j} = e_{A} − a^{n+1}. (6.5.3)

Suppose now that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, and that ∥e_{A}∥_{A} = 1. Note that ∥a^{j}∥_{A} ≤ ∥a∥_{A}^{j} for each j ≥ 0, where ∥a∥_{A}^{j} is interpreted as being equal to 1 when j = 0. If

∥a∥_{A} < 1, (6.5.4)

then ∑_{j=0}^{∞} a^{j} converges absolutely with respect to ∥ · ∥_{A}, with

∑_{j=0}^{∞} ∥a^{j}∥_{A} ≤ ∑_{j=0}^{∞} ∥a∥_{A}^{j} = (1 − ∥a∥_{A})^{−1}. (6.5.5)

This implies that ∑_{j=0}^{∞} a^{j} converges in A, with

∥∑_{j=0}^{∞} a^{j}∥_{A} ≤ ∑_{j=0}^{∞} ∥a^{j}∥_{A} ≤ (1 − ∥a∥_{A})^{−1}, (6.5.6)

as in Section 1.7.

Under these conditions, we have that

(e_{A} − a) ∑_{j=0}^{∞} a^{j} = ( ∑_{j=0}^{∞} a^{j}) (e_{A} − a) = e_{A}, (6.5.7)

because of (6.5.2). This means that e_{A} − a is invertible in A, with

(e_{A} − a)^{−1} = ∑_{j=0}^{∞} a^{j}. (6.5.8)

We also have that ∥(e_{A} − a)^{−1}∥_{A} ≤ (1 − ∥a∥_{A})^{−1}, (6.5.9)

by (6.5.6).

6.6 More on invertible elements

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}. If x, y are invertible elements of A, then it is easy to see that x y is invertible in A too, with

(x y)^{−1} = y^{−1} x^{−1}. (6.6.1)

6.6. MORE ON INVERTIBLE ELEMENTS 131

Thus the collection G(A) of invertible elements of A is a group.

Suppose that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, with ∥e_{A}∥_{A} = 1. Let x ∈ G(A) and y ∈ A be given, and observe that y = x − (x − y) = (e_{A} − (x − y) x^{−1}) x. (6.6.2) If ∥(x − y) x^{−1}∥_{A} < 1, (6.6.3) then e_{A} − (x − y) x^{−1} ∈ G(A), (6.6.4) as in the previous section. This implies that y ∈ G(A), (6.6.5) by (6.6.2).

Note that (6.6.3) holds when ∥x − y∥_{A} ∥x^{−1}∥_{A} < 1. (6.6.6) Of course, this is the same as saying that ∥x − y∥_{A} < 1/∥x^{−1}∥_{A}. (6.6.7) In particular, (6.6.8) G(A) is an open set in A, with respect to the metric on A associated to the norm.

If (6.6.3) holds, then we also get that ∥(e_{A} − (x − y) x^{−1})^{−1}∥_{A} ≤ (1 − ∥(x − y) x^{−1}∥_{A})^{−1}, (6.6.9) as in (6.5.9). It follows that

(6.6.10)  ∥y^{−1}∥_{A} = ∥x^{−1} (e − (x − y) x^{−1})^{−1}∥_{A}
≤ ∥x^{−1}∥_{A} (1 − ∥(x − y) x^{−1}∥_{A})^{−1},

using (6.6.2) in the first step. If (6.6.7) holds, then we have that ∥y^{−1}∥_{A} ≤ ∥x^{−1}∥_{A} (1 − ∥x − y∥_{A} ∥x^{−1}∥_{A})^{−1}. (6.6.11)

If y ∈ G(A), then y^{−1} − x^{−1} = y^{−1} (x − y) x^{−1}, (6.6.12) so that ∥y^{−1} − x^{−1}∥_{A} ≤ ∥y^{−1}∥_{A} ∥x − y∥_{A} ∥x^{−1}∥_{A}. (6.6.13) If (6.6.7) holds, then we can combine this with (6.6.11) to get that ∥y^{−1} − x^{−1}∥_{A} ≤ ∥x^{−1}∥_{A}^{2} (1 − ∥x − y∥_{A} ∥x^{−1}∥_{A})^{−1} ∥x − y∥_{A}. (6.6.14)

We may consider G(A) as a metric space, using the restriction to G(A) of the metric on A associated to the norm. The topology determined on G(A) by this metric is the same as the topology induced on G(A) by the topology determined on A by the metric associated to the norm, by standard arguments. The previous arguments imply that

w ↦ w^{−1} (6.6.15)

is continuous on G(A). More precisely, this mapping is continuous at any x in G(A), because of (6.6.14).

Let C_{0} be a nonnegative real number, and suppose that x, y ∈ G(A) satisfy

∥x^{−1}∥_{A}, ∥y^{−1}∥_{A} ≤ C_{0}. (6.6.16)

In this case, we have that

∥y^{−1} − x^{−1}∥_{A} ≤ C_{0}^{2} ∥x − y∥_{A}, (6.6.17)

by (6.6.13).

Note that multiplication on G(A) is continuous, as a mapping

(6.6.18) from G(A) × G(A) into G(A),

with respect to the appropriate product topology on G(A)×G(A), or a suitable product metric. This follows from the analogous statement for multiplication on A, as in Section 6.2. This implies that

(6.6.19) G(A) is a topological group,

because of the continuity of (6.6.15) on G(A).

6.7 Some additional properties of G(A)

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A} and a submultiplicative norm ∥ · ∥_{A}. In the previous two sections, we used completeness of A to get the invertibility of elements of A under certain conditions. We can use slightly different arguments to get analogous properties of invertible elements of A, even if A may not be complete.

Let x, y ∈ G(A) be given, and note that

∥e_{A}∥_{A} ≤ ∥x∥_{A} ∥x^{−1}∥_{A}. (6.7.1)

We also have that

∥y^{−1} − x^{−1}∥_{A} ≤ ∥y^{−1}∥_{A} ∥(x − y) x^{−1}∥_{A}, (6.7.2)

because of (6.6.12). It follows that

(6.7.3)  ∥y^{−1}∥_{A} ≤ ∥x^{−1}∥_{A} + ∥y^{−1} − x^{−1}∥_{A}
≤ ∥x^{−1}∥_{A} + ∥y^{−1}∥_{A} ∥(x − y) x^{−1}∥A.

This implies that

(1 − ∥(x − y) x^{−1}∥_{A}) ∥y^{−1}∥_{A} ≤ ∥x^{−1}∥_{A}. (6.7.4)

If (6.6.3) holds, then we get that (6.6.10) holds.

If (6.6.7) holds, then (6.6.11) holds, as before. We can combine this with (6.6.13) to get that (6.6.14) holds in this case. This implies that (6.6.15) is continuous on G(A), as before. It follows that G(A) is a topological group, as before.

Let {x_{j}}_{j=1}^{∞} be a sequence of elements of G(A) that converges to x ∈ A with respect to the metric associated to the norm. Suppose that there is a nonnegative real number C_{1} such that

∥x_{j}^{−1}∥_{A} ≤ C_{1} (6.7.5)

for each j. This implies that

∥x_{j}^{−1} − x_{l}^{−1}∥_{A} ≤ C_{1} ∥x_{j} − x_{l}∥_{A} (6.7.6)

for all j, l, by (6.6.17). Of course, {x_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to the metric on A associated to the norm, because it converges, by hypohtesis. It follows that {x_{j}^{−1}}_{j=1}^{∞} is a Cauchy sequence in A too, because of (6.7.6).

If A is complete with respect to the metric associated to the norm, then {x_{j}^{−1}}_{j=1}^{∞} converges to an element z of A. One can check that

x z = z x = e_{A}, (6.7.7)

so that x ∈ G(A), with x^{−1} = z. Note that

∥x^{−1}∥_{A} = ∥z∥_{A} ≤ C_{1}. (6.7.8)

This shows that {y ∈ G(A) : ∥y^{−1}∥_{A} ≤ C_{1}} (6.7.9)

is a closed set in A, with respect to the metric associated to the norm. If A is not necessarily complete, then (6.7.9) is at least relatively closed in G(A), which is to say that it is a closed set in G(A) with respect to the restriction to G(A) of the metric on A associated to the norm.

6.8 The spectrum of an element

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}. If x ∈ A, then the spectrum of x with respect to A is the set σ_{A}(x) of λ ∈ R or C, as appropriate, such that

λ e_{A} − x ∉ G(A). (6.8.1)

The complementary set of λ ∈ R or C, as appropriate, such that

λ e_{A} − x ∈ G(A) (6.8.2) is called the resolvent set of x with respect to A.

Let X be a nonempty set, and suppose for the moment that A is the algebra of all real or complex-valued functions on X. If a ∈ A, then it is easy to see that the spectrum of a with respect to A is equal to a(X).

Suppose now that A is the algebra of all bounded real or complex-valued functions on X. If a ∈ A, then one can check that the spectrum of a with respect to A is the closure of a(X) in R or C, as appropriate.

Let V be a finite-dimensional vector space over the real or complex numbers, and let T be a linear mapping from V into itself. It is well known that T is oneto-one on V if and only if T maps V onto itself, in which case T is invertible on V. If T is not invertible on V, then it follows that the kernel of T is nontrivial. This implies that the spectrum of T with respect to L(V) is the usual set of eigenvalues of T, corresponding to nonzero eigenvectors in V.

Let (A, ∥ · ∥_{A}) be a real or complex Banach algebra, with a multiplicative identity element e_{A} with norm 1. If x ∈ A, then the resolvent set of x is an open set in R or C, because G(A) is an open set in A, as in Section 6.6. This implies that (6.8.3) σ_{A}(x) is a closed set

in R or C, as appropriate.

If λ is a nonzero real or complex number, as appropriate, then

λ e_{A} − x = λ (e_{A} − λ^{−1} x). (6.8.4)

If |λ| > ∥x∥_{A}, then e_{A} − λ^{−1} x ∈ G(A), (6.8.5)

as in Section 6.5. This means that (6.8.2) holds in this case. It follows that

|λ| ≤ ∥x∥_{A} when λ ∈ σ_{A}(x). (6.8.6)

If A is a complex Banach algebra, then it is well known that

σ_{A}(x) ≠ ∅ (6.8.7)

for every x ∈ A. The basic idea is that if σ_{A}(x) = ∅, then

(λ e_{A} − x)^{−1} (6.8.8)

is a holomorphic function of λ on the complex plane with values in A. One can also check that this tends to 0 as |λ| → ∞. This would imply that (6.8.8) is a constant function of λ, which would have to be equal to 0.

More precisely, if μ is a bounded linear functional on A, then

μ((λ e_{A} − x)^{−1}) (6.8.9)

would be a complex-valued holomorphic function of λ on the complex plane. The same type of argument mentioned in the preceding paragraph implies that (6.8.9) is equal to 0 on C, by standard results from complex analysis. This implies that (6.8.8) is equal to 0 on C, by the Hahn–Banach theorem.

Suppose that A = C, considered as a Banach algebra over the real numbers, using the standard absolute value function as the norm. It is easy to see that any element of A with nonzero imaginary part has empty spectrum with respect to A.

Let A be the set of rational functions in a single variable with complex coefficients. This is a field that contains C as a subfield, and in particular A may be considered as a commutative associative algebra over C with a multiplicative identity element. One can check that the spectrum of any element of A that does not correspond to a constant function is the empty set.

6.9 Homomorphisms into R, C

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}, and let h be a homomorphism from A into R or C, as appropriate, as an algebra over itself. Observe that

h(e_{A})^{2} = h(e_{A}^{2}) = h(e_{A}), (6.9.1)

so that h(e_{A}) (h(e_{A}) − 1) = 0. This means that h(e_{A}) = 0 or

h(e_{A}) = 1. (6.9.2)

It is easy to see that h ≡ 0 on A when h(e_{A}) = 0. Let us suppose from now on in this section that h ≢ 0 on A, so that (6.9.2) holds.

Let X be a nonempty set, and suppose for the moment that A is an algebra of real or complex-valued functions on X, with respect to pointwise multiplication of functions. If x ∈ X, then h_{x}(a) = a(x) (6.9.3)

defines a homomorphism from A into R or C, as appropriate.

Using (6.9.2), we get that

h(a) ≠ 0 when a ∈ G(A). (6.9.4)

If a is any element of A, then

h(h(a) e_{A} − a) = h(a) h(e_{A}) − h(a) = 0, (6.9.5)

so that h(a) e_{A} − a ∉ G(A). (6.9.6)

This means that h(a) ∈ σ_{A}(a). (6.9.7)

Suppose now that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, with ∥e_{A}∥_{A} = 1. If a ∈ A, then |h(a)| ≤ ∥a∥_{A}, (6.9.8) because of (6.8.6) and (6.9.7). This implies that h is a bounded linear functional on A, with dual norm with respect to ∥ · ∥_{A} equal to 1, because of (6.9.2).

Let A′ be the dual space of bounded linear functionals on A, as usual. One can check that

(6.9.9) the set of nonzero algebra homomorphisms h from A into R or C,

as appropriate, is a closed set in A′, with respect to the weak^{∗} topology. It follows that

(6.9.10) (6.9.9) is compact with respect to the weak^{∗} topology,

by the Banach–Alaoglu theorem.

Suppose that A is a complex Banach algebra, and that every nonzero element of A is invertible. If x ∈ A, then σ_{A}(x) ≠ ∅, as in Section 6.8. In this case, we get that σ_{A}(x) contains exactly one element. In fact,

x = λ e_{A} (6.9.11)

for some λ ∈ C. This corresponds to a famous theorem of Gelfand and Mazur.

If A is commutative, then this can be used to get some homomorphisms from A into C. This will be discussed further in Section 6.12.

6.10 Quotient spaces

Let V be a vector space over the real or complex numbers, and let W be a linear subspace of V. Under these conditions, the corresponding quotient space V/W may be defined as a vector space over the real or complex numbers, as appropriate, in a standard way. There is also a natural quotient mapping q from V onto V/W, which is a linear mapping with kernel equal to W.

Let N_{V} be a seminorm on V. We can use this to define a nonnegative real-valued function N_{V/W} on V/W such that

N_{V/W} (q(v)) = inf{N_{V} (v − w) : w ∈ W } (6.10.1)

for every v ∈ V. More precisely, if u, v ∈ V satisfy q(u) = q(v), then u − v is an element of W, and it is easy to see that (6.10.1) is equal to the analogous definition of N_{V/W} (q(u)). Thus (6.10.1) depends only on q(v) ∈ V/W, and not on the choice of v ∈ V, so that N_{V/W} is well defined on V/W.

If v ∈ V and t ∈ R or C, as appropriate, then one can check that

N_{V/W} (t q(v)) = N_{V/W} (q(t v)) = |t| N_{V/W} (q(v)). (6.10.2)

If v_{1}, v_{2} ∈ V and w_{1}, w_{2} ∈ W, then

(6.10.3) N_{V/W} (q(v_{1}) + q(v_{2})) = N_{V/W} (q(v_{1} + v_{2}))
≤ N_{V} ((v_{1} + v_{2}) − (w_{1} + w_{2}))
≤ N_{V} (v_{1} − w_{1}) + N_{V} (v_{2} − w_{2}).

Using this, one can get that

N_{V/W} (q(v_{1}) + q(v_{2})) ≤ N_{V/W} (q_{1}(v)) + N_{V/W} (q(v_{2})). (6.10.4)

This shows that N_{V/W} is a seminorm on V/W.

Suppose now that N_{V} is a norm on V. Observe that v ∈ V satisfies

N_{V/W} (q(v)) = 0 (6.10.5)

if and only if v is in the closure of W in V, with respect to the metric on V associated to N_{V}. Suppose that

(6.10.6) W is also a closed set in V,

so that (6.10.5) holds if and only if v ∈ W. This means that (6.10.5) holds if and only if q(v) = 0 in V/W, so that

(6.10.7) N_{V/W} defines a norm on V/W.

Of course, N_{V/W} (q(v)) ≤ N_{V} (v) (6.10.8)

for every v ∈ V, by construction. This implies that q is bounded as a linear mapping from V onto V/W. More precisely, the operator norm of q is equal to 1 when W is a proper closed linear subspaces of V.

Suppose that V is a Banach space with respect to N_{V}, and let us check that V/W is a Banach space with respect to N_{V/W}. It suffices to verify that absolute convergence of a series in V/W implies convergence in V/W, as in Section 1.7. An infinite series in V/W may be expressed as

∑_{j=1}^{∞} q(v_{j}), (6.10.9)

where v_{1}, v_{2}, v_{3}, … is a sequence of elements of V. We would like to show that this series converges in V/W when

∑_{j=1}^{∞} N_{V/W} (q(v_{j})) (6.10.10)

converges as an infinite series of nonnegative real numbers.

We can choose w_{j} ∈ W such that N_{V} (v_{j} − w_{j}) is as close as we like to N_{V/W} (q(v_{j})) for each j, by the definition (6.10.1) of N_{V/W}. In particular, we can do this in such a way that

∑_{j=1}^{∞} N_{V} (v_{j} − w_{j}) (6.10.11)

converges. This implies that

∑_{j=1}^{∞}(v_{j} − w_{j}) (6.10.12)

converges in V, as in Section 1.7. It follows that

∑_{j=1}^{∞} q(v_{j} − w_{j}) (6.10.13)

converges in V/W. This series is the same as (6.10.9), because q(v_{j} −w_{j}) = q(v_{j}) for each j.

6.11 Two-sided ideals

Let A be an algebra in the strict sense over the real or complex numbers. A linear subspace I of A is said to be a two-sided ideal in A if for every a ∈ A and x ∈ I, we have that a x, x a ∈ I. (6.11.1)

Of course, the quotient A/I may be defined initially as a vector space over the real or complex numbers, as appropriate, as in the previous section. Let q be the natural quotient mapping from A onto A/I, as before.

It is well known that one can define multiplication on A/I in such a way that q(a) q(b) = q(a b) (6.11.2)

for all a, b ∈ A. One can check that the right side depends only on q(a), q(b), and not on the particular choices of a, b, because I is a two-sided ideal in A. Thus A/I may be considered as an algebra in the strict sense over R or C, as appropriate.

If A is an associative algebra, then it is easy to see that A/I is associative too. Similarly, if A is commutative, then A/I is commutative.

If A has a multiplicative identity element e_{A}, then

e_{A/I} = q(e_{A}) (6.11.3)

is the multiplicative identity element in A/I. Note that I ≠ A exactly when e_{A} is not contained in I, which means that (6.11.3) is nonzero.

Let I_{0} be a proper two-sided ideal in A. One can chow that I_{0} is contained in a maximal proper two-sided ideal in A, using Zorn’s lemma or Hausdorff’s maximality principle.

Let ∥ · ∥_{A} be a submultiplicative norm on A. If I_{1} is a two-sided ideal in A, then it is easy to see that the closure \overline{I_{1}} of I_{1} in A with respect to the metric associated to the norm is a two-sided ideal as well.

If I is a closed two-sided ideal in A, then

∥q(a)∥_{A/I} = inf{∥a − x∥_{A} : x ∈ I} (6.11.4) defines a norm on A/I, as in the previous section. If a, b ∈ A and x, y ∈ I, then

q(a) q(b) = q(a − x) q(b − y) = q((a − x) (b − y)). (6.11.5)

This implies that

(6.11.6) ∥q(a) q(b)∥_{A/I} = ∥q((a − x) (b − y))∥_{A/I}
≤ ∥(a − x) (b − y)∥_{A} ≤ ∥a − x∥_{A} ∥b − y∥_{A}.

One can use this to get that

∥q(a) q(b)∥_{A/I} ≤ ∥q(a)∥_{A/I} ∥q(b)∥_{A/I}. (6.11.7)

Suppose now that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, with a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1. If I_{1} is a proper two-sided ideal in A, then I_{1} does not contain any invertible elements of A. This means that I_{1} does not contain any elements of the open ball in A centered at e_{A} with radius 1, because the elements of that ball are invertible in A, as in Section 6.5. It follows that \overline{I_{1}} is a proper two-sided ideal in A too. In particular, a maximal proper two-sided ideal in A is a closed set in A, with respect to the metric associated to the norm.

Suppose that I is a proper closed two-sided ideal in A, and let us check that

∥e_{A/I}∥_{A/I} = ∥q(e_{A})∥_{A/I} = 1. (6.11.8)

Clearly ∥q(e_{A})∥_{A/I} ≤ ∥e_{A}∥_{A} = 1. (6.11.9)

In order to get ∥q(e_{A})∥_{A/I} ≥ 1, (6.11.10)

we need to have that ∥e_{A} − x∥_{A} ≥ 1 (6.11.11)

for every x ∈ I. This condition holds because I is a proper two-sided ideal in A, as in the preceding paragraph.

Note that A/I is a Banach space with respect to the quotient norm, as in the previous section. This means that A/I is a Banach algebra with respect to the quotient norm.

6.12 Ideals in commutative algebras

Let A be a commutative associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}. A linear subspace I of A is said to be an ideal in A if for every a ∈ A and x ∈ I we have that

a x ∈ I, (6.12.1) which is the same as saying that I is a two-sided ideal in A in this case. In this case, A/I is a commutative associative algebra over R or C, as appropriate, with a multiplicative identity element, as in the previous section.

If y ∈ A, then {a y : a ∈ A} (6.12.2)

is an ideal in A. This ideal is equal to A if and only if y is invertible in A.

Let I be a proper ideal in A. If every nonzero element of A/I is invertible in A/I, then one can check that I is maximal with respect to inclusion among proper ideals in A. Conversely, if I is maximal among proper ideals in A, then one can verify that every nonzero element of A/I is invertible.

Suppose that A is a complex Banach algebra, and that I is a maximal proper ideal in A. This implies that I is a closed set in A, and that A/I is a complex Banach algebra as well, as in the previous section. In fact, A/I is isomorphic to C, by the theorem of Gelfand and Mazur, because every nonzero element of A/I is invertible, as before.

If y ∈ A is not invertible, then (6.12.2) is a proper ideal in A, which is contained in a maximal proper ideal in A, as in the previous section. This leads to a homomorphism h from A into C such that h(e_{A}) = 1 and

h(y) = 0, (6.12.3)

as in the preceding paragrah.

If x ∈ A and λ ∈ σ_{A}(x), then λ e_{A} − x is not invertible, and there is a homomorphism h from A into C such that h(e_{A}) = 1 and

h(λ e_{A} − x) = 0, (6.12.4)

as in the previous paragraph. This implies that

h(x) = λ. (6.12.5)

Let A be a commutative ring with a multiplicative identity element e_{A}. An element x of A is said to be nilpotent if

x^{l} = 0 (6.12.6)

for some positive integer l. The set N = N (A) of all nilpotent elements of A is called the nilradical of A, as on p5 of [13]. One can check that N is an ideal in A, as in Proposition 1.7 in [13]. In fact, it is well known that N is the same as the intersection of all of the proper prime ideals in A, as in Proposition 1.8 in [13].

The Jacobson radical R = R(A) of A is defined to be the intersection of all of the maximal proper ideals in A, as on p5 of [13]. If A is a commutative Banach algebra over the real or complex numbers, then this may be simply called the radical of A, and denoted rad(A) or rad A, as in Exercise 12 on p401 of [284], and on p268 of [286]. If

rad(A) = {0}, (6.12.7) then A is said to be semisimple as a commutative Banach algebra, as on p268 of [286]. In the complex case, the radical may be described equivalently as the intersection of all of the kernels of the homomorphisms from A onto C, or as the set of x ∈ A such that σ_{A}(x) = {0}. (6.12.8)

The radical is defined a bit differently in the complex case on p27 of [11], using a result that will be mentioned in Section 6.14.

6.13 More on invertibility

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}. Suppose that x, y ∈ A commute with each other, so that x y = y x. (6.13.1)

If x is invertible in A, then y x^{−1} = x^{−1} y. (6.13.2)

Of course, (6.13.1) implies that x y commutes with x and y. Suppose that x y is invertible in A, so that (x y)^{−1} commutes with x and y, as before. It is easy to see that x and y are both invertible in A in this case, with

x^{−1} = y (x y)^{−1}, y^{−1} = x (x y)^{−1}. (6.13.3)

If e_{A} − x^{l} ∈ G(A) (6.13.4)

for some positive integer l, then

e_{A} − x ∈ G(A). (6.13.5)

This follows from (6.5.2) and the remarks in the preceding paragraph.

If x is nilpotent in A, so that x^{l} = 0 for some l ≥ 1, then (6.13.4) holds, so that (6.13.5) holds. In fact,

(e_{A} − x)^{−1} = ∑_{j=0}^{l−1} x^{j}, (6.13.6)

by (6.5.2).

Suppose that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, with ∥e_{A}∥_{A} = 1. If ∥x^{l}∥_{A} < 1 (6.13.7)

for some l ≥ 1, then (6.13.4) holds, as in Section 6.5. This implies that (6.13.5) holds, as before.

Let λ be a real or complex number, as appropriate. Suppose for the moment that |λ| > ∥x^{l}∥_{A}^{1/l} (6.13.8)

142 CHAPTER 6. ALGEBRAS AND NORMS for some l ≥ 1. This means that |λ^{l}| = |λ|^{l} > ∥x^{l}∥_{A}, so that ∥(λ^{−1} x)^{l}∥_{A} < 1. (6.13.9) It follows that e_{A} − (λ^{−1} x)^{l} ∈ G(A), (6.13.10) and thus e_{A} − λ^{−1} x ∈ G(A), (6.13.11) as before. This implies that λ e_{A} − x = λ (e_{A} − λ^{−1} x) ∈ G(A). (6.13.12)

This shows that for each l ≥ 1, |λ| ≤ ∥x^{l}∥_{A}^{1/l} when λ ∈ σ_{A}(x). (6.13.13) Put

r_{A}(x) = inf_{l≥1} ∥x^{l}∥_{A}^{1/l}. (6.13.14)

It follows that |λ| ≤ r_{A}(x) when λ ∈ σ_{A}(x). (6.13.15)

In the next section, we shall show that

lim_{j→∞} ∥x^{j}∥_{A}^{1/j} = r_{A}(x). (6.13.16)

6.14 More on r_{A}(x)

Let {a_{j}}_{j=1}^{∞} be a sequence of nonnegative real numbers that is submultiplicative, in the sense that a_{j+l} ≤ a_{j} a_{l} (6.14.1) for all j, l ≥ 1. Put

α = inf_{n≥1} a_{n}^{1/n}. (6.14.2)

We would like to check that a_{j}^{1/j} → α as j → ∞. (6.14.3) If a_{l} = 0 for some l ≥ 1, then a_{j} = 0 for all j ≥ l, α = 0, and (6.14.3) is obvious. Thus we may suppose that a_{j} > 0 for each j.

Let a positive integer m be given, and observe that a_{l m} ≤ a_{m}^{l} (6.14.4) for each l ≥ 1, so that a_{l m}^{1/(l m)} ≤ a_{m}^{1/m}. (6.14.5)

6.14. MORE ON R_{A}(X) 143

Similarly, a_{l m+r} ≤ a_{m}^{l} a_{r} (6.14.6)

for all l ≥ 0 and r ≥ 1. If j = l m + r, (6.14.7)

then we get that

a_{j}^{1/j} ≤ a_{m}^{l/j} a_{r}^{1/j} = (a_{m}^{1/m})^{(l m)/j} a_{r}^{1/j} = (a_{m})^{1/m} (a_{m})^{−r/(j m)} a_{r}^{1/j}. (6.14.8)

Note that every j ≥ 1 can be expressed as in (6.14.7) with l, r ≥ 0, r < m, and at least one of l, r positive.

One can use this to get that

lim sup_{j→∞} a_{j}^{1/j} ≤ a_{m}^{1/m}. (6.14.9)

This implies that

lim sup_{j→∞} a_{j}^{1/j} ≤ α, (6.14.10)

by the definition (6.14.2) of α. It is easy to obtain (6.14.3) from this and the definition of α.

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers, with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. If x ∈ A, then

a_{j} = ∥x^{j}∥_{A} (6.14.11)

defines a submultiplicative sequence of nonnegative real numbers. This means that (6.13.16) follows from (6.14.3).

If A is a complex Banach algebra, then it is well known that

r_{A}(x) = max{|λ| : λ ∈ σ_{A}(x)}. (6.14.12)

Remember that σ_{A}(x) is a nonempty closed and bounded subset of C in this case, as in Section 6.8. This implies that σ_{A}(x) is compact with respect to the standard Euclidean metric on C, so that the maximum on the right side of (6.14.12) is attained.

Let us give an outline of the proof of (6.14.12). Let r be the right side of (6.14.12), and note that r ≤ r_{A}(x), (6.14.13)

by (6.13.15). Thus we need only show that

r_{A}(x) ≤ r. (6.14.14)

If λ ∈ C and |λ| > r, then λ ∉ σ_{A}(x), (6.14.15)

so that λ e_{A} − x ∈ G(A).

144 CHAPTER 6. ALGEBRAS AND NORMS

Let us interpret 1/r as being +∞ when r = 0. If ζ ∈ C and |ζ| < 1/r, then e_{A} − ζ x ∈ G(A). (6.14.16) This is trivial when ζ = 0, and otherwise it follows from the fact that ζ^{−1} is not an element of σ_{A}(x), because |ζ^{−1}| = 1/|ζ| > r. Thus (e_{A} − ζ x)^{−1} (6.14.17) is an A-valued function of ζ defined on {ζ ∈ C : |ζ| < 1/r}. (6.14.18)

In fact, (6.14.17) is holomorphic on (6.14.18). Note that (6.14.17) is equal to

∑_{j=0}^{∞} ζ^{j} x^{j} (6.14.19)

when ζ is small enough, as in Section 6.5. If 0 < t < 1/r, (6.14.20) then one can show that there is a nonnegative real number C(t) such that t^{j} ∥x^{j}∥_{A} ≤ C(t) (6.14.21) for each j ≥ 1, using complex analysis. This implies that ∥x^{j}∥_{A}^{1/j} ≤ t^{−1} C(t)^{1/j} (6.14.22) for each j ≥ 1. One can use this to get that r_{A}(x) ≤ 1/t, (6.14.23) which implies (6.14.14).

6.15 Identity elements and essential ranges

Let A be an algebra in the strict sense over the real or complex numbers. Let us define A_{1} initially as a vector space over the real or complex numbers, as appropriate, by taking the direct sum of A and R or C, as appropriate. More precisely, A_{1} may be defined as the Cartesian product A × R or A × C, as appropriate, using coordinatewise addition and scalar multiplication, as in Section 5.12. We can define multiplication on A_{1} by (a, t) (b, z) = (a b + z a + t b, t z) (6.15.1) for every a, b ∈ A and t, z ∈ R or C, as appropriate. This is a bilinear mapping from A_{1} × A_{1} into A_{1}, so that A_{1} is an algebra in the strict sense over R or C, as appropriate.

By construction, a ↦ (a, 0) (6.15.2)

is an injective algebra homomorphism from A into A_{1}. We also have that

e_{A_{1}} = (0, 1) (6.15.3)

is the multiplicative identity element in A_{1}. If A is associative, then one can check that A_{1} is associative. Similarly, if A is commutative, then A_{1} is commutative.

If ∥ · ∥_{A} is a submultiplicative norm on A, then one can check that

∥(a, t)∥_{A_{1}} = ∥a∥_{A} + |t| (6.15.4)

defines a submultiplicative norm on A_{1}. Of course, ∥(a, 0)∥_{A_{1}} = ∥a∥_{A} for every a ∈ A, and ∥e_{A_{1}}∥_{A_{1}} = 1. If A is complete with respect to the metric associated to ∥ · ∥_{A}, then A_{1} is complete with respect to the metric associated to ∥ · ∥_{A_{1}}, as in Section 5.12. This corresponds to part of Proposition 2.5.4 on p58 of [11], Theorem C.3 on p470 of [151], Exercise 15 on p402 of [284], and some remarks on p228 of [286].

Let (X, M, μ) be a nonempty measure space, and note that the corresponding spaces L^{∞}(X, R), L^{∞}(X, C) are commutative associative Banach algebras over R, C, respectively, with respect to the usual L^{∞} norms. The function 1_{X} equal to 1 at every point in X is the multiplicative identity element of these algebras. Let us suppose that μ(X) > 0, so that ∥1_{X}∥_{∞} = 1. If f ∈ L^{∞}(X, R) or L^{∞}(X, C), then f is invertible in L^{∞}(X, R) or L^{∞}(X, C), as appropriate, if and only if there is a positive real number c such that

|f(x)| ≥ c (6.15.5)

for almost every x ∈ X with respect to μ. In this case, the multiplicative inverse of f is equal to 1/f almost everywhere on X with respect to μ.

If λ ∈ R or C, as appropriate, then f − λ 1_{X} is invertible in L^{∞}(X, R) or L^{∞}(X, C), as appropriate, if and only if

|f(x) − λ| ≥ c(λ) (6.15.6)

for some c(λ) > 0 and almost every x ∈ X with respect to μ. Thus the spectrum of f in L^{∞}(X, R) or L^{∞}(X, C), as appropriate, consists of the λ ∈ R or C, as appropriate, such that

μ({x ∈ X : |f(x) − λ| < r}) > 0 (6.15.7)

for every r > 0. This is called the essential range of f.

Algebras, norms, and linear mappings

7.1 Quaternions

A quaternion may be expressed as

x = x_{0} + x_{1} i + x_{2} j + x_{3} k, (7.1.1)

where x_{0}, x_{1}, x_{2}, x_{3} are real numbers. The space H of quaternions may be considered initially as a vector space over the real numbers, which may be identified with R^{4}. We may also identify x_{0} ∈ R with the quaternion x as in (7.1.1), with x_{1} = x_{2} = x_{3} = 0.

Multiplication on R can be extended to a bilinear mapping from H × H into H, with the following properties. The product of a real number and a quaternion, in either order, is the same as scalar multiplication. In particular, 1 ∈ R is the multiplicative identity element in H. We put

i^{2} = j^{2} = k^{2} = −1 (7.1.2)

and i j = −j i = k, j k = −k j = i, k i = −i k = j, (7.1.3)

as on p256 of [37]. One can check that this makes H an associative algebra over R.

If x ∈ H is as in (7.1.1), then put

x^{∗} = x_{0} − x_{1} i − x_{2} j − x_{3} k. (7.1.4)

Clearly x ↦ x^{∗} is a linear mapping from H into itself, as a vector space over R, and (x^{∗})^{∗} = x for every x ∈ H. It is easy to see that (x y)^{∗} = y^{∗} x^{∗} for every x, y ∈ H, so that x ↦ x^{∗} is an algebra involution on H.

7.2. SOME ADDITIONAL PROPERTIES OF R_{A}(X) 147

If x ∈ H is as in (7.1.1), then

x x^{∗} = x^{∗} x = x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}. (7.1.5)

Put ∥x∥_{H} = (x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2})^{1/2}, (7.1.6)

using the nonnegative square root on the right side. This corresponds exactly to the standard Euclidean norm on R^{4}, and in particular this defines a norm on H, as a vector space over R. Using this notation, (7.1.5) is the same as saying that x x^{∗} = x^{∗} x = ∥x∥_{H}^{2}. (7.1.7)

Note that ∥x^{∗}∥_{H} = ∥x∥_{H}. (7.1.8)

If x ∈ H and x ≠ 0, then ∥x∥_{H} > 0, and (7.1.7) implies that x is invertible in H, with x^{−1} = ∥x∥_{H}^{−2} x^{∗}. (7.1.9)

If x, y ∈ H, then

∥x y∥_{H}^{2} = (x y) (x y)^{∗} = x y y^{∗} x^{∗} = ∥y∥_{H}^{2} x x^{∗} = ∥x∥_{H}^{2} ∥y∥_{H}^{2}. (7.1.10)

This implies that ∥x y∥_{H} = ∥x∥_{H} ∥y∥_{H}. (7.1.11)

7.2 Some additional properties of r_{A}(x)

Let A be an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. If x ∈ A, then we can define r_{A}(x) as in (6.13.14), and (6.13.16) holds. More precisely, (6.14.11) defines a submultiplicative sequence of nonnegative real numbers, so that (6.13.16) follows from (6.14.3), as before.

Let B be another associative algebra over the real or complex numbers, as appropriate, with a submultiplicative norm ∥ · ∥_{B}. Also let ϕ be an algebra homomorphism from A into B that is bounded as a linear mapping. If x ∈ A, then ∥ϕ(x)^{j}∥_{B}^{1/j} = ∥ϕ(x^{j})∥_{B}^{1/j} ≤ ∥ϕ∥_{op,AB}^{1/j} ∥x^{j}∥_{A}^{1/j} (7.2.1)

for each j ≥ 1. This implies that

r_{B}(ϕ(x)) ≤ r_{A}(x), (7.2.2)

where the left side is defined in the same way as before. This uses the fact that C^{1/j} → 1 as j → ∞ for any positive real number C.

Similarly, if ϕ is an isomorphism from A onto B, and ϕ and its inverse are bounded as linear mappings, then

r_{B}(ϕ(x)) = r_{A}(x). (7.2.3)

There are analogous statements for opposite algebra homomorphisms and isomorphisms.

If x, y ∈ A commute, then

∥(x y)^{j}∥_{A}^{1/j} = ∥x^{j} y^{j}∥_{A}^{1/j} ≤ ∥x^{j}∥_{A}^{1/j} ∥y^{j}∥_{A}^{1/j} (7.2.4)

for each j ≥ 1. This implies that

r_{A}(x y) ≤ r_{A}(x) r_{A}(y). (7.2.5)

Suppose for the moment that A has a multiplicative identity element e_{A}. Observe that r_{A}(e_{A}) = 1, (7.2.6)

even if ∥e_{A}∥_{A} is not asked to be equal to 1. If x is an invertible element of A, then we get that 1 ≤ r_{A}(x) r_{A}(x^{−1}), (7.2.7)

by taking y = x^{−1} in (7.2.5).

As before, x ∈ A is said to be nilpotent if x^{l} = 0 for some l ≥ 1. If

r_{A}(x) = lim_{j→∞} ∥x^{j}∥_{A}^{1/j} = 0, (7.2.8)

then x is said to be quasinilpotent in A with respect to ∥ · ∥_{A}, as in Definition 1.7.4 on p20 of [11].

Suppose that x is an idempotent element of A, so that

x^{2} = x. (7.2.9)

This implies that x^{j} = x for each j ≥ 1, so that

r_{A}(x) = 1, (7.2.10)

unless x = 0.

7.3 Invertibility and subalgebras

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e = e_{A}, and let B be a subalgebra of A that contains e. Observe that any invertible element of B is also invertible as an element of A, so that G(B) ⊆ G(A). (7.3.1)

If x ∈ B, then it follows that

σ_{A}(x) ⊆ σ_{B}(x), (7.3.2)

where σ_{A}(x), σ_{B}(x) are as in Section 6.8.

Suppose that ∥ · ∥ = ∥ · ∥_{A} is a submultiplicative norm on A, and that

(7.3.3) B is a closed set in A,

with respect to the associated metric. Let {x_{j}}_{j=1}^{∞} be a sequence of invertible elements of B that converges to x ∈ B with respect to the metric associated to ∥ · ∥, and suppose that x is invertible in A. This implies that

(7.3.4) {x_{j}^{−1}}_{j=1}^{∞} converges to x^{−1}

in A, as in Section 6.7. It follows that

x^{−1} ∈ B, (7.3.5)

because of (7.3.3).

Suppose now that A is a Banach algebra with respect to ∥ · ∥. This means that B is complete with respect to the metric associated to the restriction of ∥ · ∥ to B, because of (7.3.3), as in Section 1.6. Thus B is a Banach algebra too, with respect to the restriction of ∥ · ∥ to B. If x ∈ B, then

∂σ_{B}(x) ⊆ σ_{A}(x). (7.3.6)

Here ∂σ_{B}(x) is the boundary of σ_{B}(x) in R or C, as appropriate, with respect to the standard Euclidean metric.

To see this, let λ ∈ ∂σ_{B}(x) be given. This implies that there is a sequence {λ_{j}}_{j=1}^{∞} of real or complex numbers, as appropriate, that converges to λ, with

λ_{j} e − x ∈ G(B) (7.3.7)

for each j. If λ e − x ∈ G(A), (7.3.8)

then λ e − x ∈ G(B), (7.3.9)

as in (7.3.5). Remember that σ_{B}(x) is a closed set in R or C, as appropriate, because B is a Banach algebra, as in Section 6.8. This means that λ ∈ σ_{B}(x), which is a contradiction.

This corresponds to Theorem 1.11.3 on p32 of [11], and to part of part (b) of Theorem 10.18 on p238 of [286]. This is known as the spectral permanence theorem, as in [11].

7.4 Some Volterra integral operators

Let k be a continuous real or complex-valued function on the closed triangle

Δ = {(x, y) ∈ R^{2} : 0 ≤ y ≤ x ≤ 1} (7.4.1)

in the plane. Note that k is uniformly continuous on Δ, with respect to the restriction of the standard Euclidean metric on R^{2} to Δ, because Δ is compact.

150 CHAPTER 7. ALGEBRAS, NORMS, AND LINEAR MAPPINGS If f is a continuous real or complex-valued function on the closed unit interval [0, 1], then let T_{k}(f) be the function defined on [0, 1] by (T_{k}(f))(x) = ∫_{0}^{x} k(x, y) f(y) dy, (7.4.2) as in the left side of (1.3) on p3 of [11]. In particular, let T (f) be the function defined on [0, 1] by (T (f))(x) = ∫_{0}^{x} f(y) dy, (7.4.3) as in Exercise (4) on p5 of [11], which corresponds to taking k ≡ 1 on Δ in (7.4.2).

One can check that (7.4.2) is continuous on [0, 1], as in Exercise (3) on p5 of [11]. It is easy to see that ∥T_{k}(f)∥_{sup} ≤ ∥k∥_{sup,Δ} ∥f∥_{sup}, (7.4.4) where ∥·∥_{sup} is the usual supremum norm of a continuous real or complex-valued function on [0, 1], and ∥k∥_{sup,Δ} is the supremum norm of k on Δ. This implies that T_{k} is a bounded linear mapping from C([0, 1], R) or C([0, 1], C) into itself, as appropriate, with respect to the supremum norm, as in [11].

More precisely, |(T_{k}(f))(x)| ≤ (T_{|k|}(|f|))(x) ≤ ∥k∥_{sup,Δ} (T (|f|))(x) (7.4.5) for every x ∈ [0, 1]. We can use this repeatedly to get that |(T_{k}^{n}(f))(x)| ≤ ∥k∥_{sup,Δ}^{n} (T^{n}(|f|))(x) ≤ ∥k∥_{sup,Δ}^{n} ∥f∥_{sup} (T^{n}(1))(x) (7.4.6) for every n ≥ 1 and x ∈ [0, 1]. Of course, (T^{n}(1))(x) = x^{n}/n! (7.4.7) for every n ≥ 1 and x ∈ [0, 1], by calculus. It follows that ∥T_{k}^{n}(f)∥_{sup} ≤ ∥k∥_{sup,Δ}^{n} ∥f∥_{sup}/n! (7.4.8) for every n ≥ 1, so that ∥T_{k}^{n}∥_{op} ≤ ∥k∥_{sup,Δ}^{n}/n!, (7.4.9) as in Exercise (3) on p21 of [11]. Note that equality holds in (7.4.9) when k is constant on Δ.

Alternatively, consider the n-dimensional simplex Δ_{n} = {x ∈ R^{n} : 0 ≤ x_{1} ≤ x_{2} ≤ ⋯ ≤ x_{n} ≤ 1} (7.4.10) for each positive integer n. The n-dimensional volume of Δ_{n} is the same as (T^{n}(1))(1), which is equal to 1/n!, as before. The volume of Δ_{n} can also be obtained by considering permutations of the coordinates, as in Exercise (2) on p20 of [11].

Let V be C([0, 1], R) or C([0, 1], C), as appropriate, with the supremum norm, and let A be BL(V), with the corresponding operator norm. One can use (7.4.9) to get that r_{A}(T_{k}) = 0, (7.4.11)

where r_{A} is as in Section 6.13. This means that T_{k} is quasinilpotent in A, as in Section 7.2. This is related to Exercise (4) on p21 of [11]. Note that T_{k} is not invertible on V, because (T_{k}(f))(0) = 0 for every f ∈ V.

7.5 Self-adjointness and involutions

Let W be a vector space over the complex numbers, which may also be considered as a vector space over the real numbers. If W_{0} ⊆ W is a linear subspace of W as a vector space over the real numbers, then we may refer to W_{0} as a real-linear subspaces of W.

Let A be an algebra in the strict sense over the real or complex numbers, and let x ↦ x^{∗} (7.5.1)

be an algebra involution on A, as in Section 6.4. Remember that this means that (7.5.1) is an opposite algebra isomorphism from A onto itself such that

(x^{∗})^{∗} = x (7.5.2)

for every x ∈ A. In the complex case, we may also be interested in conjugatelinear involutions, as before.

Let us say that x ∈ A is self-adjoint with respect to (7.5.1) if

x^{∗} = x, (7.5.3)

and anti-self-adjoint if x^{∗} = −x. (7.5.4)

If A is complex and (7.5.1) is conjugate-linear, then x ∈ A is anti-self-adjoint if and only if i x is self-adjoint. In this case, self-adjoint elements of A are also said to be Hermitian.

The spaces of self-adjoint and anti-self-adjoint elements of A are linear subspaces of A in the real case, and when the involution is complex-linear in the complex case. If A is complex and the involution is conjugate-linear, then the spaces of self-adjoint and anti-self-adjoint elements of A are real-linear subspaces of A.

If x is any element of A, then

(1/2) (x + x^{∗}) (7.5.5)

is self-adjoint with respect to (7.5.1), and

(1/2) (x − x^{∗}) (7.5.6) is anti-self-adjoint. Note that x is equal to the sum of (7.5.5) and (7.5.6). We also have that x^{∗} x (7.5.7)

is self-adjoint.

If x x^{∗} = x^{∗} x, (7.5.8)

then one may say that x is normal with respect to (7.5.1). This terminology is perhaps most commonly used when A is complex, and the involution is conjugate-linear.

Suppose that A has a multiplicative identity element e_{A}. Note that

e_{A}^{∗} = e_{A}, (7.5.9)

as in Section 6.4. Suppose that A is associative as well. If x ∈ A is invertible, then it is easy to see that x^{∗} is invertible, with

(x^{∗})^{−1} = (x^{−1})^{∗}. (7.5.10)

Consider U(A) = {x ∈ A : x x^{∗} = x^{∗} x = e_{A}}. (7.5.11)

Equivalently, U(A) consists of the invertible elemets x of A such that

x^{−1} = x^{∗}. (7.5.12)

One can check that (7.5.13) U(A) is a subgroup of G(A).

If A = BL(V) for some Hilbert space V, with the Hilbert space adjoint as the involution, then U(A) consists of the unitary transformations from V onto itself. This is related to some remarks in Sections 2.10 and 3.5.

7.6 Some remarks about idempotents

Let A be an algebra in the strict sense over the real or complex numbers. An element a of A is said to be an idempotent if

a a = a. (7.6.1)

If V is a vector space over the real or complex numbers, then an idempotent element of the algebra of linear mappings from V into itself is the same as a projection on V. These will be discussed further in Section 8.2.

Suppose for the moment that ∥ · ∥_{A} is a submultiplicative norm on A. If a is an idempotent element of A, then

∥a∥_{A} = ∥a a∥_{A} ≤ ∥a∥_{A}^{2}. (7.6.2)

7.7. THE C^{∗} IDENTITY 153

If a ≠ 0, then we get that ∥a∥_{A} ≥ 1. (7.6.3)

Suppose for the moment again that x ↦ x^{∗} is an algebra involution on A, which may be conjugate-linear in the complex case. If a is an idempotent element of A, then a^{∗} = (a a)^{∗} = a^{∗} a^{∗}, (7.6.4)

so that a^{∗} is idempotent as well.

Suppose now that A is an associative algebra with a multiplicative identity element e_{A}. In particular, e_{A} is an idempotent element of A. If an idempotent element a of A is invertible in A, then

a = e_{A}. (7.6.5)

If a is any idempotent element of A, then

a (e_{A} − a) = (e_{A} − a) a = 0. (7.6.6)

This implies that (e_{A} − a)^{2} = e_{A} − a, (7.6.7)

so that e_{A} − a is idempotent in A too. If e_{A} − a is invertible in A, then e_{A} − a = e_{A}, as in the preceding paragraph, so that a = 0.

Let λ ∈ R or C be given, as appropriate. Observe that

λ e_{A} − a = λ (e_{A} − a) + (λ − 1) a. (7.6.8)

If λ ≠ 0, 1, then λ^{−1} (e_{A} − a) + (λ − 1)^{−1} a (7.6.9)

defines an element of A. One can check that this is the multiplicative inverse of λ e_{A} − a in this case.

This shows that σ_{A}(a) ⊆ {0, 1}, (7.6.10)

where σ_{A}(a) is as in Section 6.8. We also get that equality holds unless a = 0 or e_{A}.

7.7 The C^{∗} identity

Let A be an algebra in the strict sense over the real or complex numbers with an involution x ↦ x^{∗}, which may be conjugate-linear in the complex case. Also let ∥ · ∥_{A} be a submultiplicative norm on A. Thus

∥x^{∗} x∥_{A} ≤ ∥x^{∗}∥_{A} ∥x∥_{A} (7.7.1)

for every x ∈ A. If ∥x^{∗}∥_{A} = ∥x∥_{A}, (7.7.2) then (7.7.1) is the same as saying that

∥x^{∗} x∥_{A} ≤ ∥x∥_{A}^{2}. (7.7.3)

Suppose that ∥ · ∥_{A} satisfies the C^{∗} identity

∥x^{∗} x∥_{A} = ∥x∥_{A}^{2} (7.7.4)

for every x ∈ A. This implies that

∥x∥_{A}^{2} ≤ ∥x∥_{A} ∥x^{∗}∥_{A} (7.7.5)

for every x ∈ A. It follows that

∥x∥_{A} ≤ ∥x^{∗}∥_{A} (7.7.6)

when x ≠ 0, and which is trivial when x = 0. If we replace x with x^{∗}, then we get that ∥x^{∗}∥_{A} ≤ ∥x∥_{A}. (7.7.7)

This shows that (7.7.2) holds for every x ∈ A under these conditions.

More precisely, if ∥x∥_{A}^{2} ≤ ∥x^{∗} x∥_{A} (7.7.8)

for every x ∈ A, then (7.7.5) holds for every x ∈ A. In this case, the argument in the preceding paragraph shows that (7.7.2) holds for every x ∈ A. Using this and (7.7.1), we get that (7.7.3) holds for every x ∈ A. Combining this with (7.7.8), we get (7.7.4). This corresponds to Remark 2.9.1 on p75 of [11], and to part (a) of Exercise 5 on p300 of [291].

As another variant, if (7.7.2) holds, then (7.7.4) is the same as saying that

∥x^{∗} x∥_{A} = ∥x^{∗}∥_{A} ∥x∥_{A} (7.7.9)

for every x ∈ A. This is mentioned on p276 of [286]. Note that ∥x x^{∗}∥_{A} is normally considered in [286], rather than ∥x^{∗} x∥_{A}. Of course, the arguments for the two versions are very similar, and they are related using the involution.

If x is self-adjoint or anti-self-adjoint in A, then (7.7.4) is the same as saying that ∥x x∥_{A} = ∥x∥_{A}^{2}. (7.7.10)

Note that (x x)^{∗} = x^{∗} x^{∗} = x x (7.7.11)

in both cases. If x is a self-adjoint idempotent element of A, then (7.7.10) implies that ∥x∥_{A} = ∥x x∥_{A} = ∥x∥_{A}^{2}. (7.7.12)

This means that ∥x∥_{A} = 1 (7.7.13)

when x ≠ 0.

Suppose for the moment that A has a multiplicative identity element e_{A}. Remember that e_{A} is self-adjoint, as in Sections 6.4 and 7.5. It follows that

∥e_{A}∥_{A} = 1, (7.7.14)

as in (7.7.13), unless e_{A} = 0, which would mean that A = {0}. This corresponds to part (b) of Exercise (2) on p49 of [11].

Suppose from now on in this section that A is associative. If x ∈ A is self-adjoint, then ∥x^{2^{l}}∥_{A} = ∥x∥_{A}^{2^{l}} (7.7.15)

for every nonnegative integer l, because of (7.7.10) and (7.7.11). This also works when x is anti-self-adjoint. In both cases, one can use this to get that

∥x^{j}∥_{A} = ∥x∥_{A}^{j} (7.7.16)

for each j ≥ 1.

If x is any element of A, then x^{∗} x is self-adjoint, so that

∥(x^{∗} x)^{j}∥_{A} = ∥x^{∗} x∥_{A}^{j} = ∥x∥_{A}^{2 j} (7.7.17)

for each j ≥ 1. If x is normal in A, in the sense that x commutes with x^{∗}, then

(x^{∗} x)^{j} = (x^{∗})^{j} x^{j} = (x^{j})^{∗} x^{j} (7.7.18)

for each j. This means that

∥(x^{∗} x)^{j}∥_{A} = ∥(x^{j})^{∗} x^{j}∥_{A} = ∥x^{j}∥_{A}^{2} (7.7.19)

for each j. It follows that (7.7.16) holds in this case as well.

7.8 Continuous vector-valued functions

Let X, Y be nonempty metric or topological spaces, and let C(X, Y) be the space of continuous mappings from X into Y. Of course, if X is equipped with the discrete metric or topology, then every mapping from X into Y is continuous.

Suppose from now on in this section that (Y, d_{Y}) is a metric space. A mapping f from X into Y is said to be bounded if f(X) is a bounded set in Y, which means that it is contained in a ball in Y. Let B(X, Y) be the set of all bounded mappings from X into Y. One can define the supremum metric on B(X, Y) in a standard way, using d_{Y}. If Y is complete as a metric space with respect to d_{Y}, then it is well known that

(7.8.1) B(X, Y) is complete with respect to the supremum metric.

Let C_{b}(X, Y) = B(X, Y) ∩ C(X, Y) (7.8.2) be the space of all bounded continuous mappings from X into Y. If X is compact, then every continuous mapping f from X into Y is bounded. More precisely, f(X) is a compact subset of Y, which is bounded in particular.

One can check that

(7.8.3) C_{b}(X, Y) is a closed set in B(X, Y),

with respect to the supremum metric. This basically corresponds to the fact that if {f_{j}}_{j=1}^{∞} is a sequence of continuous mappings from X into Y that converges uniformly to a mapping f from X into Y, then

(7.8.4) f is continuous on X

too. If Y is complete, then it follows that C_{b}(X, Y) is complete with respect to the supremum metric, as in Section 1.6.

If X is a metric space, then we let UC(X, Y) be the space of all uniformly continuous mappings from X into Y. If X is compact, then it is well known that every continuous mapping from X into Y is uniformly continuous. If X is equipped with the discrete metric, then every mapping from X into Y is uniformly continuous.

Let

UC_{b}(X, Y) = B(X, Y) ∩ UC(X, Y) = C_{b}(X, Y) ∩ UC(X, Y) (7.8.5)

be the space of bounded uniformly continuous mappings from X into Y. One can verify that (7.8.6) UC_{b}(X, Y) is a closed set in B(X, Y),

with respect to the supremum metric. As before, this basically corresponds to the fact that if {f_{j}}_{j=1}^{∞} is a sequence of uniformly continuous mappings from X into Y that converges uniformly to a mapping f from X into Y, then

(7.8.7) f is uniformly continuous on X

as well. If Y is complete, then we get that UC_{b}(X, Y) is complete with respect to the supremum metric, as before.

Suppose now that Y is a vector space over the real or complex numbers with a norm ∥ · ∥_{Y}, using the metric on Y associated to ∥ · ∥_{Y}. If X is any nonempty topological or metric space again, then one can check that C(X, Y) is a linear subspace of the space of all Y -valued functions on X, with respect to pointwise addition and scalar multiplication.

In this case, B(X, Y) is the same as ℓ^{∞}(X, Y), as in Section 2.3, on which the supremum metric is the same as the metric associated to the supremum norm, as before. The space

C_{b}(X, Y) = C(X, Y) ∩ ℓ^{∞}(X, Y) (7.8.8)

of all bounded continuous mappings from X into Y is a linear subspace of each of C(X, Y) and ℓ^{∞}(X, Y). The restriction of the supremum norm on ℓ^{∞}(X, Y) to C_{b}(X, Y) defines a norm on C_{b}(X, Y), which may be denoted ∥ · ∥_{∞}, ∥ · ∥_{sup}, or ∥ · ∥_{C_{b}(X,Y)}. If Y is a Banach space, then we get that

(7.8.9) C_{b}(X, Y) is a Banach space

with respect to the supremum norm as well.

If X is a metric space, then one can verify that UC(X, Y) is a linear subspace of C(X, Y). Similarly, UC_{b}(X, Y) is a linear subspace of C_{b}(X, Y) and UC(X, Y). If Y is a Banach space, then UC_{b}(X, Y) is a Banach space with respect to the supremum norm too.

7.9 Algebra-valued functions

Let X be a nonempty set, and let A be an algebra over the real or complex numbers in the strict sense. The space of all A-valued functions on X is an algebra in the strict sense over R or C, as appropriate, with respect to pointwise multiplication of functions. If A is associative or commutative, then the algebra of all A-valued functions on X has the same property. If A has a multiplicative identity element e_{A}, then the function on X equal to e_{A} at every point is the multiplicative identity element in the algebra of all A-valued functions on X. If A is asociative and A has a multiplicative identity element e_{A}, then an A-valued function f on X is invertible as an A-valued function on X if and only if

(7.9.1) f(x) is invertible in A for every x ∈ X.

Suppose that ∥ · ∥_{A} is a submultiplicative norm on A, and remember that ℓ^{∞}(X, A) is the space of all bounded A-valued functions on X, as in Section 2.3. If f, g ∈ ℓ^{∞}(X, A), then it is easy to see that their product f(x) g(x) is bounded on X too, with

∥f g∥_{∞} ≤ ∥f∥_{∞} ∥g∥_{∞}. (7.9.2)

Thus ℓ^{∞}(X, A) is a subalgebra of the algebra of all A-valued functions on X. If A has a multiplicative identity element e_{A}, then the function on X equal to e_{A} at every point is bounded, with supremum norm equal to ∥e_{A}∥_{A}. If A is also associative, then f ∈ ℓ^{∞}(X, A) is invertible in ℓ^{∞}(X, A) if and only if (7.9.1) holds, and (7.9.3) f(x)^{−1} is bounded on X.

If f is a bounded real or complex-valued function on X, then one can check that ∥f^{l}∥_{∞} = ∥f∥_{∞}^{l} (7.9.4)

for every positive integer l. Of course, this uses the standard absolute value function on the real or complex numbers.

Suppose now that X is a metric or topological space. If f, g are continuous A-valued functions on X, then one can check that their product f(x) g(x) is continuous on X as well. This implies that the space C(X, A) of all continuous A-valued functions on X is a subalgebra of the space of all A-valued functions on X. If A has a multiplicative identity element e_{A}, then the function on X equal to e_{A} at every point is continuous, and thus is the multiplicative identity element in C(X, A). If A is associative as well, then f ∈ C(X, A) is invertible in C(X, A) if and only if (7.9.1) holds, because of continuity of taking inverses on G(A), as in Section 6.7.

Similarly, the space

C_{b}(X, A) = C(X, A) ∩ ℓ^{∞}(X, A) (7.9.5)

of all bounded continuous A-valued functions on X is a subalgebra of each of C(X, A) and ℓ^{∞}(X, A). If A has a multiplicative identity element e_{A}, then the function on X equal to e_{A} at every point is the multiplicative identity element in C_{b}(X, A). If A is also associative, then f ∈ C_{b}(X, A) is invertible in C_{b}(X, A) if and only if (7.9.1) and (7.9.3) hold, because of continuity of taking inverses on G(A).

Suppose that X is a metric space, and that f, g are bounded uniformly continuous A-valued functions on X. One can check that

(7.9.6) f g is uniformly continuous on X,

using (7.10.7). This implies that UC_{b}(X, A) is a subalgebra of C_{b}(X, A).

Suppose that A has a multiplicative identity element e_{A}, so that the function on X equal to e_{A} at every point is the multiplicative identity element in UC_{b}(X, A). If A is associative, and f ∈ UC_{b}(X, A) satisfies (7.9.1) and (7.9.3), then one can verify that

(7.9.7) f(x)^{−1} is uniformly continuous on X,

using (7.10.12).

7.10 Bounded Lipschitz functions

Let (X, d_{X}), (Y, d_{Y}) be nonempty metric spaces. Also let {f_{j}}_{j=1}^{∞} be a sequence of Lipschitz mappings from X into Y that converges pointwise to a mapping f from X into Y. Suppose that there is a nonnegative real number C such that

Lip_{X,Y} (f_{j}) ≤ C (7.10.1)

for each j, where Lip_{X,Y} (f_{j}) is the minimal Lipschitz constant of f_{j}, as in Section 2.1. Under these conditions, one can check that f is Lipschitz, with

Lip_{X,Y} (f) ≤ C. (7.10.2)

Let W be a vector space over the real or complex numbers with a norm ∥ · ∥_{W}. Consider the space

Lip_{b}(X, W) = Lip(X, W) ∩ C_{b}(X, W) (7.10.3) of all bounded Lipschitz mappings from X into W. This is a linear subspace of each of Lip(X, W) and C_{b}(X, W).

If f is a Lipschitz mapping from X into W, then we let Lip(f) be the minimal Lipschitz constant of f, as in Section 2.1 again. This defines a seminorm on Lip(X, W), as before. If t is a nonnegative real number and f ∈ Lip_{b}(X, W), then put (7.10.4) ∥f∥_{Lip_{b},t} = ∥f∥_{Lip_{b}(X,W),t} = ∥f∥_{sup} + t Lip(f).

It is easy to see that this defines a norm on Lip_{b}(X, W).

If W is a Banach space and t > 0, then

(7.10.5) Lip_{b}(X, W) is a Banach space with respect to ∥ · ∥_{Lip_{b,t}}.

To see this, let {f_{j}}_{j=1}^{∞} be a sequence of elements of Lip_{b}(X, W) that is a Cauchy sequence with respect to the metric associated to ∥·∥_{Lip_{b},t}. In particular, {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to the supremum metric, so that {f_{j}}_{j=1}^{∞} converges uniformly to a bounded continuous W -valued function f on X. One can use the fact that {f_{j}}_{j=1}^{∞} is bounded with respect to ∥ · ∥_{Lip_{b},t} to get that f is Lipschitz on X, as in (7.10.2). One can also use the Cauchy condition for {f_{j}}_{j=1}^{∞} with respect to the metric associated to ∥ · ∥_{Lip_{b},t} to get that {f_{j}}_{j=1}^{∞} converges to f with respect to this metric.

Let A be an algebra in the strict sense over the real or complex numbers, with a submultiplicative norm ∥ · ∥_{A}. If f, g are A-valued functions on X, then

f(x) g(x) − f(w) g(w) = (f(x) − f(w)) g(x) + f(w) (g(x) − g(w)) (7.10.6)

for every x, w ∈ X, so that

(7.10.7) ∥f(x) g(x) − f(w) g(w)∥_{A} ≤ ∥f(x) − f(w)∥_{A} ∥g(x)∥_{A}
+∥f(w)∥_{A} ∥g(x) − g(w)∥_{A}.

If f, g ∈ Lip_{b}(X, A), then it follows that f g is Lipschitz on A too, with

(7.10.8) Lip(f g) ≤ Lip(f) ∥g∥_{sup} + ∥f∥_{sup} Lip(g).

This implies that Lip_{b}(X, A) is a subalgebra of C_{b}(X, A). One can also check that (7.10.9) ∥f g∥_{Lip_{b},t} ≤ ∥f∥_{Lip_{b},t} ∥g∥_{Lip_{b},t}

for each t ≥ 0.

Suppose that A has a multiplicative identity element e_{A}, so that the function 1_{X} e_{A} equal to e_{A} at every point in X is the multiplicative identity element in Lip_{b}(X, A). Note that ∥1_{X} e_{A}∥_{Lip_{b},t} = ∥e_{A}∥_{A} (7.10.10)

for each t ≥ 0.

Suppose that A is also associative, and that f is an A-valued function on X such that f(x) is invertible in A for every x ∈ X. Observe that

f(x)^{−1} − f(w)^{−1} = f(x)^{−1} (f(w) − f(x)) f(w)^{−1} (7.10.11)

160 CHAPTER 7. ALGEBRAS, NORMS, AND LINEAR MAPPINGS

for every x, w ∈ X, so that ∥f(x)^{−1} − f(w)^{−1}∥_{A} ≤ ∥f(x)^{−1}∥_{A} ∥f(w) − f(x)∥_{A} ∥f(w)^{−1}∥_{A}. (7.10.12) If f(x)^{−1} is bounded on X, and f is Lipschitz on X, then it follows that f(x)^{−1} is Lipschitz on X, with Lip(f(·)^{−1}) ≤ ∥f(·)^{−1}∥_{sup}^{2} Lip(f). (7.10.13)

Suppose now that f is a real or complex-valued function on X that is bounded and Lipschitz. If l ≥ 2 is an integer, then one can check that Lip(f^{l}) ≤ l ∥f∥_{sup}^{l−1} Lip(f), (7.10.14) using (7.10.8). This implies that ∥f∥_{sup}^{l} = ∥f^{l}∥_{sup} ≤ ∥f^{l}∥_{Lip_{b},t} ≤ ∥f∥_{sup}^{l} + t l ∥f∥_{sup}^{l−1} Lip(f) (7.10.15) for each t ≥ 0. One can use this to get that

lim_{l→∞} ∥f^{l}∥_{Lip_{b},t}^{1/l} = ∥f∥_{sup} (7.10.16)

for each t > 0.

7.11 Bilipschitz conditions and invertibility

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces, and let f be a mapping from X into Y. Suppose that there is a positive real number c such that c d_{X}(x, w) ≤ d_{Y} (f(x), f(w)) (7.11.1) for every x, w ∈ X. This implies in particular that f is one-to-one on X, so that the inverse mapping f^{−1} may be defined on the image f(X) of X under f in Y. More precisely, (7.11.1) is the same as saying that (7.11.2) f^{−1} is Lipschitz with constant 1/c on f(X), with respect to the restriction of d_{Y} to f(X).

If f is also Lipschitz on X, then f is said to be a bilipschitz mapping from X into Y. In this case, if X is complete as a metric space, then it is easy to see that (7.11.3) f(X) is complete, with respect to the restriction of d_{Y} to f(X). This implies that (7.11.4) f(X) is a closed set in Y, as in Section 1.6. If f(X) is dense in Y, then it follows that f(X) = Y. (7.11.5)

Suppose now that Y is a vector space over the real or complex numbers with a norm ∥ · ∥_{Y}, and that d_{Y} is the metric on Y associated to ∥ · ∥_{Y}. Let f be a mapping from X into Y that satisfies (7.11.1) for some c > 0 again, and let g be a mapping from X into Y such that

(7.11.6) f − g is Lipschitz on X.

If x, w ∈ X, then

c d_{X}(x, w) ≤ ∥f(x) − f(w)∥_{Y}
(7.11.7)  ≤ ∥(f(x) − g(x)) − (f(w) − g(w))∥_{Y} + ∥g(x) − g(w)∥_{Y}
≤ Lip_{X,Y} (f − g) d_{X}(x, w) + ∥g(x) − g(w)∥_{Y} ,

where Lip_{X,Y} (f − g) is the minimal Lipschitz constant of f − g on X, as in Section 2.1. This implies that

(c − Lip_{X,Y} (f − g)) d_{X}(x, w) ≤ ∥g(x) − g(w)∥_{Y} . (7.11.8)

If Lip_{X,Y} (f − g) < c, (7.11.9)

then this is the same type of condition as before.

Suppose that X is also a vector space over the real or complex numbers, as appropriate, with a norm ∥ · ∥_{X}, and that d_{X} is the metric on X associated to ∥ · ∥_{X}. If f is a linear mapping from X into Y, then (7.11.1) implies that

c ∥x∥_{X} ≤ ∥f(x)∥_{Y} (7.11.10)

for every x ∈ X. Conversely, this condition implies that

c ∥x − w∥_{X} ≤ ∥f(x − w)∥_{Y} = ∥f(x) − f(w)∥_{Y} (7.11.11)

for every x, w ∈ X, which is the same as (7.11.1) in this case. Note that f(X) is a linear subspace of Y, and that (7.11.10) is the same as saying that f^{−1} is bounded as a linear mapping from f(X) into X, with respect to the restriction of ∥ · ∥_{Y} to f(X), and with operator norm less than or equal to 1/c.

Suppose that g is a linear mapping from X into Y such that

f − g ∈ BL(X, Y). (7.11.12)

If f satisfies (7.11.10) for some c > 0, then

(7.11.13)  c ∥x∥_{X} ≤ ∥f(x) − g(x)∥_{Y} + ∥g(x)∥_{Y}
≤ ∥f − g∥_{op,X,Y} ∥x∥_{X} + ∥g(x)∥_{X}

for every x ∈ X. This implies that

(c − ∥f − g∥_{op,XY}) ∥x∥_{X} ≤ ∥g(x)∥_{Y} (7.11.14)

162 CHAPTER 7. ALGEBRAS, NORMS, AND LINEAR MAPPINGS for every x ∈ X, which is the same type of condition as before when ∥f − g∥_{op,XY} < c. (7.11.15) Of course, this could also be obtained from the analogous statements for Lipschitz mappings, but the arguments are a bit simpler in this case.

Suppose that f satisfies (7.11.1) for some c > 0, and that (7.11.6) holds. If x ∈ X, then we get that ∥f(x) − g(x)∥_{Y} ≤ ∥f − g∥_{op,XY} ∥x∥_{X} ≤ c^{−1} ∥f − g∥_{op,XY} ∥f(x)∥_{Y} . (7.11.16) Suppose that (7.11.15) holds as well, which is the same as saying that c^{−1} ∥f − g∥_{op,XY} < 1. (7.11.17) If f(X) = Y, then it follows that (7.11.18) g(X) is dense in Y, as in Section 4.6. This implies that g(X) = Y when X is complete, as before.

Similarly, if f(X) is dense in Y, then (7.11.18) holds. Indeed, if x ∈ X and y ∈ Y, then

(7.11.19) ∥y − g(x)∥_{Y} ≤ ∥y − f(x)∥_{Y} + ∥f(x) − g(x)∥_{Y}
≤ ∥y − f(x)∥_{Y} + c^{−1} ∥f − g∥_{op,XY} ∥f(x)∥_{Y} ,

using (7.11.16) in the second step. This implies that

(7.11.20) ∥y − g(x)∥_{Y} ≤ (1 + c^{−1} ∥f − g∥_{op,XY}) ∥y − f(x)∥_{Y}
+c^{−1} ∥f − g∥_{op,XY} ∥y∥_{Y} .

One can use this to get that the criterion for density in Section 4.6 holds in this case too.

7.12 The contraction mapping theorem

Let (X, d_{X}) be a nonempty metric space, and let ϕ be a mapping from X into itself. Suppose that ϕ is a contraction on X, which means that ϕ is Lipschitz with Lip(ϕ) < 1. (7.12.1) If X is complete, then the contraction mapping theorem states that (7.12.2) ϕ has a unique fixed point in X, which is to say that there is a unique x ∈ X such that ϕ(x) = x. (7.12.3)

7.12. THE CONTRACTION MAPPING THEOREM 163 The uniqueness of the fixed point can be verified directly, and does not use completeness of X.

To get the existence of the fixed point, let x_{1} ∈ X be given, and let {x_{j}}_{j=1}^{∞} be the sequence of elements of X defined recursively by x_{j+1} = ϕ(x_{j}) (7.12.4) for each j. One can show that (7.12.5) {x_{j}}_{j=1}^{∞} is a Cauchy sequence in X, using (7.12.1). This implies that {x_{j}}_{j=1}^{∞} converges in X, because X is complete, and we put

x = lim_{j→∞} x_{j}. (7.12.6)

One can check that (7.12.3) holds, using (7.12.4), and the fact that ϕ is continuous on X.

Suppose now that X is a Banach space over the real or complex numbers, with norm ∥ · ∥_{X}, and with d_{X} equal to the metric associated to the norm. Put ψ(x) = x − ϕ(x) (7.12.7) for each x ∈ X, which defines a mapping from X into itself. More precisely, ψ is Lipschitz on X, with Lip(ψ) ≤ 1 + Lip(ϕ), (7.12.8) as in Section 2.1. In fact, (7.12.9) ψ is bilipschitz on X, as in the previous section. We would like to show that ψ(X) = X, (7.12.10) using the contraction mapping theorem.

Let y ∈ X be given, and put ϕ_{y}(x) = ϕ(x) + y (7.12.11) for each x ∈ X. Note that ϕ_{y} is a Lipschitz mapping from X into itself, with Lip(ϕ_{y}) = Lip(ϕ). (7.12.12) Thus (7.12.13) ϕ_{y} is a contraction on X, because ϕ is a contraction, by hypothesis. The contraction mapping theorem implies that ϕ_{y} has a unique fixed point x(y) in X, so that ϕ(x(y)) + y = ϕ_{y}(x(y)) = x(y). (7.12.14)

This implies that ψ(x(y)) = y, (7.12.15)

so that (7.12.10) holds.

If ϕ is a bounded linear mapping on X, then (7.12.1) is the same as saying that ∥ϕ∥_{op} < 1. (7.12.16)

This gives another way to look at the invertibility of ψ as a bounded linear mapping on X under these conditions.

7.13 The open mapping theorem

Let X, Y be metric spaces, or topological spaces. A mapping f from X into Y is said to be open at a point x ∈ X if for every open set U ⊆ X with x ∈ U there is an open set V ⊆ Y such that

f(x) ∈ V ⊆ f(U). (7.13.1)

We simply say that f is an open mapping if for every open set U ⊆ X,

(7.13.2) f(U) is an open set in Y.

If f is an open mapping, then f is clearly open at every x ∈ X. Conversely, if f is open at every x ∈ X, then it is easy to see that f is an open mapping.

Suppose for the moment that

(7.13.3) f is a one-to-one mapping from X onto Y.

In this case, f is an open mapping at x ∈ X if and only if

(7.13.4) f^{−1} is continuous at f^{−1}(x).

Similarly, f is an open mapping if and only if f^{−1} is continuous.

Suppose for the moment again that

X is compact (7.13.5)

and (7.13.6) Y is Hausdorff,

and note that (7.13.6) holds when Y is a metric space. If E ⊆ X is a closed set, then it is well known that E is compact, because X is compact. If f is a continuous mapping from X into Y, then it follows that f(E) is compact in Y. This implies that (7.13.7) f(E) is a closed set in Y,

because Y is Hausdorff. If f is also a one-to-one mapping from X onto Y, then we get that (7.13.8) f^{−1} is continuous under these conditions.

Now let V, W be vector spaces, both real or both complex, with norms ∥·∥_{V}, ∥ · ∥_{W}, respectively. Also let T be a linear mapping from V into W. If

(7.13.9) T is an open mapping at 0,

then it is easy to see that T is an open mapping at every point in V, so that

(7.13.10) T is an open mapping from V into W.

Note that T (V) = W (7.13.11)

in this case.

Let us use B_{V} (v, r), B_{W} (w, r) for the open balls in V, W centered at v ∈ V, w ∈ W with radius r > 0 with respect to the metrics associated to the norms, respectively. If T is open at 0, then there is an r_{1} > 0 such that

B_{W} (0, r_{1}) ⊆ T (B_{V} (0, 1)). (7.13.12)

Conversely, this condition implies that

B_{W} (0, r_{1} r) ⊆ T (B_{V} (0, r)) (7.13.13)

for every r > 0, because of linearity. It follows that T is an open mapping at 0.

Suppose that V, W are Banach spaces, and that T is a bounded linear mapping from V onto W. Under these conditions, the open mapping theorem states that (7.13.14) T is an open mapping.

It suffices to show that (7.13.12) holds for some r_{1} > 0, as in the preceding paragraph.

Observe that

⋃_{j=1}^{∞} T (B_{V} (0, j)) = W, (7.13.15)

because T maps ⋃_{j=1}^{∞} B_{V} (0, j) = V onto W, by hypothesis. In particular, this means that

⋃_{j=1}^{∞} \overline{T (B_{V} (0, j))} = W. (7.13.16)

Because W is a Banach space, we can use the Baire category theorem to get that (7.13.17) \overline{T (B_{V} (0, j))} has nonempty interior in W

for some j.

More precisely, this implies that \overline{T (B_{V} (0, j))} contains an open set that contains an element of T (B_{V} (0, j)). One can use this to get that

B_{W} (0, r_{0}) ⊆ \overline{T (B_{V} (0, 2 j))} (7.13.18) for some r_{0} > 0.

Using this and linearity, one can check that there is a positive real number C such that for each w ∈ W there is a v ∈ V such that

∥v∥_{V} ≤ C ∥w∥_{W} (7.13.19)

and ∥w − T (v)∥_{W} is as small as we like.

We can repeat the process and approximate w −T (v) in the same way. Continuing in this manner, we get an infinite series in V that converges absolutely and thus converges in V, because V is a Banach space. By construction, T sends the sum of this series to w. The norm of the sum of the series is less than or equal to the sum of the norms, which can be estimated by C ∥w∥_{W} plus an arbitrarily small positive real number.

7.14 Open mappings and quotient spaces

Let V be a vector space over the real or complex numbers with a norm ∥ · ∥_{V}. Also let V_{0} be a closed linear subspace of V, and let V/V_{0} be the corresponding quotient space, with the quotient norm ∥ · ∥_{V/V_{0}} defined as in Section 6.10. One can check that

(7.14.1) the natural quotient mapping q_{0} from V onto V/V_{0} is an open mapping,

with respect to the metrics associated to the norms. More precisely, q_{0} maps the open ball in V centered at 0 with radius r > 0 onto the open ball in V/V_{0} centered at 0 with radius r, because of the way that ∥ · ∥_{V/V_{0}} is defined.

Let W be another vector space over the real or complex numbers, as appropriate, and with a norm ∥ · ∥_{W}. If T_{0} is a linear mapping from V/V_{0} into W, then T = T_{0} ∘ q_{0} (7.14.2)

defines a linear mapping from V into W such that

V_{0} ⊆ ker T. (7.14.3)

If T_{0} is bounded as a linear mapping from V/V_{0} into W, then it follows that T is a bounded linear mapping from V into W, with

∥T ∥_{op,V W} ≤ ∥T_{0}∥_{op,(V/V_{0})W} . (7.14.4)

In fact, one can check that

∥T_{0}∥_{op,(V/V_{0})W} ≤ ∥T ∥_{op,V W} (7.14.5)

too. This means that

∥T_{0}∥_{op,(V/V_{0})W} = ∥T ∥_{op,V W} . (7.14.6)

7.15. THE CLOSED GRAPH THEOREM 167

Conversely, if T is a linear mapping from V into W that satisfies (7.14.3), then there is a unique linear mapping T_{0} from V/V_{0} into W such that (7.14.2) holds, by standard arguments. If T is a bounded linear mapping from V into W, then one can verify that (7.14.7) T_{0} is bounded on V/V_{0}, and that (7.14.5) holds. We also have that (7.14.4) holds, so that (7.14.6) holds, as before.

If T is any bounded linear mappng from V into W, then the kernel of T is a closed linear subspace of V. Thus we can take V_{0} = ker T (7.14.8) in the previous paragraphs. This leads to a bounded linear mapping T_{0} from V/V_{0} into W as in (7.14.2). In this case, we also have that ker T_{0} = {0}, (7.14.9) by construction.

Of course, T (V) = T_{0}(V/V_{0}). (7.14.10) Suppose that T maps V onto W, so that T_{0} maps V/V_{0} onto W. Suppose that V and W are Banach spaces as well, and remember that V/V_{0} is a Banach space too, as in Section 6.10. The open mapping theorem implies that (7.14.11) T_{0}^{−1} is bounded as a linear mapping from W onto V/V_{0}.

7.15 The closed graph theorem

Let X, Y be metric spaces, or topological spaces, and let f be a mapping from X into Y. The graph of f is the subset {(x, f(x)) : x ∈ X} (7.15.1) of the Cartesian product X ×Y. If f is continuous, and X, Y are metric spaces, then one can check that (7.15.2) the graph of f is a closed set in X × Y, with respect to a suitable metric on X × Y as in Section 5.11. This also works when X, Y are topological spaces, using the product topology on X × Y, if Y is Hausdorff.

Suppose for the moment that (7.15.3) X and Y are compact, so that X × Y is compact, by Tychonoff’s theorem. If the graph of f is a closed set in X × Y, then it is compact as well. Let p be the restriction of the obvious coordinate projection from X × Y onto X to the graph of f, so that

p((x, f(x))) = x (7.15.4)

for every x ∈ X. This is a one-to-one mapping from the graph of f onto X. It is easy to see that (7.15.5) p is continuous,

with respect to the restriction of a suitable metric on X × Y to the graph of f when X and Y are metric spaces, or with respect to the topology induced on the graph of f by the product topology on X ×Y when X and Y are topological spaces. It follows that (7.15.6) p^{−1} is continuous

when X, Y are metric spaces, and when X, Y are topological spaces and X is Hausdorff, as in Section 7.13. This implies that

(7.15.7) f is continuous

under these conditions.

Now let V, W be vector spaces, both real or both complex, and let T be a linear mapping from V into W. As before, the graph of T is the subset

{(v, T (v)) : v ∈ V } (7.15.8)

of the Cartesian product V × W. Remember that V × W may be considered as a vector space over the real or complex numbers, as appropriate, with respect to coordinatewise addition and scalar multiplication. It is easy to see that the graph of T is a linear subspace of V × W.

Let ∥ · ∥_{V}, ∥ · ∥_{W} be norms on V, W, respectively, which can be used to get a suitable norm on V × W, as in Section 5.12. If T is a bounded linear mapping from V into W, then

(7.15.9) the graph of T is a closed set in V × W

with respect to the metric associated to such a norm, as before.

The condition that the graph of T be a closed set in V × W is equivalent to saying that if

(7.15.10) {(v_{j}, T (v_{j}))}_{j=1}^{∞} is a sequence of elements of the graph of T that converges to (v, w) ∈ V × W,

then (v, w) is an element of the graph of T, so that

w = T (v). (7.15.11)

Thus the graph of T is a closed set in V × W if and only if for every sequence {v_{j}}_{j=1}^{∞} of elements of V that converges to v ∈ V with respect to the metric associated to ∥ · ∥_{V}, and for which {T (v_{j})}_{j=1}^{∞} converges to w ∈ W with respect to the metric associated to ∥ · ∥_{W}, we have that (7.15.11) holds. One can also use linearity of T to reduce to the case where v = 0.

Suppose that V and W are Banach spaces, and that the graph of T is a closed set in V × W. Under these econditions, the closed graph theorem states that (7.15.12) T is a bounded linear mapping from V into W.

To see this, note that

(7.15.13) V × W is a Banach space with respect to a suitable norm,

as in Section 5.12. It follows that

(7.15.14) the graph of T is a Banach space

with respect to the restriction of the norm on V × W to the graph of T, as in Section 1.6.

Let p be the restriction of the obvious coordinate projection from V × W onto V to the graph of T, as before. This is a bounded linear mapping from the graph of T onto V with respect to the appropriate norms. More precisely, p is a one-to-one mapping from the graph of T onto V, and the open mapping theorem implies that

(7.15.15) p^{−1} is a bounded linear mapping from V onto the graph of T.

This means that T is a bounded linear mapping from V into W.

Bilinear forms and linear mappings

8.1 Invertible mappings and product spaces

Let X, Y be metric spaces, or topological spaces, and consider the Cartesian products X × Y, Y × X. If X, Y are metric spaces, then we can get suitable metrics on X × Y, Y × X as in Section 5.11. We may as well use the same way to get these metrics on X × Y and Y × X, so that (x, y) ↦ (y, x) (8.1.1) is an isometry from X × Y onto Y × X. Otherwise, if X, Y are topological spaces, then we can use the product topologies on X × Y and Y × X, and (8.1.1) is a homeomorphism.

If f is a one-to-one mapping from X onto Y, then the graph {(y, f^{−1}(y)) : y ∈ Y } (8.1.2) of f^{−1} in Y × X is the same as {(f(x), x) : x ∈ X}. (8.1.3) This corresponds to the graph (7.15.1) of f under the mapping (8.1.1). In particular, this means that the graph of f is a closed set in X × Y if and only if (8.1.4) the graph of f^{−1} is a closed set in Y × X.

Let V, W be vector spaces, both real or both complex, so that V × W and W ×V may be considered as real or complex vector spaces, as appropriate, with respect to coordinatewise addition and scalar multiplication. Of course, (v, w) ↦ (w, v) (8.1.5) defines an isomorphism from V × W onto V × W, as real or complex vector spaces, as appropriate. If T is a one-to-one linear mapping from V onto W, then the graph {(w, T^{−1}(w)) : w ∈ W } (8.1.6)

of T^{−1} is a linear subspace of W × V. This is the same as

{(T (v), v) : v ∈ V }. (8.1.7)

This corresponds to the graph (7.15.8) of T under the mapping (8.1.5), as before.

Let ∥ · ∥_{V}. ∥ · ∥_{W} be norms on V, W, respectively, which can be used to get suitable norms on V × W and W × V, as in Section 5.12. We may as well use the same way to get norms on each of V × W and W × V, so that (8.1.5) is an isometric linear mapping from V × W onto W × V. If T is a bounded linear mapping from V into W, then the graph of T is a closed set in V × W, as in Section 7.15.

If T is a one-to-one bounded linear mapping from V onto W, then it follows that (8.1.8) the graph of T^{−1} is a closed set in W × V.

If V, W are Banach spaces, then the conclusion of the closed graph theorem implies that T^{−1} is a bounded linear mapping from W onto V. Of course, this is the same as the version of the open mapping theorem that was used to get the closed graph theorem.

Let V be a vector space over the real or complex numbers again, and let V_{1}, V_{2} be linear subspaces of V. Thus V_{1} × V_{2} may be considered as a vector space over the real or complex numbers, as appropriate, as well, and

(v_{1}, v_{2}) ↦ v_{1} + v_{2} (8.1.9)

defines a linear mapping from V_{1} × V_{2} into V. This mapping sends V_{1} × V_{2} onto

V_{1} + V_{2} = {v_{1} + v_{2} : v_{1} ∈ V_{2}, v_{2} ∈ V_{2}}, (8.1.10)

and the kernel of this linear mapping is equal to

{(v_{1}, v_{2}) ∈ V_{1} × V_{2} : v_{1} = −v_{2}}. (8.1.11)

In particular, the kernel of (8.1.9) is equal to {0} if and only if

V_{1} ∩ V_{2} = {0}. (8.1.12)

Let ∥ · ∥_{V} be a norm on V again, whose restrictions to V_{1}, V_{2} define norms on those spaces. We can use these norms to get a suitable norm on V_{1} × V_{2}, as in Section 5.12. It is easy to see that (8.1.9) is a bounded linear mapping from V_{1} × V_{2} with respect to such a norm. Suppose that (8.1.10) is equal to V, and that (8.1.12) holds, so that (8.1.9) is a one-to-one mapping from V_{1} × V_{2} onto V. We would like to have conditions under which

the inverse of (8.1.9) is bounded as a linear mapping

(8.1.13)

from V onto V_{1} × V_{2}.

172 CHAPTER 8. BILINEAR FORMS AND LINEAR MAPPINGS

A necessary condition for this to hold is that (8.1.14) V_{1}, V_{2} be closed sets in V, with respect to the metric associated to ∥ · ∥_{V}, because V_{1} × {0}, {0} × V_{2} are closed sets in V_{1} × V_{2}. Suppose that this condition holds, and that V is a Banach space. This implies that V_{1}, V_{2} are Banach spaces with respect to the restrictions of ∥ · ∥_{V} to those spaces, as in Section 1.6. It follows that V_{1} × V_{2} is a Banach space as well, as in Section 5.12. In this case, we get (8.1.13) from the open mapping theorem.

8.2 Projections on vector spaces

Let V be a vector space over the real or complex numbers. A linear mapping P from V into itself is said to be a projection if P ∘ P = P (8.2.1) on V, so that P (P (v)) = P (v) (8.2.2) for every v ∈ V. This implies that (I − P) ∘ P = P ∘ (I − P) = 0. (8.2.3) where I = I_{V} is the identity map on V. It follows that (I − P) ∘ (I − P) = I − P, (8.2.4) so that I − P is a projection on V as well.

Observe that P (V) ⊆ ker(I − P) (8.2.5) and (I − P)(V) ⊆ ker P, (8.2.6) by (8.2.3). The opposite inclusions can be verfiied directly. This means that P (V) = ker(I − P) (8.2.7) and (I − P)(V) = ker P. (8.2.8)

One can use (8.2.7) and (8.2.8) to get that (ker P) + P (V) = V (8.2.9) and (ker P) ∩ P (V) = {0}. (8.2.10)

We may consider (ker P) × P (V) (8.2.11)

as a vector space over the real or complex numbers, as appropriate, with respect to coordinatewise addition and scalar multiplication, as usual. Using (8.2.9) and (8.2.10), we get that (v, w) ↦ v + w (8.2.12)

defines a one-to-one linear mapping from (8.2.11) onto V, as in the previous section. The inverse mapping is given by

u ↦ ((I − P)(u), P (u)). (8.2.13)

Let ∥·∥_{V} be a norm on V, whose restrictions to ker P, P (V) define norms on those spaces. This can be used to get a suitable norm on (8.2.11), as in Section 5.12. Note that (8.2.12) is bounded as a linear mapping from (8.2.11) into V, as in the previous section. The inverse mapping is bounded if and only if

(8.2.14) P is bounded on V,

because it can be given as in (8.2.13). Of course, if P is bounded on V, then I − P is bounded on V, and (8.2.7), (8.2.8) are closed sets in V.

If V_{1}, V_{2} are linear subspaces of V such that V_{1} + V_{2} = V and V_{1} ∩ V_{2} = {0}, then (8.2.12) defines a one-to-one linear mapping from V_{1} ×V_{2} onto V, as in the previous section. In this case, it is easy to see that there is a unique projection P on V with ker P = V_{1} (8.2.15)

and P (V) = V_{2}. (8.2.16)

If V is a Banach space, and V_{1}, V_{2} are closed sets in V, then the inverse of (8.2.12) as a linear mapping from V_{1} × V_{2} onto V is bounded, with respect to a suitable norm on V_{1} × V_{2}, as before. This means that (8.2.14) holds, which corresponds to a simplification of part (b) of Theorem 5.16 on p126 of [286]. More precisely, the proof in [286] uses the closed graph theorem a bit more directly.

8.3 Orthogonal projections

Let (V, ⟨·, ·⟩_{V}) be an inner product space over the real or complex numbers, with corresponding norm ∥ · ∥_{V}. Also let W be a linear subspace of V, and suppose that for each v ∈ V there is an element P_{W} (v) of W such that

⟨v − P_{W} (v), w⟩_{V} = 0 (8.3.1)

for every w ∈ W. Remember that P_{W} (v) is uniquely determined by these properties, as in Section 2.12. It is easy to see that P_{W} is linear, using uniqueness. If v ∈ W, then we have that P_{W} (v) = v, (8.3.2) by uniqueness.

It follows that P_{W} is a projection on V, as in the previous section. This is the orthogonal projection of V onto W. If W has finite dimension, then one can use an orthonormal basis for W to get P_{W}, as in Section 2.12. If V is a Hilbert space, and W is a closed linear subspace of V, then one can get P_{W} as in Sections 2.14 or 2.15.

Note that P_{W} is a bounded linear mapping from V into itself with respect to ∥ · ∥_{V}, with operator norm less than or equal to 1, as in Section 2.12. More precisely, ∥P_{W} ∥_{op} = 1 (8.3.3)

when W ≠ {0}, and otherwise P_{W} = 0. We also have that

W = P_{W} (V) = ker(I − P_{W}), (8.3.4)

as in the previous section. This implies that

(8.3.5) W is a closed set in V,

with respect to the metric associated to ∥ · ∥_{V}, because I − P_{W} is a bounded linear mapping from V into itself, as before.

Remember that the orthogonal complement of W in V is the closed linear subspace of V defined by

(8.3.6) W^{⊥} = {u ∈ V : ⟨u, w⟩_{V} = 0 for every w ∈ W },

as in Section 2.15. Note that

W^{⊥} = ker P_{W} , (8.3.7)

as before. This implies that

(I − P_{W})(V) = W^{⊥}, (8.3.8)

as in the previous section. Of course, (8.3.1) is the same as saying that

(I − P_{W})(V) ⊆ W^{⊥}, (8.3.9)

and the opposite inclusion can be obtained from (8.3.7).

Put P_{W^{⊥}} = I − P_{W} . (8.3.10)

If v ∈ V, then P_{W^{⊥}}(v) ∈ W^{⊥}, and

⟨v − P_{W^{⊥}}(v), u⟩_{V} = ⟨P_{W} (v), u⟩_{V} = 0 (8.3.11)

for every u ∈ W^{⊥}, because P_{W} (v) ∈ W. These properties determine P_{W^{⊥}}(v) uniquely, as in Section 2.12 again. This means that P_{W^{⊥}} is the orthogonal projection of V onto W^{⊥}, as before.

Let V_{1}, V_{2} be linear subspaces of V that are orthogonal to each other with respect to ⟨·, ·⟩_{V} on V, which is to say that

⟨v_{1}, v_{2}⟩_{V} = 0 (8.3.12)

for every v_{1} ∈ V_{1} and v_{2} ∈ V_{2}. Equivalently, this means that

V_{1} ⊆ V_{2}^{⊥}, V_{2} ⊆ V_{1}^{⊥}. (8.3.13)

In particular, this implies that V_{1} ∩ V_{2} = {0}. Suppose that V_{1} + V_{2} = V, so that every element of V can be expressed in a unique way as a sum of elements of V_{1} and V_{2}. This leads to a unique linear mapping P from V into itself such that P (v_{1} + v_{2}) = v_{2} (8.3.14)

for every v_{1} ∈ V_{1} and V_{2} ∈ V_{2}.

It is easy to see that P is the same as the orthogonal projection P_{V_{2}} of V onto V_{2} under these conditions. Similarly,

(I − P)(v_{1} + v_{2}) = v_{1} (8.3.15)

for every v_{1} ∈ V_{1} and v_{2} ∈ V_{2}, and I − P is the orthogonal projection P_{V_{1}} of V onto V_{1}. We also have that

V_{1} = V_{2}^{⊥}, V_{2} = V_{1}^{⊥} (8.3.16)

in this case, as in Section 2.15. In particular, this implies that V_{1}, V_{2} are closed sets in V, as before. Remember that P is bounded as a linear mapping from V into itself, as mentioned earlier.

8.4 Projections and distances

Let V be a vector space over the real or complex numbers, and let V_{1}, V_{2} be linear subspaces of V such that V_{1} ∩ V_{2} = {0} and V_{1} + V_{2} = V. This means that every element of V can be expressed in a unique way as v_{1} + v_{2} for some v_{1} ∈ V_{1} and v_{2} ∈ V_{2}, which leads to a unique linear mapping P from V into itself that sends v_{1} +v_{2} to v_{2}. This is the unique projection on V with kernel V_{1} that maps V onto V_{2}, as in Section 8.2. Similarly, I −P is the unique projection on V with kernel V_{2} that maps V onto V_{1}, and which sends v_{1} + v_{2} to v_{1} for every v_{1} ∈ V_{1}, v_{2} ∈ V_{2}. Any projection on V may be considered in this way, as before.

Let ∥ · ∥_{V} be a norm on V. Observe that P is a bounded linear mapping from V into itself with respect to ∥ · ∥_{V} if and only if there is a nonnegative real number C_{2} such that ∥v_{2}∥_{V} ≤ C_{2} ∥v_{1} + v_{2}∥_{V} (8.4.1)

for all v_{1} ∈ V_{1} and v_{2} ∈ V_{2}. Similarly, I − P is bounded on V if and only if there is a C_{1} ≥ 0 such that

∥v_{1}∥_{V} ≤ C_{1} ∥v_{1} + v_{2}∥_{V} (8.4.2)

176 CHAPTER 8. BILINEAR FORMS AND LINEAR MAPPINGS for all v_{1} ∈ V_{1} and v_{2} ∈ V_{2}. Of course, P is bounded on V if and only if I − P is bounded on V.

Note that (8.4.1) is the same as saying that ∥v_{2}∥_{V} ≤ C_{2} ∥v_{2} − v_{1}∥_{V} (8.4.3) for all v_{1} ∈ V_{1} and v_{2} ∈ V_{2}. Equivalently, this means that ∥v_{2}∥_{V} ≤ C_{2} dist(v_{2}, V_{1}) (8.4.4) for all v_{2} ∈ V_{2}, where dist(v_{2}, V_{1}) is as in Section 2.11, using the metric on V associated to ∥ · ∥_{V}. Similarly, (8.4.2) is the same as saying that ∥v_{1}∥_{V} ≤ C_{1} ∥v_{1} − v_{2}∥_{V} (8.4.5) for all v_{1} ∈ V_{1} and v_{2} ∈ V_{2}. This means that ∥v_{1}∥_{V} ≤ C_{1} dist(v_{1}, V_{2}) (8.4.6) for all v_{1} ∈ V_{1}.

Of course, dist(v, V_{1}) ≤ ∥v∥_{V} (8.4.7) for every v ∈ V, because 0 ∈ V_{2}. If (8.4.4) holds with C_{2} = 1, then we get that ∥v_{2}∥_{V} = dist(v_{2}, V_{1}) (8.4.8) for all v_{2} ∈ V_{2}. Similarly, dist(v, V_{2}) ≤ ∥v∥_{V} (8.4.9) for every v ∈ V. If (8.4.6) holds with C_{1} = 1, then ∥v_{1}∥_{V} = dist(v_{1}, V_{2}) (8.4.10) for all v_{1} ∈ V_{1}.

If v ∈ V, then (8.4.11) dist(v, V_{1}) = dist(P (v), V_{1}), because v − P (v) ∈ V_{1}, by construction. Thus (8.4.4) implies that ∥P (v)∥_{V} ≤ C_{2} dist(v, V_{1}) (8.4.12) for every v ∈ V, because P (v) ∈ V_{2}. Conversely, this implies (8.4.4), because P (v_{2}) = v_{2} for every v_{2} ∈ V_{2}. Similarly, (8.4.13) dist(v, V_{2}) = dist(v − P (v), V_{2}) for every v ∈ V, because P (v) ∈ V_{2}. It follows that (8.4.6) holds if and only if ∥v − P (v)∥_{V} ≤ C_{1} dist(v, V_{2}) (8.4.14) for every v ∈ V, because v − P (v) ∈ V_{1}.

Observe that

(8.4.15) dist(v, V_{1}) = dist(P (v), V_{1}) ≤ ∥P (v)∥_{V}

for every v ∈ V, using (8.4.11) in the first step, and the analogue of (8.4.7) with P (v) in place of v in the second step. If (8.4.12) holds with C_{2} = 1, then we have that ∥P (v)∥_{V} = dist(v, V_{1}) (8.4.16)

for every v ∈ V. Similarly,

(8.4.17) dist(v, V_{2}) = dist(v − P (v), V_{2}) ≤ ∥v − P (v)∥_{V}

for every v ∈ V. If (8.4.14) holds with C_{1} = 1, then

∥v − P (v)∥_{V} = dist(v, V_{2}) (8.4.18)

for every v ∈ V.

Suppose for the moment that ⟨·, ·⟩_{V} is an inner product on V, and that ∥·∥_{V} is the corresponding norm on V. If V_{1}, V_{2} are orthogonal to each other with respect to ⟨·, ·⟩_{V}, then (8.4.1) and (8.4.2) hold with C_{1} = C_{2} = 1. Conversely, if (8.4.1) holds with C_{2} = 1, or (8.4.2) holds with C_{1} = 1, then one can check that V_{1} and V_{2} are orthogonal to each other with respect to ⟨·, ·⟩_{V}. This is similar to an argument mentioned in Section 2.15.

Let v_{0} be any nonzero element of V, and let λ_{0} be a linear functional on V such that λ_{0}(v_{0}) = 1. (8.4.19)

Put P_{0}(v) = λ_{0}(v) v_{0} (8.4.20)

for each v ∈ V, which defines a linear mapping from V into itself. Note that

P_{0}(v_{0}) = v_{0}, (8.4.21)

so that P_{0} is a projection from V onto the linear subspace spanned by v_{0}. We also have that ker P_{0} = ker λ_{0}, (8.4.22)

by construction.

Let ∥ · ∥_{V} be a norm on V again. If λ_{0} is a bounded linear functional on V with respect to ∥ · ∥_{V}, then P_{0} is bounded as a linear mapping from V into itself, with ∥P_{0}∥_{op} = ∥λ_{0}∥_{V′} ∥v_{0}∥_{V} . (8.4.23)

The Hahn–Banach theorem implies that there is such a λ_{0} for which the right side of (8.4.23) is equal to 1, as in Section 3.10.

178 CHAPTER 8. BILINEAR FORMS AND LINEAR MAPPINGS

8.5 Symmetric bilinear forms

Let V be a vector space over the real or complex numbers, and let b be a bilinear form on V, which is to say a bilinear mapping from V × V into R or C, as appropriate. We say that b is symmetric on V if b(v, w) = b(w, v) (8.5.1) for all v, w ∈ V, and that b is antisymmetric on V if b(v, w) = −b(w, v) (8.5.2) for all v, w ∈ V. If b is any bilinear form on V, then (1/2) (b(v, w) + b(w, v)) (8.5.3) is a symmetric bilinear form on V, and (1/2) (b(v, w) − b(w, v)) (8.5.4) is an antisymmetric bilinear form on V. Of course, b(v, w) is the same as the sum of (8.5.3) and (8.5.4).

If b is a bilinear form on V, then b(v + w, v + w) = b(v, v) + b(v, w) + b(w, v) + b(w, w) (8.5.5) for all v, w ∈ V. Equivalently, (1/2) (b(v, w) + b(w, v)) = (1/2) (b(v + w, v + w) − b(v, v) − b(w, w)), (8.5.6) which is a polarization identity. If b is antisymmetric on V, then b(v, v) = 0 (8.5.7) for every v ∈ V. Conversely, this condition implies that b is antisymmetric on V, because of (8.5.5). If b is symmetric on V, then (8.5.6) shows that b is determined by the values of b(u, u), u ∈ V.

Suppose now that V is a complex vector space, and that b is a sesquilinear form on V, which is to say a sesquilinear mapping from V × V into C. Note that \overline{b(w, v)} (8.5.8) is a sesquilinear form on V as well. If b(v, w) = \overline{b(w, v)} (8.5.9) for every v, w ∈ V, then b is said to be Hermitian symmetric, or a Hermitian form on V. If b(v, w) = −\overline{b(w, v)} (8.5.10) for every v, w ∈ V, then one might say that b is Hermitian antisymmetric on V, which is equivalent to i b being Hermitian symmetric on V. If b is any sesquilinear form on V, then

(1/2) (b(v, w) + \overline{b(w, v)}) (8.5.11)

is Hermitian symmetric on V,

(1/2) (b(v, w) − \overline{b(w, v)}) (8.5.12)

is Hermitian antisymmetric on V, and b(v, w) is the same as the sum of (8.5.11) and (8.5.12).

If b is a sesquilinear form on V, then (8.5.5) holds, as before. We also have that b(v + i w, v + i w) = b(v, v) − i b(v, w) + i b(w, v) + b(w, w) (8.5.13)

for all v, w ∈ V, which is another polarization identity. Using (8.5.5) and (8.5.13), we get that b can be obtained from the values of b(u, u), u ∈ V.

If b is Hermitian symmetric on V, then

b(v, v) ∈ R (8.5.14)

for every v ∈ V. Conversely, one can check that this condition implies that b is Hermitian symmetric on V, using (8.5.5) and (8.5.13). Alternatively, observe that if v = w, then (8.5.11) and (8.5.12) are equal to the real and imaginary parts of b(v, v), respectively. If (8.5.14) holds for each v ∈ V, then (8.5.12) is equal to 0 when v = w, which implies that (8.5.12) is equal to 0 for all v, w ∈ V, as in the preceding paragraph. As another approach, (8.5.14) implies that

(8.5.15) b(v, w) and \overline{b(v, w)} are the same

when v = w. It follows that (8.5.15) holds for all v, w ∈ V, because b(v, w) and \overline{b(w, v)} are both sesquilinear forms on V, as before. This is the argument used in the proof of Theorem 3 on p13 of [140].

8.6 Self-adjoint linear operators

Let (V, ⟨·, ·⟩) be an inner product space over the real or complex numbers, with the corresponding norm ∥ · ∥. A linear mapping T from V into itself is said to be symmetric with respect to ⟨·, ·⟩ if

⟨T (v), w⟩ = ⟨v, T (w)⟩ (8.6.1)

for every v, w ∈ V. If ⟨T (v), w⟩ = −⟨v, T (w)⟩ (8.6.2)

for every v, w ∈ V, then T is said to be antisymmetric on V with respect to ⟨·, ·⟩. In the complex case, this is the same as saying that i T is symmetric on V.

Suppose for the moment that V is a Hilbert space, and that T is a bounded linear mapping from V into itself. If w ∈ V, then there is a unique element T^{∗}(w) of V such that ⟨T (v), w⟩ = ⟨v, T^{∗}(w)⟩ (8.6.3)

for every v ∈ V, as in Section 3.5. This defines a bounded linear mapping from V into itself, as before. Under these conditions, (8.6.1) is the same as saying that T^{∗} = T. (8.6.4)

This means that T is self-adjoint on V, or equivalently that T is Hermitian when V is complex. Similarly, (8.6.2) is the same as saying that

T^{∗} = −T, (8.6.5)

so that T is anti-self-adjoint on V. In the complex case, this happens exactly when i T is self-adjoint on V. Of course, the Hilbert space adjoint defines an algebra involution on BL(V), so that this terminology corresponds to that in Section 7.5.

If V is any inner product space, and T is any linear mapping from V into itself, then b_{T} (v, w) = ⟨T (v), w⟩ (8.6.6)

defines a bilinear form on V in the real case, and a sesquilinear form on V in the complex case. Remember that T is bounded as a linear mapping from V into itself if and only if b_{T} is bounded as a bilinear or sesquilinear form on V, as appropriate, as in Section 5.15.

If V is a Hilbert space, and T is symmetric or antisymmetric on V, then T is bounded on V. This corresponds to a theorem of Hellinger and Toeplitz on p110 of [286]. More precisely, if V is any inner product space, and T is symmetric or antisymmetric on V, then one can check that

(8.6.7) the graph of T is a closed set in V × V.

If V is a Hilbert space, then the closed graph theorem implies that T is bounded, as in Section 7.15.

Alternatively, if V is any inner product space again, and T is symmetric or antisymmetric on V, then one can verify that

(8.6.8) b_{T} (v, w) is separately continuous in each of v and w.

If V is a Hilbert space, then it follows that b_{T} is bounded as a bilinear or sesquilinear form on V, as in Section 5.14. In the complex case, we may consider b_{T} as a real-bilinear mapping from V × V into C, as in Section 5.15.

Another approach to the boundedness of T under these conditions will be mentioned in the next section.

If V is a real inner product space, then

b_{T} (w, v) = ⟨T (w), v⟩ = ⟨v, T (w)⟩ (8.6.9) for every v, w ∈ V. This implies that T is symmetric or antisymmetric on V if and only if b_{T} is symmetric or antisymmetric as a bilinear form on V, respectively. In particular, T is antisymmetric on V if and only if

⟨T (v), v⟩ = b_{T} (v, v) = 0 (8.6.10)

for every v ∈ V, as in the previous section.

Similarly, if V is a complex inner product space, then

\overline{b_{T} (w, v)} = \overline{⟨T (w), v⟩} = ⟨v, T (w)⟩ (8.6.11)

for every v, w ∈ V. It follows that T is symmetric or antisymmetric on V if and only if b_{T} is Hermitian symmetric or antisymmetric as a sesquilinear form on V, respectively. This means that T is symmetric on V if and only if

⟨T (v), v⟩ = b_{T} (v, v) ∈ R (8.6.12)

for every v ∈ V, as in the previous section. This corresponds to part (a) of Exercise (2) on p45 of [11], and to the first part of Theorem 2 on p41 of [140].

Suppose that P_{W} is the orthogonal projection of V onto a linear subspace W, as in Section 8.3. If v_{1}, v_{2} ∈ V, then

⟨P_{W} (v_{1}), v_{2}⟩ = ⟨P_{W} (v_{1}), P_{W} (v_{2})⟩ = ⟨v_{1}, P_{W} (v_{2})⟩, (8.6.13)

using (8.3.1) in both steps. This implies that P_{W} is symmetric with respect to ⟨·, ·⟩_{V} on V. If V is a Hilbert space, then this means that P_{W} is self-adjoint on V.

Conversely, let P be a projection on V, as in Section 8.2. Suppose that P is symmetric with respect to ⟨·, ·⟩ on V, so that

⟨P (v), w⟩ = ⟨v, P (w)⟩ (8.6.14)

for every v, w ∈ V. This implies that

⟨v, P (w)⟩ = 0 (8.6.15)

when P (v) = 0, so that ker P and P (V) are orthogonal to each other in V. It follows that P is the orthogonal projection of V onto P (V), as in Section 8.3.

8.7 Some related continuity arguments

Let V, W be vector spaces, both real or both complex, and with norms ∥ · ∥_{V}, ∥·∥_{W}, respectively. Also let T be a linear mapping from V into W, and remember that T is bounded if and only if it is continuous at 0, as in Section 2.2. It is well known that this happens if and only if for every sequence {v_{j}}_{j=1}^{∞} of elements of V that converges to 0 with respect to the metric associated to ∥ · ∥_{V}, we have that

lim_{j→∞} T (v_{j}) = 0 (8.7.1)

with respect to the metric on W associated to ∥ · ∥_{W}.

Of course, (8.7.1) implies that

(8.7.2) {T (v_{j})}_{j=1}^{∞} is bounded in W,

with respect to the metric associated to ∥ · ∥_{W}. In fact, it is well known that T is continuous at 0 when (8.7.2) holds for all sequences {v_{j}}_{j=1}^{∞} of elements of V that converges to 0. This corresponds to a simplification of part of Theorem 1.32 on p23 of [286].

Remember that T is bounded as a linear mapping from V into W when

∥T (v)∥_{W} (8.7.3)

is bounded on a ball in V of positive radius centered at 0, as in Section 2.2. If this does not happen, then one can get a sequence {v_{j}}_{j=1}^{∞} of elements of V that converges to 0 such that (8.7.2) does not hold.

Alternatively, let {u_{j}}_{j=1}^{∞} be a sequence of elements of V that converges to 0. One can find a sequence {r_{j}}_{j=1}^{∞} of positive real numbers such that r_{j} → ∞ as j → ∞ and

lim_{j→∞} r_{j} u_{j} = 0 (8.7.4)

in V. This is a simplification of part (b) of Theorem 1.28 on p21 of [286]. Here we can take r_{j} = ∥u_{j}∥_{V}^{−1/2} (8.7.5)

when u_{j} ≠ 0, and r_{j} = j otherwise. If (8.7.2) holds with v_{j} = r_{j} u_{j}, then it follows that T (u_{j}) = r_{j}^{−1} T (r_{j} u_{j}) → 0 as j → ∞ (8.7.6)

in W.

Note that (8.7.2) holds when

(8.7.7) {T (v_{j})}_{j=1}^{∞} converges to 0 weakly in W,

as in Section 4.9. If (8.7.7) holds for every sequence {v_{j}}_{j=1}^{∞} of elements of V that converges to 0, then it follows that T is continuous at 0.

Let Z be a subset of the dual space W′ of bounded linear functionals on W, and suppose that (8.7.8) Z separates points in W.

This means that for each nonzero w ∈ W there is a μ ∈ Z such that

μ(w) ≠ 0. (8.7.9)

Suppose also that for every μ ∈ Z,

(8.7.10) μ ∘ T is a bounded linear functional on V,

so that μ ∘ T is continuous on V. Under these conditions, one can check that

(8.7.11) the graph of T is a closed set in V × W.

If V and W are Banach spaces, then it follows that T is continuous, by the closed graph theorem, as in Section 7.15.

This basically corresponds to a simplification of Theorem 5.1 on p110 of [286], and its proof. That result considers other types of topological vector spaces, and all continuous linear functionals on W. However, the proof works as well for sets of continuous linear functionals on W that separate points.

More precisely, (8.7.10) is the same as saying that for every sequence {v_{j}}_{j=1}^{∞} of elements of V that converges to 0,

lim_{j→∞} μ(T (v_{j})) = 0. (8.7.12)

If we take Z = W′ here, then (8.7.12) reduces to (8.7.7), and we can use the earlier argument, without asking V or W to be complete.

Let (V, ⟨·, ·⟩) be an inner product space over the real or complex numbers, and let T be a linear mapping from V itself. Suppose that T is symmetric or antisymmetric with respect to ⟨·, ·⟩. If {v_{j}}_{j=1}^{∞} is a sequence of elements of V that converges to 0 with respect to the metric associated to the corresponding norm on V, then

⟨T (v_{j}), w⟩ = ±⟨v_{j}, T (w)⟩ → 0 as j → ∞. (8.7.13)

This corresponds to (8.7.12), where W = V, and Z is the linear subspace of V′ = W′ consisting of linear functionals of the form μ_{w}(v) = ⟨v, w⟩, w ∈ V. Note that Z separates points in V.

If V is a Hilbert space, then it follows that T is continuous, as before. More precisely, every continuous linear functional on V is in Z in this case, so that (8.7.13) implies (8.7.7). Otherwise, one can use the closed graph theorem, as before.

8.8 Bounded symmetric bilinear forms

Let V be a vector space over the real or complex numbers, and let b be a symmetric bilinear form on V. Observe that

b(v, w) = (1/4) (b(v + w, v + w) − b(v − w, v − w)) (8.8.1)

for every v, w ∈ V. This is another type of polarization identity, which can also be used to show that b is determined by the values of b(u, u), u ∈ V.

Let ⟨·, ·⟩_{V} be an inner product on V, and let ∥ · ∥_{V} be the corresponding norm on V. Suppose that |b(u, u)| ≤ C ∥u∥_{V}^{2} (8.8.2)

for some C ≥ 0 and every u ∈ V. If v, w ∈ V, then we get that

|b(v, w)| ≤ (C/4) (∥v + w∥_{V}^{2} + ∥v − w∥_{V}^{2} ) = (C/2) (∥v∥_{V}^{2} + ∥w∥_{V}^{2} ), (8.8.3)

using (8.8.1) in the first step, and the parallelogram law in the second step. This implies that |b(v, w)| ≤ C (8.8.4) when ∥v∥_{V} , ∥w∥_{V} ≤ 1. It follows that

|b(v, w)| ≤ C ∥v∥_{V} ∥w∥_{V} (8.8.5)

for every v, w ∈ V, by standard arguments. Of course, (8.8.5) implies (8.8.2). This corresponds to the analogue of Theorem 3 on p33 of [140] in the real case.

Suppose now that V is a complex vector space, and that b is a sesquilinear form on V. One can check that

b(v, w) = (1/4) ∑_{l=0}^{3} i^{l} b(v + i^{l} w, v + i^{l} w) (8.8.6)

for every v, w ∈ V. This is another polarization identity, which corresponds to Exercise (1) on p45 of [11], and to Theorem 1 on p12 of [140]. This can be used to show that b is determined by the values b(u, u), u ∈ V, as in Theorem 2 on p13 of [140].

If b is Hermitian symmetric on V, then

b(u, u) ∈ R (8.8.7)

for every u ∈ V, as in Section 8.5. If (8.8.7) holds for every u ∈ V, then (8.8.6) implies that

Re b(v, w) = (1/4) (b(v + w, v + w) − b(v − w, v − w)) (8.8.8)

and Im b(v, w) = (1/4) (b(v + i w, v + i w) − b(v − i w, v − i w)) (8.8.9)

for every v, w ∈ V. This can be used to get that b is Hermitian symmetric on V. Alternatively, if (8.8.7) holds, then b(v, w) and \overline{b(w, v)} are sesquilinear forms on V that agree when v = w, which implies that they agree for all v, w ∈ V, as in the preceding paragraph. This corresponds to the proof of Theorem 3 on p13 of [140].

Let ⟨·, ⟩_{V} be an inner product on V again, with corresponding norm ∥ · ∥_{V}, and suppose that (8.8.2) holds for some C ≥ 0 and every u ∈ V. If b is Hermitian symmetric on V, then we can use (8.8.8) to get that

(8.8.10)  | Re b(v, w)| ≤ (C/4) (∥v + w∥_{V}^{2} + ∥v − w∥_{V}^{2} )
= (C/2) (∥v∥_{V}^{2} + ∥w∥_{V}^{2} )

for every v, w ∈ V. If ∥v∥_{V} , ∥w∥_{V} ≤ 1, then it follows that

| Re b(v, w)| ≤ C. (8.8.11)

If a is a complex number with |a| ≤ 1, then we get that

| Re(a b(v, w))| = | Re b(a v, w)| ≤ C (8.8.12)

when ∥v∥_{V} , ∥w∥_{V} ≤ 1. This implies that (8.8.4) holds. One can use this to get (8.8.5), as before. This corresponds to Theorem 3 on p33 of [140].

If b is not necessarily Hermitian symmetric on V, then we can still use (8.8.6) to get that

|b(v, w)| ≤ (1/4) ∑_{l=0}^{3} |b(v + i^{l} w, v + i^{l} w)| (8.8.13)

for every v, w ∈ V. Thus (8.8.2) implies that

|b(v, w)| ≤ (C/4) ∑_{l=0}^{3} ∥v + i^{l} w∥_{V}^{2} (8.8.14)

for every v, w ∈ V. The sum on the right is the same as

(∥v + w∥_{V}^{2} + ∥v − w∥_{V}^{2} ) + (∥v + i w∥_{V}^{2} + ∥v − i w∥_{V}^{2} ), (8.8.15)

which is equal to 4 ∥v∥_{V}^{2} + 4 ∥w∥_{V}^{2} , (8.8.16)

by the parallelogram law. Combining this with (8.8.14), we get that

|b(v, w)| ≤ C (∥v∥_{V}^{2} + ∥w∥_{V}^{2} ) (8.8.17)

for every v, w ∈ V.

It follows that |b(v, w)| ≤ 2 C (8.8.18)

when ∥v∥_{V} , ∥w∥_{V} ≤ 1. As before, one can use this and standard arguments to get that |b(v, w)| ≤ 2 C ∥v∥_{V} ∥w∥_{V} (8.8.19)

for every v, w ∈ V. This corresponds to Theorem 2 on p33 of [140].

8.9 Bounded symmetric linear mappings

Let (V, ⟨·, ·⟩_{V}) be an inner product space over the real or complex numbers, with the corresponding norm ∥ · ∥_{V}. Also let T be a bounded linear mapping from V into itself with respect to ∥ · ∥_{V} that is symmetric with respect to ⟨·, ·⟩_{V}, as in Section 8.6. Thus b_{T} (v, w) = ⟨T (v), w⟩_{V} (8.9.1)

is a bounded symmetric bilinear form on V in the real case, and a bounded Hermitian symmetric sesquilinear form on V in the complex case.

Suppose that V ≠ {0}, and consider

sup{|b_{T} (u, u)| : u ∈ V, ∥u∥_{V} = 1}. (8.9.2)

This is the same as the smallest nonnegative real number C such that (8.8.2) holds, with b = b_{T}. Similarly,

sup{|b_{T} (v, w)| : v, w ∈ V, ∥v∥_{V} = ∥w∥_{V} = 1} (8.9.3) is the same as the smallest nonnegative real number C such that (8.8.5) holds, with b = b_{T}. Clearly (8.9.2) is less than or equal to (8.9.3), and in fact they are equal to each other, as in the previous section.

We also have that (8.9.3) is equal to the operator norm of T on V. This is very similar to a remark at the beginning of Section 3.5. It follows that (8.9.2) is equal to ∥T ∥_{op} too. This corresponds to the second part of Theorem 2 on p41 of [140].

Put α(T) = inf{⟨T (v), v⟩_{V} : v ∈ V, ∥v∥_{V} = 1} (8.9.4)

and β(T) = sup{⟨T (v), v⟩_{V} : v ∈ V, ∥v∥_{V} = 1}. (8.9.5)

These are real numbers with α(T) ≤ β(T) and

max(−α(T), β(T)) = sup{|⟨T (v), v⟩_{V} | : v ∈ V, ∥v∥_{V} = 1}. (8.9.6)

This means that max(−α(T), β(T)) = ∥T ∥_{op}, (8.9.7)

as in the previous paragraph.

Note that α(−T) = −β(T), β(−T) = −α(T). (8.9.8)

If a ∈ R, then α(a I + T) = a + α(T) (8.9.9)

and β(a I + T) = a + β(T), (8.9.10)

where I = I_{V} is the identity mapping on V.

8.10 Nonnegative bilinear forms and operators

Let V be a vector space over the real numbers, and let b be a symmetric bilinear form on V. If b(v, v) ≥ 0 (8.10.1)

for every v ∈ V, then b is said to be nonnegative on V. Under these conditions, it is well known that

|b(v, w)| ≤ b(v, v)^{1/2} b(w, w)^{1/2} (8.10.2)

for all v, w ∈ V, which is the analogue of the Cauchy–Schwarz inequality in this case.

Similarly, let V be a vector space over the complex numbers, and let b be a Hermitian symmetric sesquilinear form on V. Thus b(v, v) ∈ R for every v ∈ V, as in Section 8.5. If (8.10.1) holds for every v ∈ V, then b is said to be nonnegative on V. It is well known that (8.10.2) holds in this case too, 8.10. NONNEGATIVE BILINEAR FORMS AND OPERATORS 187 which is another version of the Cauchy–Schwarz inequality. This corresponds to Proposition 12.1 on p114 of [333].

To get (8.10.2) in the complex case, we start by observing that 0 ≤ b(v + a w, v + a w) = b(v, v) + 2 Re(\overline{a} b(v, w)) + |a|^{2} b(w, w) (8.10.3) for every v, w ∈ V and a ∈ C. This implies that 2 t |b(v, w)| ≤ b(v, v) + t^{2} b(w, w) (8.10.4) for all nonnegative real numbers t, using a suitable choice of a with |a| = t. This also works in the real case, with some simplifications. One can get (8.10.2) from (8.10.4) in much the same way as when b is an inner product, except that one should be a bit more careful about b(v, v) or b(w, w) being 0 even if v or w is not zero, as appropriate.

Now let (V, ⟨·, ·⟩_{V}) be a real or complex inner product space, with the corresponding norm ∥ · ∥_{V}. Also let T be a linear mapping from V into itself that is symmetric with respect to ⟨·, ·⟩_{V}, as in Section 8.6. If ⟨T (v), v⟩_{V} ≥ 0 (8.10.5) for every v ∈ V, then T is said to be nonnegative with respect to ⟨·, ·⟩_{V} on V. Of course, this is the same as saying that b_{T} (v, w) = ⟨T (v), w⟩_{V} (8.10.6) is nonnegative as a symmetric bilinear form on V in the real case, or as a Hermitian symmetric sesquilinear form on V in the complex case. In both cases, we get that |⟨T (v), w⟩_{V} | ≤ ⟨T (v), v⟩_{V}^{1/2} ⟨T (w), w⟩_{V}^{1/2} (8.10.7) for every v, w ∈ V, as in (8.10.2).

If T is any bounded linear mapping from V into itself with respect to ∥ · ∥_{V}, then |⟨T (w), w⟩_{V} | ≤ ∥T (w)∥_{V} ∥w∥_{V} ≤ ∥T ∥_{op} ∥w∥_{V}^{2} (8.10.8) for every w ∈ V. If T is symmetric and nonnegative on V as well, then |⟨T (v), w⟩_{V} | ≤ ∥T ∥_{op}^{1/2} ⟨T (v), v⟩^{1/2} ∥w∥_{V} (8.10.9) for every v, w ∈ V, by (8.10.7). This implies that ∥T (v)∥_{V} ≤ ∥T ∥_{op}^{1/2} ⟨T (v), v⟩_{V}^{1/2} (8.10.10) for every v ∈ V.

Suppose that c_{0} ∥v∥_{V} ≤ ∥T (v)∥_{V} (8.10.11) for some c_{0} > 0 and every v ∈ V. If T is invertible on V, with bounded inverse, then this holds with c_{0} = 1/∥T^{−1}∥_{op}. If T is bounded, symmetric, and nonnegative on V too, then we can combine (8.10.10) and (8.10.11) to get that

c_{0}^{2} ∥v∥_{V}^{2} ≤ ∥T ∥_{op} ⟨T (v), v⟩_{V} (8.10.12)

for every v ∈ V.

Suppose that V ≠ {0}, and note that T is nonnegative on V exactly when

α(T) ≥ 0, (8.10.13)

where the left side is as in (8.9.4). If T is invertible on V, with bounded inverse, then α(T) > 0, (8.10.14)

by (8.10.12). This means that T does not have a bounded inverse on V when

α(T) = 0. (8.10.15)

Suppose that V is a Hilbert space, and let (W, ⟨·, ·⟩_{W}) be another real or complex Hilbert space, as appropriate, with associated norm ∥ · ∥_{W}. Also let R be a bounded linear mapping from V into W, so that the adjoint R^{∗} of R is a bounded linear mapping from W into V, as in Section 3.5. Thus R^{∗} ∘ R is a bounded linear mapping from V into itself, which is self-adjoint. If v ∈ V, then

⟨(R^{∗} ∘ R)(v), v⟩_{V} = ⟨R(v), R(v)⟩_{W} = ∥R(v)∥_{W}^{2} , (8.10.16)

so that R^{∗} ∘ R is nonnegative on V. Similarly, if V = W is an inner product space, and R is a linear mapping from V into itself that is symmetric with respect to ⟨·, ·⟩_{V}, then R ∘ R is symmetric and nonnegative with respect to ⟨·, ·⟩.

8.11 Some remarks about normal operators

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers, with corresponding norm ∥ · ∥_{V}. If a bounded linear mapping T from V into itself commutes with its adjoint, then one may say that T is normal, as in Section 7.5. This terminology is perhaps most commonly used in the complex case, as before.

If T is a bounded linear mapping from V into itself and v ∈ V, then

∥T (v)∥_{V}^{2} = ⟨T (v), T (v)⟩_{V} = ⟨T^{∗}(T (v)), v⟩_{V} (8.11.1)

and ∥T^{∗}(v)∥_{V}^{2} = ⟨T^{∗}(v), T^{∗}(v)⟩_{V} = ⟨T (T^{∗}(v)), v⟩_{V} . (8.11.2)

If T is normal, then it follows that

∥T (v)∥_{V} = ∥T^{∗}(v)∥_{V} . (8.11.3)

8.11. SOME REMARKS ABOUT NORMAL OPERATORS 189 In particular, this means that ker T = ker T^{∗}, (8.11.4) as in part (b) of Theorem 12.12 on p298 of [286]. Thus T is injective if and only if T^{∗} is injective under these conditions.

Conversely, if (8.11.3) holds for every v ∈ V, then ⟨T^{∗}(T (v)), v⟩_{V} = ⟨T (T^{∗}(v)), v⟩_{V} (8.11.5) for every v ∈ V. This implies that ⟨T^{∗}(T (v)), w⟩_{V} = ⟨T (T^{∗}(v)), w⟩_{V} (8.11.6) for every v, w ∈ V, using polarization identities, as in Sections 8.5 or 8.8. More precisely, in the real case, this also uses the fact that T^{∗} ∘ T and T ∘ T^{∗} are self-adjoint on V, to get that both sides of (8.11.6) are symmetric bilinear forms on V. It is easy to see that T and T^{∗} commute on V, using (8.11.6). This corresponds to Theorem 1 on p42 of [140], and to part (a) of Theorem 12.12 on p298 of [286].

If T is any bounded linear mapping from V into itself, then one can check that ker T^{∗} = T (V)^{⊥} (8.11.7) as in Theorem 12.10 on p298 of [286]. If T is normal, then we get that ker T = T (V)^{⊥}, (8.11.8) because of (8.11.4).

Suppose that T is a bounded linear mapping from V into itself such that c ∥v∥_{V} ≤ ∥T (v)∥_{V} (8.11.9) for some c > 0 and all v ∈ V. This implies that T (V) is a closed set in V, because V is complete with respect to the metric associated to ∥ · ∥_{V}, as in Section 7.11. If T is normal, then we also have that T (V) is dense in V, because of (8.11.8). This means that T is a one-to-one linear mapping from V onto itself wtih bounded inverse in this case.

Suppose now that V is complex, and that T is a bounded self-adjoint linear mapping from V into itself. Also let λ ∈ C be given, and observe that (λ I − T)^{∗} = \overline{λ} I − T^{∗} = \overline{λ} I − T, (8.11.10) where I = I_{V} is the identity mapping on V, as usual. In particular, λ I − T is normal on V.

If v ∈ V, then ⟨(λ I − T)(v), v⟩_{V} = λ ⟨v, v⟩_{V} − ⟨T (v), v⟩_{V} = λ ∥v∥_{V}^{2} − ⟨T (v), v⟩_{V} . (8.11.11)

Note that ⟨T (v), v⟩_{V} ∈ R, (8.11.12)

because T is self-adjoint on V. This implies that

Im⟨(λ I − T)(v), v⟩_{V} = (Im λ) ∥v∥_{V}^{2} . (8.11.13)

It follows that | Im λ| ∥v∥_{V}^{2} ≤ ∥(λ I − T)(v)∥_{V} ∥v∥_{V} , (8.11.14)

by the Cauchy–Schwarz inequality. This means that

| Im λ| ∥v∥_{V} ≤ ∥(λ I − T)(v)∥_{V} , (8.11.15)

which is trivial when v = 0.

If Im λ ≠ 0, then we get that λ I − T is a one-to-one linear mapping from V onto itself with bounded inverse, as before. This shows that the spectrum of T is contained in the real line, with respect to the algebra of bounded linear mappings from V into itself. This corresponds to Theorem 1 on p54 of [140].

8.12 Positivity and invertibility

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers, with corresponding norm ∥ · ∥_{V}, and let T be a bounded self-adjoint linear mapping from V into itself. Suppose for the moment that there is a positive real number c such that c ∥v∥_{V}^{2} ≤ ⟨T (v), v⟩_{V} (8.12.1)

for every v ∈ V. This implies that

c ∥v∥_{V}^{2} ≤ ∥T (v)∥_{V} ∥v∥_{V} (8.12.2)

for every v ∈ V, by the Cauchy–Schwarz inequality. It follows that T is a oneto-one mapping from V onto itself, with bounded inverse, as in the previous section.

Suppose from now on in this section that V ≠ {0}, and let α(T) be as in Section 8.9. If α(T) > 0, (8.12.3)

then T has a bounded inverse on V, as in the preceding paragraph.

Suppose for the moment again that T is nonnegative on V, and let a be a positive real number. Thus a I + T is self-adjoint on V, and

α(a I + T) = a + α(T) ≥ a > 0. (8.12.4)

This implies that a I + T is a one-to-one linear mapping from V onto itself with bounded inverse, as before. It follows that the spectrum of T is contained in the set of nonnegative real numbers. This uses the fact that the spectrum of T is contained in the real line in the complex case, as in the previous section.

8.12. POSITIVITY AND INVERTIBILITY 191

Let T be any bounded self-adjoint linear mapping from V into itself again. If λ is a real number such that λ < α(T), (8.12.5) then α(T − λ I) = α(T) − λ > 0, (8.12.6) so that T − λ I has a bounded inverse on V. Of course, this is the same as saying that λ I − T has a bounded inverse on V.

Similarly, suppose that λ ∈ R satisfies λ > β(T), (8.12.7) where β(T) is as in Section 8.9. This implies that α(λ I − T) = λ − β(T) > 0, (8.12.8) so that λ I − T has a bounded inverse on V.

Let σ(T) = σ_{BL(V)}(T) be the spectrum of T with respect to the algebra of bounded linear mappings from V into itself, as in Section 6.8. Using the remarks in the previous two paragraphs, we get that σ(T) ⊆ [α(T), β(T)]. (8.12.9) This also uses the fact that σ(T) ⊆ R in the complex case, as before.

Observe that α(T − α(T) I) = α(T) − α(T) = 0, (8.12.10) so that T − α(T) I does not have a bounded inverse on V, as in Section 8.10. Equivalently, this means that α(T) I − T does not have a bounded inverse on V, so that α(T) ∈ σ(T). (8.12.11) Similarly, α(β(T) I − T) = β(T) − β(T) = 0, (8.12.12) so that β(T) I − T does not have a bounded inverse on V, and thus β(T) ∈ σ(T). (8.12.13)

In particular, we get that max{|λ| : λ ∈ σ(T)} = max(−α(T), β(T)). (8.12.14) This implies that max{|λ| : λ ∈ σ(T)} = ∥T ∥_{op}, (8.12.15) as in Section 8.9. This corresponds to Theorem 2 on p55 of [140].

8.13 Polynomials and associative algebras

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}. Suppose that

p(t) = ∑_{j=0}^{n} c_{j} t^{j} (8.13.1)

is a polynomial in a single variable with real or complex coefficients, as appropriate. If x ∈ A, then

p̃(x) = ∑_{j=0}^{n} c_{j} x^{j} (8.13.2)

defines an element of A, where x^{0} is interpreted as being equal to e_{A}, as usual. This element is often simply denoted p(x), but it is sometimes helpful to use different notation, as on p243 of [286].

The space of all such polynomials p is a commutative associative algebra over the real or complex numbers, as appropriate, with respect to the usual definition of multiplication. It is easy to see that

p ↦ p̃(x) (8.13.3)

defines a homomorphism from this algebra into A.

Let us check that p(σ_{A}(x)) ⊆ σ_{A}(p̃(x)), (8.13.4)

where σ_{A}(y) is the spectrum of y ∈ A, as in Section 6.8. This is at least part of a version of the spectral mapping theorem. Let λ ∈ σ_{A}(x) be given, and let us verify that p(λ) ∈ σ_{A}(p̃(x)). (8.13.5)

It is well known and not difficult to show that

p(t) − p(λ) = (t − λ) q(t) (8.13.6)

for some polynomial q(t) with real or complex coefficients, as appropriate. This implies that p̃(x) − p(λ) e_{A} = (x − λ e_{A}) q̃(x). (8.13.7)

Note that x−λ e_{A} and q̃(x) commute with each other. If the left side of (8.13.7) is invertible in A, then each of the factors on the right side is invertible as well, as in Section 6.13. In particular, if x−λ e_{A} is not invertible, then p̃(x)−p(λ) e_{A} is not invertible.

We would like to have that

σ_{A}(p̃(x)) ⊆ p(σ_{A}(x)) (8.13.8)

under suitable conditions, which would imply that

p(σ_{A}(x)) = σ_{A}(p̃(x)). (8.13.9)

If p is a constant, then (8.13.8) holds when σ_{A}(x) ≠ ∅. In particular, this works when A is a complex Banach algebra, as in Section 6.8.

Suppose now that p is not a constant. Let μ ∈ C be given, and observe that

p(t) − μ = c ∏_{j=1}^{n}(t − λ_{j}) (8.13.10)

for some complex numbers c and λ_{1}, …, λ_{n}, with c ≠ 0, by the fundamental theorem of algebra. Suppose that A is complex, so that

p̃(x) − μ e_{A} = c ∏_{j=1}^{n}(x − λ_{j} e_{A}). (8.13.11)

If μ ∈ σ_{A}(p̃(x)), then it follows that λ_{j} ∈ σ_{A}(x) for some j. This means that μ = p(λ_{j}) is an element of p(σ_{A}(x)).

Suppose that A is real, so that p(t) has real coefficients, and let μ ∈ R be given. In this case,

p(t) − μ = c ∏_{j=1}^{r}(t − λ_{j}) ∏_{l=1}^{m}((t − a_{l})^{2} + b_{l}^{2}) (8.13.12)

for some real numbers c, λ_{1}, …, λ_{l}, a_{1}, …, a_{m}, and b_{1}, …, b_{m}, with b_{1}, …, b_{m} and c not equal to 0. This is the same as (8.13.10), with n = r + 2 m, arranged so that the previous λ_{j}’s in R are listed first, and the remaining λ_{j}’s are of the form a_{l} ± b_{l} i. Thus

p̃(x) − μ e_{A} = c ∏_{j=1}^{r}(x − λ_{j} e_{A}) ∏_{l=1}^{m}((x − a_{l} e_{A})^{2} + b_{l}^{2} e_{A}). (8.13.13)

Suppose that (x − a e_{A})^{2} + b^{2} e_{A} (8.13.14)

is invertible in A for all a, b ∈ R with b ≠ 0. If μ ∈ σ_{A}(p̃(x)), then we get that λ_{j} ∈ σ_{A}(x) for some j. This implies that μ = p(λ_{j}) ∈ p(σ_{A}(x)), as before.

Let (V, ⟨·, ·⟩_{V}) be a real Hilbert space, with associated norm ∥ · ∥_{V}, and with V ≠ {0}. If A = B(V) and x ∈ BL(V) is self-adjoint, then σ_{A}(x) ≠ ∅, and (8.13.14) is invertible in A for all a, b ∈ R with b ≠ 0, as in the previous section. This means that (8.13.9) holds for all polynomials p with real coefficients, as before.

If A is a complex Banach algebra and x ∈ A, then one can define f̃(x) ∈ A for complex-valued holomorphic functions f defined on open subsets of the complex plane that contain σ_{A}(x). This is discussed in Section 1.12 of [11], and beginning on p240 of [286]. Part (b) of Theorem 10.28 on p244 of [286] is the spectral mapping theorem for functions of this type.

8.14 Polynomials of self-adjoint operators

Let (V, ⟨·, ·⟩_{V}) be a real or complex Hilbert space, with associated norm ∥ · ∥_{V}, and with V ≠ {0}. Also let T be a bounded self-adjoint linear mapping from V into itself, and let p be a polynomial in a single variable with real coefficients. Thus p̃(T) may be defined as a bounded linear mapping from V into itself as in the previous section, and it is easy to see that p̃(T) is self-adjoint as well. As in (8.13.9), we have that σ(p̃(T)) = p(σ(T)), (8.14.1)

where σ(·) = σ_{BL(V)}(·) is the spectrum with respect to the algebra of bounded linear mappings from V into itself.

Remember that

∥p̃(T)∥_{op} = max{|μ| : μ ∈ σ(p̃(T))}, (8.14.2)

as in (8.12.15). This means that

∥p̃(T)∥_{op} = max{|μ| : μ ∈ p(σ(T))}, (8.14.3)

because of (8.14.1). Of course, this is the same as saying that

∥p̃(T)∥_{op} = max{|p(λ)| : λ ∈ σ(T)}. (8.14.4)

This corresponds to Theorem 3 on p55 of [140].

Let C(σ(T), R) be the space of continuous real-valued functions on σ(T), with respect to the restriction of the standard Euclidean metric on R to σ(T). If f ∈ C(σ(T), R), then the supremum norm of f may be denoted ∥f∥_{sup}, as usual, or ∥f∥_{sup,σ(T)}, to indicate the role of σ(T). We would like to define a mapping f ↦ f̃(T) (8.14.5)

from C(σ(T), R) into BL(V) that agrees with the previous definition of f̃(T) when f is the restriction to σ(T) of a polynomial with real coefficients, and with other nice properties.

Remember that BL(V) is an associative algebra over the real or complex numbers, depending on whether V is real or complex. We may consider BL(V) as an associative algebra over the real numbers in both cases. The mapping (8.14.5) is a homomorphism from C(σ(T), R) into BL(V), as associative algebras over R, and an isometry with respect to the supremum norm on C(σ(T), R) and the operator norm on BL(V).

The continuous real-valued functions on σ(T) obtained from restrictions of polynomials with real coefficients to σ(T) form a subalgebra of C(σ(T), R). This subalgebra is dense in C(σ(T), R) with respect to the supremum metric, by the Stone–Weierstrass theorem. More precisely, it is well known that a continuous real-valued function on a closed set in R can be extended to a continuous function on R. One could use this to reduce to Weierstrass’ approximation theorem for continuous functions on a closed interval in R.

This leads to a unique bounded linear mapping from C(σ(T), R) into BL(V), as a Banach space over the real numbers, as in Section 2.2. This extension will be expressed as in (8.14.5), to be compatible with the previous notation for polynomials. One can check that this extension is an algebra homomorphism, because of the analogous property for polynomials. Similarly, this extension is an isometry, because of (8.14.4).

Note that

(8.14.6)  f̃(T) is self-adjoint

for every f ∈ C(σ(T), R), because of the analogous property of polynomials. If

f(λ) ≠ 0 (8.14.7)

for each λ ∈ σ(T), then g = 1/f is a continuous real-valued function on σ(T) too. This implies that

(8.14.8)  f̃(T) is invertible in BL(V),

with inverse equal to g̃(T).

8.15 Some functions on associative algebras

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}. If a ∈ A and p is a polynomial in a single variable with real or complex coefficients, as appropriate, then p̃(a) may be defined as an element of A as in Section 8.13. We have also seen some other situations in which it may be reasonable to define p̃(a) for other real or complexvalued functions p defined on or around σ_{A}(a). We would like to consider some other aspects of this in this section.

If p̃(a) ∈ A can be defined for some suitable functions p, then a related question is whether p̃(a) depends only on the restriction of p to σ_{A}(a). To put it another way, if p = 0 on σ_{A}(a), (8.15.1)

then do we have that p̃(a) = 0? (8.15.2)

If a is a nilpotent element of A, for instance, then it is easy to see that

σ_{A}(a) = {0}, (8.15.3)

using a remark in Section 6.13. If p is a polynomial with real or complex coefficients, then (8.15.1) is the same as saying that p(0) = 0 in this case. If p(t) = t, then p̃(a) = a, so that (8.15.2) would mean that a = 0.

Let X be a nonempty set, and suppose for the moment that A is the algebra of all real or complex-valued functions on X. If a ∈ A, then

σ_{A}(a) = a(X), (8.15.4) as mentioned in Section 6.8. If p is a real or complex-valued function on a(X), as appropriate, then p̃(a) = p ∘ a (8.15.5)

defines another real or complex-valued function on X. Note that this agrees with the definition of p̃(a) in Section 8.13 when p is a polynomial.

If A is the algebra of bounded real or complex-valued functions on X and a ∈ A, then σ_{A}(a) is the closure \overline{a(X)} of a(X) in R or C, as appropriate, as in Section 6.8. If p is a bounded real or complex-valued function on a(X), then (8.15.5) defines a bounded real or complex-valued function on X, as appropriate.

If X is a metric space or a topological space, A is the algebra of continuous real or complex-valued functions on X, and a ∈ A, then (8.15.4) holds again. If p is a continuous real or complex-valued function on a(X), with respect to the restriction of the standard Euclidean metric on R or C, then (8.15.5) defines a continuous real or complex-valued function on X, as appropriate.

Similarly, if A is an algebra of functions on some space with some additional properties, then one may need additional properties of p to get that (8.15.5) is an element of A.

Let V be a vector space over the real or complex numbers, and let T be a linear mapping from V into itself. In some cases, we may be able to define p̃(T) for suitable functions p on or around the spectrum of T. Some basic examples of this are given by multiplication operators on various spaces of functions.

If A is a Banach algebra and p is defined by a power series, then p̃(a) can be defined for suitable a ∈ A, as in Section 9.14.

Algebras, operators, and power series

9.1 Eigenvalues and eigenvectors

Let V be a vector space over the real or complex numbers, and let T be a linear mapping from V into itself. As usual, v ∈ V is said to be an eigenvector of T with eigenvalue λ ∈ R or C, as appropriate, when T (v) = λ v. (9.1.1) However, λ ∈ R or C, as appropriate, is normally considered to be an eigenvalue of T only when it is the eigenvalue associated to a nonzero eigenvector of T.

The set σ_{p}(T) of all eigenvalues of T in this sense is called the point spectrum of T. Equivalently, this is the set of λ ∈ R or C, as appropriate, such that ker(λ I − T) ≠ {0}. (9.1.2) Note that σ_{p}(T) ⊆ σ_{L(V)}(T), (9.1.3) where σ_{L(V)}(T) is the spectrum of T with respect to the algebra L(V) of all linear mappings from V into itself. If V has finite dimension, then σ_{p}(T) = σ_{L(V)}(T). (9.1.4)

Let q be a polynomial in a single variable with real or complex coefficients, as appropriate. If v ∈ V is an eigenvector of T with eigenvalue λ, then (q̃(T))(v) = q(λ) v. (9.1.5) This implies that q(λ) ∈ σ_{p}(q̃(T)) (9.1.6) when v ≠ 0, so that q(σ_{p}(T)) ⊆ σ_{p}(q̃(T)). (9.1.7)

If q is not a constant, then we would like to have that

σ_{p}(q̃(T)) ⊆ q(σ_{p}(T)) (9.1.8)

under suitable conditions. In the complex case, this can be obtained from the same type of argument as in Section 8.13.

More precisely, if μ ∈ C, then one can use the fundamental theorem of algebra to express q(t) − μ as the product of a nonzero complex number and finitely many linear factors t − λ_{j}. If

μ ∈ σ_{p}(q̃(T)), (9.1.9)

then q̃(T) − μ I is not injective on V. This would imply that one of the corresponding factors T − λ_{j} I of q̃(T) − μ I is not injective. This means that λ_{j} is an eigenvalue of T for some j. It follows that

μ ∈ q(σ_{p}(T)), (9.1.10)

because q(λ_{j}) = μ for each j, by construction.

In the real case, one can use the same type of argument as before as well. This works when (T − a I)^{2} + b^{2} I (9.1.11)

is injective on V for all a, b ∈ R with b ≠ 0. This condition holds when (V, ⟨·, ·⟩_{V}) is a real inner product space, and T is symmetric with respect to ⟨·, ·⟩_{V}.

Let (V, ⟨·, ·⟩_{V}) be a complex inner product space with corresponding norm ∥ · ∥_{V}, and suppose that T is symmetric with respect to ⟨·, ·⟩_{V}. If v ∈ V is an eigenvector of T with eigenvalue λ ∈ C, then

λ ∥v∥_{V}^{2} = ⟨T (v), v⟩_{V} = ⟨v, T (v)⟩_{V} = \overline{λ} ∥v∥_{V}^{2} . (9.1.12)

This implies that λ ∈ R when v ≠ 0.

9.2 Eigenvalues of bounded linear mappings

Let V be a real or complex vector space with a norm ∥ · ∥_{V}, let T be a bounded linear mapping from V into itself, and let σ_{BL(V)}(T) be the spectrum of T with respect to the algebra BL(V) of all bounded linear mappings from V into itself. Note that σ_{L(V)}(T) ⊆ σ_{BL(V)}(T), (9.2.1)

because BL(V) ⊆ L(V), as in Section 7.3. In particular,

σ_{p}(T) ⊆ σ_{BL(V)}(T). (9.2.2)

It is easy to see that

|λ| ≤ ∥T ∥_{op} when λ ∈ σ_{p}(T). (9.2.3)

More precisely, if λ ∈ σ_{p}(T), then λ^{l} ∈ σ_{p}(T^{l}) for every positive integer l, so that |λ|^{l} = |λ^{l}| ≤ ∥T^{l}∥_{op}. (9.2.4)

Equivalently, |λ| ≤ ∥T^{l}∥_{op}^{1/l}, so that

|λ| ≤ r_{BL(V)}(T), (9.2.5)

where r_{BL(V)}(T) is as in Section 6.13.

If V is a Banach space, then

σ_{BL(V)}(T) = σ_{L(V)}(T), (9.2.6)

by the open mapping theorem. In this case, BL(V) is a Banach algebra with respect to the operator norm, and |λ| ≤ r_{BL(V)}(T) for every λ ∈ σ_{BL(V)}(T), as in Section 6.13.

Suppose for the moment that V is a complex Banach space, and that T is bounded on V. If q is a complex-valued holomorphic function defined on an open subset U of C such that

σ_{BL(V)}(T) ⊆ U, (9.2.7)

then q̃(T) may be defined as a bounded linear mapping from V into itself, as mentioned in Section 8.13. The analogues of (9.1.5) and (9.1.7) in this case are given in parts (a) and (b) of Theorem 10.33 on p247 of [286].

Part (d) of that theorem says that (9.1.8) holds when q is not constant on any connected component of U. More precisely, part (c) of that theorem says that if (9.1.9) holds and q is not identically equal to μ on any connected component of U, then (9.1.10) holds.

Let (V, ⟨·, ·⟩_{V}) be a real or complex Hilbert space, and let T be a bounded linear mapping from V into itself which is normal, in the sense that T commutes with T^{∗}. In the real case, we have that

ker(T − λ I) = ker(T^{∗} − λ I) (9.2.8)

for every λ ∈ R, as in Section 8.11. This means that T and T^{∗} have the same eigenvalues, and with the same eigenvectors. Similarly, in the complex case, if λ ∈ C, then ker(T − λ I) = ker(T^{∗} − \overline{λ} I). (9.2.9)

It follows that λ is an eigenvalue of T if and only if \overline{λ} is an eigenvaue of T^{∗}, with the same eigenvectors.

Suppose now that T is self-adjoint, and let f be a continuous real-valued function on the spectrum σ_{BL(V)}(T) of T with respect to BL(V). If v ∈ V is an eigenvector of T with eigenvalue λ, then

(f̃(T))(v) = f(λ) v. (9.2.10)

This can be obtained from the analogous statement for polynomials, by approximating f by polynomials uniformly on σ_{BL(V)}(T). This implies that

f(σ_{p}(T)) ⊆ σ_{p}(f̃(T)), (9.2.11)

as before.

9.3 Approximate eigenvalues

Let V, W, and Z be vector spaces, all real or all complex, and with norms ∥·∥_{V}, ∥ · ∥_{W}, and ∥ · ∥_{Z}, respectively. Also let T_{1} be a linear mapping from V into W, and let T_{2} be a linear mapping from W into Z. Suppose that there are positive real numbers c_{1}, c_{2} such that

∥T_{1}(v)∥_{W} ≥ c_{1} ∥v∥_{V} (9.3.1)

for every v ∈ V, and ∥T_{2}(w)∥_{Z} ≥ c_{2} ∥w∥_{W} (9.3.2)

for every w ∈ W. Under these conditions, we get that

∥(T_{2} ∘ T_{1})(v)∥_{Z} = ∥T_{2}(T_{1}(v))∥_{Z} ≥ c_{2} ∥T_{1}(v)∥_{W} ≥ c_{1} c_{2} ∥v∥_{V} (9.3.3)

for every v ∈ V.

Suppose that V ≠ {0}, and let T be a bounded linear mapping from V into itself. Also let λ be a real or complex number, as appropriate. If T − λ I has a bounded inverse on V, then

∥(T − λ I)(v)∥_{V} ≥ c ∥v∥_{V} (9.3.4)

for some c > 0 and all v ∈ V, as in Section 7.11.

If there is no c > 0 such that (9.3.4) holds, then λ is said to be an approximate eigenvalue of T on V, as on p51 of [140]. Equivalently, this means that there is a sequence {v_{j}}_{j=1}^{∞} of elements of V such that

∥v_{j}∥_{V} = 1 for each j (9.3.5)

and ∥(T − λ I)(v_{j})∥_{V} = ∥T (v_{j}) − λ v_{j}∥_{V} → 0 as j → ∞. (9.3.6)

Of course, if λ is an eigenvalue of T, then λ is an approximate eigenvalue of T.

The set of approximate eigenvalues of T is known as the approximate point spectrum of T, and may be denoted σ_{ap}(T). Thus

σ_{p}(T) ⊆ σ_{ap}(T) ⊆ σ_{BL(V)}(T). (9.3.7)

One can check that

|λ| ≤ ∥T ∥_{op} when λ ∈ σ_{ap}(T). (9.3.8)

Alternatively, if |λ| > ∥T ∥_{op}, then (9.3.4) holds with

c = |λ| − ∥T ∥_{op}, (9.3.9)

as in Section 7.11.

The set of λ ∈ R or C, as appropriate, such that (9.3.4) holds for some c > 0 is an open set, as in Section 7.11. Equivalently,

(9.3.10) σ_{ap}(T) is a closed set

in R or C, as appropriate, with respect to the standard Euclidean metric.

Suppose that λ ∈ σ_{ap}(T), and let {v_{j}}_{j=1}^{∞} be a sequence of elements of V that satisfies (9.3.5) and (9.3.6). If l is a positive integer, then one can verify that ∥T^{l}(v_{j}) − λ^{l} v_{j}∥_{V} → 0 as j → ∞. (9.3.11)

This means that λ^{l} ∈ σ_{ap}(T^{l}). (9.3.12)

It follows that |λ|^{l} = |λ^{l}| ≤ ∥T^{l}∥_{op}, as in (9.3.8). This implies that

|λ| ≤ r_{BL(V)}(T), (9.3.13)

where r_{BL(V)}(T) is as in Section 6.13.

9.4 More on approximate eigenvalues

Let us continue with the same notation and hypotheses as in the previous section. If q is a polynomial in a single variable with real or complex coefficients, as appropriate, then we get that

∥(q̃(T))(v_{j}) − q(λ) v_{j}∥_{V} → 0 as j → ∞. (9.4.1)

This implies that q(λ) ∈ σ_{ap}(q̃(T)), (9.4.2)

so that q(σ_{ap}(T)) ⊆ σ_{ap}(q̃(T)). (9.4.3)

If q is not constant, then we would like to get that

σ_{ap}(q̃(T)) ⊆ q(σ_{ap}(T)) (9.4.4)

under suitable conditions. In the complex case, we can use the same type of argument as in Sections 8.13 and 9.1, as follows.

Let μ ∈ C be given, and remember that q(t) − μ can be factored into the product of a nonzero complex number and finitely many linear factors t − λ_{j}, by the fundametal theorem of algebra. If each of the factors T − λ_{j} I satisfies a condition like (9.3.4), then q̃(T) − μ I satisfies the same type of condition. This uses the remark about compositions at the beginning of the previous section.

If μ ∈ σ_{ap}(q̃(T)), (9.4.5)

then at least one of the factors T − λ_{j} I does not satisfy a condition like (9.3.4). This implies that λ_{j} is an approximate eigenvalue of T for some j. This means that μ ∈ q(σ_{ap}(T)), (9.4.6)

because q(λ_{j}) = μ for each j.

In the real case, suppose that for every a, b ∈ R with b ≠ 0 there is a positive real number c_{a,b} such that

∥((T − a I)^{2} + b^{2} I)(v)∥_{V} ≥ c_{a,b} ∥v∥_{V} (9.4.7)

for all v ∈ V. This permits one to use the same type of argument as in Sections 8.13 and 9.1 again. If (V, ⟨·, ·⟩_{V}) is a real inner product space, and T is symmetric with respect to ⟨·, ·⟩_{V}, then (9.4.7) holds with c_{a,b} = b^{2}.

Suppose for the moment that V is a complex Banach space, and that q is a holomorphic function defined on an open subset U of C with

σ_{BL(V)}(T) ⊆ U. (9.4.8)

Thus q̃(T) may be defined as a bounded linear mapping from V into itself, as mentioned in Sections 8.13 and 9.2. One can check that (9.4.1) and (9.4.3) hold in this case, using essentially the same argument as in the proofs of parts (a) and (b) of Theorem 10.33 on p247 of [286].

Similarly, if (9.4.5) holds and q is not identically equal to μ on any connected component of U, then one can verify that (9.4.6) holds, using essentially the same argument as in the proof of part (c) of that theorem. If q is not constant on any connected component of U, then it follows that (9.4.4) holds, as in part (d) of that theorem.

9.5 Inner products and σ_{ap}(T)

Let us continue with the same notation and hypotheses as in the previous two sections. Suppose now that (V, ⟨·, ·⟩_{V}) is a real or complex inner product space with corresponding norm ∥ · ∥_{V}. If λ ∈ σ_{ap}(T), and {v_{j}}_{j=1}^{∞} is a sequence of elements of V that satisfies (9.3.5) and (9.3.4), then

lim_{j→∞}⟨T (v_{j}), v_{j}⟩_{V} = λ. (9.5.1)

In the complex case, we also have that

lim_{j→∞}⟨v_{j}, T (v_{j})⟩_{V} = \overline{λ}. (9.5.2)

If T is symmetric with respect to ⟨·, ·⟩_{V}, then it follows that λ ∈ R.

9.5. INNER PRODUCTS AND σ_{AP} (T) 203

Alternatively, in the complex case, if T is symmetric with respect to ⟨·, ·⟩_{V}, then ∥(T − λ I)(v)∥_{V} ≥ | Im λ| ∥v∥_{V} (9.5.3)

for every λ ∈ C and v ∈ V, as in Section 8.11. If Im λ ≠ 0, then it follows that λ ∉ σ_{ap}(T).

Suppose that V is a real or complex Hilbert space, and that T is normal, in the sense that T commutes with T^{∗}. Of course, this implies that T − λ I is normal too. This implies that

∥(T − λ I)(v)∥_{V} = ∥(T − λ I)^{∗}(v)∥_{V} (9.5.4)

for every v ∈ V, as in Section 8.11. Thus (9.3.4) holds for some c > 0 if and only if (T − λ I)^{∗} has the analogous property. Similarly, (9.3.6) holds for some sequence {v_{j}}_{j=1}^{∞} of elements of V if and only if

∥(T − λ I)^{∗}(v_{j})∥_{V} → 0 as j → ∞. (9.5.5)

In particular, this means that λ is an approximate eigenvalue of T if and only if λ is an approximate eigenvalue of T^{∗} in the real case, and if and only if \overline{λ} is an approximate eigenvalue of T^{∗} in the complex case. In both cases, one can use the same sequence {v_{j}}_{j=1}^{∞} for T and T^{∗}.

If T is normal, so that T − λ I is normal as well, then (9.3.4) implies that T − λ I has a bounded inverse on V, as in Section 8.11. This means that

σ_{ap}(T) = σ_{BL(V)}(T) (9.5.6)

in this case, as in Theorem 2 on p51 of [140].

Suppose that T is self-adjoint on V, and let f be a continuous real-valued function on σ_{B(V)}(T). Suppose also that λ ∈ σ_{ap}(T), and let {v_{j}}_{j=1}^{∞} be a sequence of elements of V that satisfies (9.3.5) and (9.3.6). Under these conditions, one can verify that

∥(f̃(T))(v_{j}) − f(λ) v_{j}∥_{V} → 0 as j → ∞, (9.5.7)

by approximating f uniformly on σ_{BL(V)}(T) by polynomials with real coefficients, and using the analogous statement (9.4.1) for such polynomials. This means that f(λ) is an approximate eigenvalue of f̃(T). It follows that

f(σ_{BL(V)}(T)) ⊆ σ_{BL(V)}(f̃(T)), (9.5.8)

because of (9.5.6).

Remember that f̃(T) is invertible in BL(V) when f ≠ 0 at every point in σ_{BL(V)}(T), as in Section 8.14. Similarly, f̃(T) − λ I is invertible in BL(V) when f ≠ λ at every point in σ_{BL(V)}(T). This implies that

σ_{BL(V)}(f̃(T)) ⊆ f(σ_{BL(V)}(T)). (9.5.9)

This means that f(σ_{BL(V)}(T)) = σ_{BL(V)}(f̃(T)), (9.5.10)

because of (9.5.8).

9.6 Invertibility and dual linear mappings

Let V, W be vector spaces, both real or both complex, and remember that V^{alg}, W^{alg} are the algebraic dual spaces of V, W, respectively, as in Section 3.1. If T is a linear mapping from V into W, then T^{alg} is the corresponding dual linear mapping from W^{alg} into V^{alg}, as in Section 3.13. If T is a one-to-one linear mapping from V onto W, then it is easy to see that T^{alg} is a one-to-one linear mapping from W^{alg} onto V^{alg}, with

(T^{alg})^{−1} = (T^{−1})^{alg}. (9.6.1)

More precisely, T^{−1} is a linear mapping from W onto V, and the corresponding dual linear mapping (T^{−1})^{alg} maps V^{alg} onto W^{alg}.

If T is any linear mapping from V into W, then

(9.6.2) ker T^{alg} = {λ ∈ W^{alg} : T (V) ⊆ ker λ}.

This implies that ker T^{alg} = {0} if and only if T (V) = W. If T^{alg} is a one-to-one mapping from V^{alg} onto W^{alg}, then we would like to check that T is a one-to-one mapping from V onto W. It suffices to verify that T is injective, because the previous statement implies that T is surjective.

Let (T^{alg})^{alg} be the dual linear mapping from (V^{alg})^{alg} into (W^{alg})^{alg} associated to T^{alg}, as in Section 3.15. Remember that there are natural one-to-one linear mappings from V, W into (V^{alg})^{alg}, (W^{alg})^{alg}, respectively, as in Section 3.14. We have also seen that (T^{alg})^{alg} corresponds to T, using these linear mappings. If (T^{alg})^{alg} is injective, then it follows that T is injective. If T^{alg} is invertible, then (T^{alg})^{alg} is invertible, as before, which implies in particular that (T^{alg})^{alg} is injective.

Suppose now that ∥·∥_{V}, ∥·∥_{W} are norms on V, W, respectively, and remember that V′, W′ are the corresponding dual spaces of bounded linear functionals on V, W, respectively, with their associated dual norms, as in Section 3.1. If T is a bounded linear mapping from V into W, then T′ is the corresponding dual linear mapping from W′ into V′, which is a bounded linear mapping with the same operator norm as T, as in Section 3.13. If T is a one-to-one bounded linear mapping from V onto W with bounded inverse, then T′ is a one-to-one bounded linear mapping from W′ onto V′ with bounded inverse, and

(T′)^{−1} = (T^{−1})′. (9.6.3)

As before, T^{−1} is a bounded linear mapping from W onto V, and (T^{−1})′ maps V′ onto W′.

If T is any bounded linear mapping from V into W, then

ker T′ = {λ ∈ W′ : T (V) ⊆ ker λ}. (9.6.4)

Thus ker T′ = {0} (9.6.5)

9.7. SUBADDITIVITY OF R_{A}(X) 205

when (9.6.6) T (V) is dense in W,

with respect to the metric associated to ∥ · ∥_{W}. Conversely, one can check that (9.6.5) implies (9.6.6), using the Hahn–Banach theorem. More precisely, if T (V) is not dense in W, then the closure \overline{T (V)} of T (V) in W is a closed linear subspace of W. Under these conditions, the Hahn–Banach theorem implies that there is a nonzero element λ of W′ that is in the kernel of T′.

Remember that V′, W′ are Banach spaces with respect to the appropriate dual norms, as in Sections 2.2 and 3.1. If T′ is a one-to-one linear mapping from W′ onto V′, then the inverse of T′ is a bounded linear mapping from V′ onto W′, by the open mapping theorem, as in Section 7.13. In this case, we would like to check that c ∥v∥_{V} ≤ ∥T (v)∥_{W} (9.6.7)

for some c > 0 and all v ∈ V.

Let T′′ be the dual linear mapping from V′′ into W′′ associated to T′, as in Section 3.15 again. This is a one-to-one bounded linear mapping from V′′ onto W′′ with bounded inverse, because of the corresponding properties of T′, as before. In particular, there is a positive real number c such that

c ∥L∥_{V′′} ≤ ∥T′′(L)∥_{W′′} (9.6.8)

for every L ∈ V′′, as in Section 7.11. Remember that the natural mappings from V, W into V′′, W′′ are isometries, as in Section 3.14. One can use this to get (9.6.7) from (9.6.8), because T′′ corresponds to T with respect to these embeddings, as in Section 3.15.

If V is a Banach space, then (9.6.7) implies that T (V) is a closed set in W, as in Section 7.11. If T′ is one-to-one on W′, then T (V) is dense in W, as before, and we get that T (V) = W. If T′ is a one-to-one bounded linear mapping from W′ onto V′ with bounded inverse, then it follows that T is a one-to-one bounded linear mapping from V onto W with bounded inverse, when V is a Banach space.

9.7 Subadditivity of r_{A}(x)

Let A be an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. If x ∈ A, then

r_{A}(x) = inf_{l≥1} ∥x^{l}∥_{A}^{1/l} = lim_{j→∞} ∥x^{j}∥_{A}^{1/j}, (9.7.1)

as in Sections 6.13, 6.14, and 7.2. If x, y ∈ A commute with each other, then we would like to show that

r_{A}(x + y) ≤ r_{A}(x) + r_{A}(y). (9.7.2)

This corresponds to part of Exercise 12 on p289 of [286] when A is a complex Banach algebra. One can reduce to the case where A is commutative, by considering the closed subalgebra generated by x, y. This permits one to use a characterization of r_{A}(x) in terms of complex homomorphisms on A.

Let us consider a more direct approach here. Let R_{x}, R_{y} be positive real numbers such that r_{A}(x) < R_{x}, r_{A}(y) < R_{y}. (9.7.3)

It suffices to show that r_{A}(x + y) ≤ R_{x} + R_{y}. (9.7.4)

Let n be a positive integer, and observe that

(x + y)^{n} = ∑_{j=0}^{n} \binom{n}{j}x^{j} y^{n−j}, (9.7.5)

by the binomial theorem. Using (9.7.3), we get that there are positive integers L_{x}, L_{y} such that ∥x^{l}∥_{A} < R_{x}^{l} (9.7.6)

when l ≥ L_{x}, and ∥y^{l}∥_{A} < R_{y}^{l} (9.7.7)

when l ≥ L_{y}. We shall only be concerned with large n, and in particular we may as well take n to be larger than L_{x} + L_{y}. Put

(9.7.8)  Σ_{1}(n) = ∑_{j=0}^{L_{x}−1} \binom{n}{j} x^{j} y^{n−j},

(9.7.9)  Σ_{2}(n) = ∑_{j=L_{x}}^{n−L_{y}} \binom{n}{j} x^{j} y^{n−j},

(9.7.10)  Σ_{3}(n) = ∑_{j=n−L_{y}+1}^{n} \binom{n}{j} x^{j} y^{n−j},

so that (x + y)^{n} = Σ_{1}(n) + Σ_{2}(n) + Σ_{3}(n). (9.7.11)

Observe that

(9.7.12) ∥(x + y)^{n}∥_{A} ≤ ∥Σ_{1}(n)∥_{A} + ∥Σ_{2}(n)∥_{A} + ∥Σ_{3}(n)∥_{A}
≤ 3 max(∥Σ_{1}(n)∥_{A}, ∥Σ_{2}(n)∥_{A}, ∥Σ_{3}(n)∥_{A}).

Thus

∥(x + y)^{n}∥_{A}^{1/n} ≤ 3^{1/n} max(∥Σ_{1}(n)∥_{A}^{1/n}, ∥Σ_{2}(n)∥_{A}^{1/n}, ∥Σ_{3}(n)∥_{A}^{1/n}). (9.7.13)

We also have that

∥Σ_{2}(n)∥_{A} ≤ ∑_{j=L_{x}}^{n−L_{y}} \binom{n}{j} ∥x^{j}∥_{A} ∥y^{n−j}∥_{A} ≤ ∑_{j=L_{x}}^{n−L_{y}} \binom{n}{j} R_{x}^{j} R_{y}^{n−j}

(9.7.14)  ≤ ∑_{j=0}^{n} \binom{n}{j} R_{x}^{j} R_{y}^{n−j} = (R_{x} + R_{y})^{n},

using (9.7.6) and (9.7.7) in the second step. This implies that

∥Σ_{2}(n)∥_{A}^{1/n} ≤ R_{x} + R_{y}. (9.7.15)

If n ≥ L_{x} + L_{y} and j ≤ L_{x} − 1, then n − j ≥ L_{y} + 1, and we can use (9.7.7) to get that

∥Σ_{1}(n)∥_{A} ≤ ∑_{j=0}^{L_{x}−1} \binom{n}{j} ∥x^{j}∥_{A} R_{y}^{n−j}. (9.7.16)

This implies that

∥Σ_{1}(n)∥_{A}^{1/n} ≤ ( ∑_{j=0}^{L_{x}−1} \binom{n}{j} ∥x^{j}∥_{A} R_{y}^{−j})^{1/n} R_{y}. (9.7.17)

Let us reexpress Σ_{3}(n) as

Σ_{3}(n) = ∑_{l=0}^{L_{y}−1} \binom{n}{l} x^{n−l} y^{l}. (9.7.18)

If n ≥ L_{x} + L_{y} and l ≤ L_{y} − 1, then n − l ≥ L_{x} + 1, and we can use (9.7.6) to get that

∥Σ_{3}(n)∥_{A} ≤ ∑_{l=0}^{L_{y}−1} \binom{n}{l} R_{x}^{n−l} ∥y^{l}∥_{A}. (9.7.19)

This implies that

∥Σ_{3}(n)∥_{A}^{1/n} ≤ ( ∑_{l=0}^{L_{y}−1} \binom{n}{l} R_{x}^{−l} ∥y^{l}∥_{A})^{1/n} R_{x}. (9.7.20)

One can use (9.7.13), (9.7.15), (9.7.17) and (9.7.20) to get that

r_{A}(x + y) = lim_{n→∞} ∥(x + y)^{n}∥_{A}^{1/n} ≤ R_{x} + R_{y}. (9.7.21)

9.8 Sums of nonnegative sums

Let X be a nonempty set, and let f be a nonnegative real-valued function on X. If E ⊆ X, then ∑_{x∈E} f(x) may be defined as a nonnegative extended real number as in Section 2.4, and we have that

∑_{x∈E} f(x) ≤ ∑_{x∈X} f(x). (9.8.1)

In particular, if f is summable on X, then f is summable on E.

In some cases, it may be convenient to allow sums to be taken of nonnegative extended real numbers. A sum of this type is considered to be +∞ whenever any of the terms is equal to +∞.

208 CHAPTER 9. ALGEBRAS, OPERATORS, AND POWER SERIES

Let I be a nonempty set, and let E_{j} be a subset of X for each j ∈ I. Put

E = ⋃_{j∈I} E_{j}, (9.8.2)

and let us check that

∑_{x∈E} f(x) ≤ ∑_{j∈I} ( ∑_{x∈E_{j}} f(x)). (9.8.3)

More precisely, the sum on the right may be considered as the sum of a nonnegative extended real-valued function on I. It suffices to verify that if A is a finite subset of E, then

∑_{x∈A} f(x) ≤ ∑_{j∈I} ( ∑_{x∈E_{j}} f(x)). (9.8.4)

In fact, if B is a finite subset of I such that

A ⊆ ⋃_{j∈B} E_{j}, (9.8.5)

then

∑_{x∈A} f(x) ≤ ∑_{j∈B} ( ∑_{x∈A∩E_{j}} f(x)) ≤ ∑_{j∈B} ( ∑_{x∈E_{j}} f(x)). (9.8.6)

Suppose now that the E_{j}’s are pairwise-disjoint, so that E_{j} ∩ E_{l} = ∅ (9.8.7) when j ≠ l. In order to show that

∑_{x∈E} f(x) = ∑_{j∈I} ( ∑_{x∈E_{j}} f(x)), (9.8.8)

it is enough to check that

∑_{j∈I} ( ∑_{x∈E_{j}} f(x)) ≤ ∑_{x∈E} f(x). (9.8.9)

To get this, it suffices to have that

∑_{j∈B} ( ∑_{j∈E_{j}} f(x)) ≤ ∑_{x∈E} f(x). (9.8.10)

This can be obtained from an analogous statement about sums over unions of two disjoint sets mentioned in Section 2.4. Alternatively, one can observe that if A_{j} is a finite subset of E_{j} for each j ∈ B, then

∑_{j∈B} ( ∑_{x∈A_{j}} f(x)) = ∑ ; x∈⋃_{j∈B} A_{j} f(x) ≤ ∑_{x∈E} f(x). (9.8.11)

Let Y, Z be nonempty sets, and suppose that

X = Y × Z. (9.8.12)

Consider the iterated sums

∑_{y∈Y} ( ∑_{z∈Z} f(y, z)) (9.8.13)

and

∑_{z∈Z} ( ∑_{y∈Y} f(y, z)). (9.8.14)

Each of these iterated sums is equal to ∑_{x∈X} f(x), as in (9.8.8). This corresponds to considering X as the union of the pairwise-disjoint sets {y} × Z, y ∈ Y, or Y × {z}, z ∈ Z. In particular, (9.8.13) and (9.8.14) are equal to each other.

9.9 Sums of generalized sums

Let X be a nonempty set, let W be a vector space over the real or complex numbers with a norm ∥ · ∥_{W}, and let f be a W -valued function on X. Suppose that ∑_{x∈X} f(x) satisfies the generalized Cauchy condition, as in Section 2.7. If E is a nonempty subset of X, then it is easy to see that

∑_{x∈E} f(x) (9.9.1)

satisfies the generalized Cauchy condition as well. If W is a Banach space, then it follows that (9.9.1) converges in the generalized sense, as in Section 2.7 again. Similarly, if f has bounded finite sums on X, as in Section 2.9, then the restriction of f to E has bounded finite sums too, with BF S norm less than or equal to the BF S norm of f on X.

Let us suppose from now on in this section that W is a Banach space. Let I be a nonempty set, and let {E_{j}}_{j∈I} be a pairwise-disjoint family of nonempty subsets of X. Thus

∑_{x∈E_{j}} f(x) (9.9.2)

converges in the generalized sense for each j ∈ I, as in the preceding paragraph. Put E = ⋃_{j∈I} E_{j}. If I has only finitely many elements, then one can check that

∑_{j∈I} ( ∑_{x∈E_{j}} f(x)) = ∑_{x∈E} f(x). (9.9.3)

If I has infinitely many elements, then we would like to check that the sum over I in the left side of (9.9.3) converges in the generalized sense, and is equal to the right side of (9.9.3). Of course, if B ⊆ I has only finitely many elements, then

∑_{j∈B} ( ∑_{x∈E_{j}} f(x)) = ∑ ; x∈⋃_{j∈B} E_{j} f(x), (9.9.4)

as in the preceding paragraph. Let ϵ > 0 be given, and remember that there is a finite set A_{1}(ϵ) ⊆ X such that

∥∑_{x∈C} f(x)∥_{W} < ϵ (9.9.5)

for every finite set C ⊆ X \ A_{1}(ϵ), as in Section 2.7. Put

B_{1}(ϵ) = {j ∈ I : A_{1}(ϵ) ∩ E_{j} ≠ ∅}, (9.9.6)

which is a finite subset of I. If B_{0} ⊆ I \ B_{1}(ϵ) is a finite set, then one can verify that

∥ ∑_{j∈B_{0}} ( ∑_{x∈E_{j}} f(x))∥_{W} = ∥ ∑ ; x∈⋃_{j∈B_{0}} E_{j} f(x)∥_{W} ≤ ϵ, (9.9.7)

using (9.9.5).

This shows that the sum over I in the left side of (9.9.3) satisfies the generalized Cauchy condition. Similarly, one can show that for suitable finite subsets B of I, the right side of (9.9.4) can be approximated by finite sums that also approximate the right side of (9.9.3). This is a bit simpler when E_{j} has only finitely many elements for each j, so that ⋃_{j∈B} E_{j} is finite when B is finite.

Suppose for the moment that ∥f(x)∥_{W} is summable as a nonnegative realvalued on X. In this case,

∥ ∑_{x∈E_{j}} f(x)∥_{W} ≤ ∑_{x∈E_{j}} ∥f(x)∥_{W} (9.9.8)

for each j ∈ I. We also have that

∑_{j∈I} ( ∑_{x∈E_{j}} ∥f(x)∥_{W} ) = ∑_{x∈E} ∥f(x)∥_{W} , (9.9.9)

as in the previous section. It follows that the left side of (9.9.8) is summable as a nonnegative real-valued function on I. If W = R or C, with the standard absolute value function as the norm, then (9.9.3) can be obtained from the analogous statement in the previous section, by expressing f as a linear combination of nonnegative real-valued summable functions on X.

9.10 Cauchy products

Let ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l} be infinite series of real or complex numbers, and put

c_{n} = ∑_{j=0}^{n} a_{j} b_{n−j} (9.10.1)

for each nonnegative integer n. The series ∑_{n=0}^{∞} c_{n} is called the Cauchy product of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l}, and it is easy to see that

∑_{n=0}^{∞} c_{n} = ( ∑_{j=0}^{∞} a_{j}) ( ∑_{l=0}^{∞} b_{l}), (9.10.2)

at least formally. More precisely, both sides of (9.10.2) are formally equal to

∑_{(j,l)∈(Z_{+}∪{0})^{2}} a_{j} b_{l}, (9.10.3)

where (Z_{+} ∪ {0})^{2} = (Z_{+} ∪ {0}) × (Z_{+} ∪ {0}). If a_{j} = 0 for all but finitely many j, and b_{l} = 0 for all but finitely many l, then one can check c_{n} = 0 for all but finitely many n. In this case, each of these infinite sums reduces to a finite sum, and both sides of (9.10.2) are equal to (9.10.3).

Suppose for the moment that a_{j}, b_{l} are nonnegative real numbers for each j, l, respectively, so that c_{n} is a nonnegative real number for each n. Under these conditions, the infinite sums mentioned in the preceding paragraph may be defined as nonnegative extended real numbers as in Section 2.4. It is easy to see that the left side of (9.10.2) is equal to (9.10.3) in this sense, as in Section 9.8. One can also check that (9.10.2) holds, or equivalently that the right side of (9.10.2) is equal to (9.10.3), where the right side of (9.10.2) is interpreted as being equal to 0 when either of the factors is equal to 0. In particular, if ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge, then ∑_{n=0}^{∞} c_{n} converges, (9.10.2) holds, and (9.10.3) is finite and equal to both sides of (9.10.2).

If the a_{j}’s and b_{l}’s are any real or complex numbers, then

|c_{n}| ≤ ∑_{j=0}^{n} |a_{j}| |b_{n−j}| (9.10.4)

for each n ≥ 0. Note that the right side is the same as the nth term of the Cauchy product of ∑_{j=0}^{∞} |a_{j}| and ∑_{l=0}^{∞} |b_{l}|. If ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge absolutely, then it follows that ∑_{n=0}^{∞} c_{n} converges absolutely, with

∑_{n=0}^{∞} |c_{n}| ≤ ( ∑_{j=0}^{∞} |a_{j}|) ( ∑_{l=0}^{∞} |b_{l}|). (9.10.5)

We also have that

∑_{(j,l)∈(Z_{+}∪{0})^{2}} |a_{j}| |b_{l}| = ( ∑_{j=0}^{∞} |a_{j}|) ( ∑_{l=0}^{∞} |b_{l}|), (9.10.6)

as before. It follows that f(j, l) = a_{j} b_{l} (9.10.7)

is summable as a real or complex-valued function on (Z_{+} ∪{0})^{2}, which is to say that it is an element of ℓ^{1}((Z_{+} ∪{0})^{2}, R) or ℓ^{1}((Z_{+} ∪{0})^{2}, C), as appropriate.

This means that the sum (9.10.3) converges in the generalized sense, as in Section 2.8. Using this, it is easy to see that the sum is equal to the left side of (9.10.2), as in the previous section. One can also check that (9.10.3) is equal to the right side of (9.10.2), by considering the sum over (Z_{+} ∪ {0})^{2} as an iterated sum over each variable. Of course, this implies that (9.10.2) holds. Alternatively, one can verify (9.10.2) by approximating ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} by finite subsums, using (9.10.5).

If the a_{j}’s and b_{l}’s are real numbers, then the absolute convergence of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} implies that these series may be expressed as differences of convergent series of nonnegative real numbers. This permits one to reduce (9.10.2) to the analogous statement in that case. Similarly, if the a_{j}’s and b_{l}’s are complex numbers, then one can reduce to the real case by considering their real and imaginary parts.

9.11 Cauchy products and bilinear mappings

Let V, W, and Z be vector spaces all real or all complex, and let B be a bilinear mapping from V × W into Z. Also let ∑_{j=0}^{∞} v_{j} and ∑_{l=0}^{∞} w_{l} be infinite series of elements of V and W, respectively. Put

z_{n} = ∑_{j=0}^{n} B(v_{j}, w_{n−j}) (9.11.1)

for each nonnegative integer n. The series ∑_{n=0}^{∞} z_{n} may be described as the Cauchy product of ∑_{j=0}^{∞} v_{j} and ∑_{l=0}^{∞} w_{l} with respect to B. It is easy to see that

∑_{n=0}^{∞} z_{n} = B( ∑_{j=0}^{∞} v_{j}, ∑_{l=0}^{∞} w_{l}), (9.11.2)

at least formally.

In fact, both sides of (9.11.2) correspond to formally expressing

∑_{(j,l)∈(Z_{+}∪{0})^{2}} B(v_{j}, w_{l}) (9.11.3)

as an iterated sum. If v_{j} = 0 for all but finitely many j, and w_{l} = 0 for all but finitely many l, then z_{n} = 0 for all but finitely many n, each of these infinite sums reduces to a finite sum, and both sides of (9.11.2) are equal to (9.11.3).

Let ∥ · ∥_{V}, ∥ · ∥_{W}, and ∥ · ∥_{Z} be norms on V, W, and Z, respectively, and suppose that B is bounded with respect to these norms, so that

∥B(v, w)∥_{Z} ≤ C ∥v∥_{V} ∥w∥_{W} (9.11.4)

for some nonnegative real number C and all v ∈ V, w ∈ W. This implies that

∥z_{n}∥_{Z} ≤ C ∑_{j=0}^{n} ∥v_{j}∥_{V} ∥w_{n−j}∥_{W} (9.11.5)

for every n ≥ 0. The sum on the right is the same as the nth term of the Cauchy product of the series ∑_{j=0}^{∞} ∥v_{j}∥_{V} and ∑_{l=0}^{∞} ∥w_{l}∥_{W}. Suppose from now on in this section that ∑_{j=0}^{∞} v_{j} and ∑_{l=0}^{∞} w_{l} converge absolutely with respect to ∥·∥_{V} and ∥ · ∥_{W}, respectively. This means that ∑_{n=0}^{∞} z_{n} converges absolutely with respect to ∥ · ∥_{Z}, with

∑_{n=0}^{∞} ∥z_{n}∥_{Z} ≤ C ( ∑_{j=0}^{∞} ∥v_{j}∥_{V} ) ( ∑_{l=0}^{∞} ∥w_{l}∥_{W} ), (9.11.6)

as in the previous section.

Similarly,

(9.11.7) ∑_{(j,l)∈(Z_{+}∪{0})^{2}} ∥B(v_{j}, w_{l})∥_{Z} ≤ C ∑_{(j,l)∈(Z_{+}∪{0})^{2}} ∥v_{j}∥_{V} ∥w_{l}∥_{W}

= C ( ∑_{j=0}^{∞} ∥v_{j}∥_{V} ) ( ∑_{l=0}^{∞} ∥w_{l}∥_{W} ).

Using the absolute convergence of ∑_{j=0}^{∞} v_{j} and ∑_{l=0}^{∞} w_{l} with respect to ∥ · ∥_{V} and ∥ · ∥_{W}, respectively, we get that

g(j, l) = B(v_{j}, w_{l}) (9.11.8)

is summable as a Z-valued function on (Z_{+}∪{0})^{2} with respect to ∥·∥_{Z}, which is to say that it is an element of ℓ^{1}((Z_{+} ∪{0})^{2}, Z), as in Section 2.6. Suppose that V, W, and Z are complete with respect to the metrics associated to their norms, so that absolutely convergent series in these spaces converge, as in Section 1.7. We also get that (9.11.3) converges in the generalized sense, as in Section 2.8. The sum is equal to the left side of (9.11.2), as in Section 9.9.

Similarly, (9.11.3) is equal to the right side of (9.11.2), because the sum over (Z_{+} ∪{0})^{2} can be expressed as an iterated sum. As before, one can get (9.11.2) more directly by approximating ∑_{j=0}^{∞} v_{j} and ∑_{l=0}^{∞} w_{l} by finite subsums, using (9.11.6). Note that the completeness of Z is only a convenience here, because the convergence of the sums in Z can be obtained from the convergence of ∑_{j=0}^{∞} v_{j} and ∑_{l=0}^{∞} w_{l} in V and W, respectively.

9.12 Radius of convergence

Let

∑_{j=0}^{∞} a_{j} z^{j} (9.12.1)

be a power series with real or complex coefficients. Also let E_{0} be the set of nonnegative real numbers r such that

(9.12.2) {|a_{j}| r^{j}}_{j=0}^{∞} is a bounded sequence, and let E_{1} be the set of r ≥ 0 such that

∑_{j=0}^{∞} |a_{j}| r^{j} (9.12.3)

converges. Note that 0 ∈ E_{1}, and that E_{1} ⊆ E_{0}. If 0 ≤ r_{1} < r_{0} and r_{0} ∈ E_{0}, then one can check that r_{1} ∈ E_{1}. It follows that

sup E_{0} = sup E_{1}, (9.12.4)

where the suprema are interpreted as being +∞ when E_{0}, E_{1} are unbounded.

Let ρ be the common value in (9.12.4), which is the radius of convergence of the power series (9.12.1). If z ∈ C and |z| < ρ, then

|z| ∈ E_{1}, (9.12.5)

and (9.12.1) converges absolutely, by the comparison test. If |z| > ρ, then

|z| ∉ E_{0}, (9.12.6)

and (9.12.1) does not converge. It is easy to see that ρ is uniquely determined by these two properties.

If r ∈ E_{1}, then (9.12.1) converges absolutely when |z| ≤ r, by the comparison test. This implies that

f(z) = ∑_{j=0}^{∞} a_{j} z^{j} (9.12.7)

defines a complex-valued function on the closed disk

{z ∈ C : |z| ≤ r}. (9.12.8)

We also get that the sequence of partial sums

∑_{j=0}^{n} a_{j} z^{j} (9.12.9)

converges to f uniformly on (9.12.8), by a well-known criterion for uniform convergence due to Weierstrass. It follows that f is continuous on (9.12.8), with respect to the standard Euclidean metric on C and its restriction to (9.12.8), because polynomials define continuous functions on C. More precisely, one can use the same argument to get that f is uniformly continuous on (9.12.8), because polynomials are uniformly continuous on bounded subsets of C.

Suppose that ρ > 0, so that (9.12.7) defines a complex-valued function on

{z ∈ C : |z| < ρ}. (9.12.10)

It is easy to see that f is continuous on (9.12.10), because the restriction of f to (9.12.8) is continuous when 0 ≤ r < ρ, as in the preceding paragraph. Of course, it is well known that f is holomorphic on (9.12.10). Conversely, it is well known that any holomorphic function on an open disk in C centered at 0 of positive radius can be represented by a unique power series in this way, where the radius of convergence of the power series is at least the radius of the disk.

Let

∑_{l=0}^{∞} b_{l} z^{l} (9.12.11)

be another power series with resl or complex coefficients. Put c_{n} = ∑_{j=0}^{n} a_{j} b_{n−j} for each nonnegative integer n, as in Section 9.10, and observe that

c_{n} z^{n} = ∑_{j=0}^{n}(a_{j} z^{j}) (b_{n−j} z^{n−j}) (9.12.12)

for each n ≥ 0. Thus

∑_{n=0}^{∞} c_{n} z^{n} (9.12.13)

is the Cauchy product of (9.12.1) and (9.12.11). If (9.12.1) and (9.12.11) converge absolutely for some z ∈ C, then it follows that (9.12.13) converges absolutely too, and that

∑_{n=0}^{∞} c_{n} z^{n} = ( ∑_{j=0}^{∞} a_{j} z^{j}) ( ∑_{l=0}^{∞} b_{l} z^{l}), (9.12.14)

as before.

9.13 Cauchy products and Banach algebras

Let A be an algebra in the strict sense over the real or complex numbers, and let ∑_{j=0}^{∞} v_{j}, ∑_{l=0}^{∞} w_{l} be infinite series of elements of A. Put

z_{n} = ∑_{j=0}^{n} v_{j} w_{n−j} (9.13.1)

for each nonnegative integer n, so that ∑_{n=0}^{∞} z_{n} is the Cauchy product of ∑_{j=0}^{∞} v_{j} and ∑_{l=0}^{∞} w_{l} with respect to multiplication on A, as in Section 9.11. As before,

∑_{n=0}^{∞} z_{n} = ( ∑_{j=0}^{∞} v_{j}) ( ∑_{l=0}^{∞} w_{l}), (9.13.2)

at least formally. In particular, if v_{j} = 0 for all but finitely many j, and w_{l} = 0 for all but finitely many l, then z_{n} = 0 for all but finitely many n. In this case, the infinite series reduce to finite sums that satisfy (9.13.2).

Let ∥ · ∥ be a submultiplicative norm on A, and note that

∥z_{n}∥_{A} ≤ ∑_{j=0}^{n} ∥v_{j}∥_{A} ∥w_{n−j}∥_{A} (9.13.3)

for each n ≥ 0. If ∑_{j=0}^{∞} v_{j}, ∑_{l=0}^{∞} w_{l} converge absolutely with respect to ∥ · ∥_{A}, then ∑_{n=0}^{∞} z_{n} converges absolutely with respect to ∥ · ∥_{A}, with

∑_{n=0}^{∞} ∥z_{n}∥_{A} ≤ ( ∑_{j=0}^{∞} ∥v_{j}∥_{A}) ( ∑_{l=0}^{∞} ∥w_{l}∥_{A}). (9.13.4)

If A is complete with respect to metric associated to ∥ · ∥_{A}, then we have seen that (9.13.2) holds under these conditions.

Suppose now that (A, ∥·∥_{A}) is a Banach algebra with a multiplicative identity element e_{A}. Let ∑_{j=0}^{∞} a_{j} z^{j} be a power series with real or complex coefficients, as appropriate. If x ∈ A and

∑_{j=0}^{∞} |a_{j}| ∥x^{j}∥_{A} (9.13.5)

converges as an infinite series of nonnegative real numbers, then

∑_{j=0}^{∞} a_{j} x^{j} (9.13.6)

converges absolutely with respect to ∥·∥_{A}, and thus converges in A, as in Section 1.7. In particular, if

∑_{j=0}^{∞} |a_{j}| ∥x∥_{A}^{j} (9.13.7)

converges, then (9.13.5) converges, by the comparison test.

Let ∑_{l=0}^{∞} b_{l} z^{l} be another power series with real or complex coefficients, as appropriate, and let ∑_{n=0}^{∞} c_{n} z^{n} be the Cauchy product of ∑_{j=0}^{∞} a_{j} z^{j} and ∑_{l=0}^{∞} b_{l} z^{l}. Suppose that

∑_{l=0}^{∞} |b_{l}| ∥x^{l}∥_{A} (9.13.8)

converges as an infinite series of nonnegative real numbers too, so that

∑_{l=0}^{∞} b_{l} x^{l} (9.13.9)

converges absolutely with respect to ∥ · ∥_{A}. Note that

c_{n} x^{n} = ∑_{j=0}^{n}(a_{j} x^{j}) (b_{n−j} x^{n−j}) (9.13.10)

for each n ≥ 0, so that

∑_{n=0}^{∞} c_{n} x^{n} (9.13.11)

is the Cauchy product of (9.13.6) and (9.13.9) with respect to multiplication on A. Using (9.13.10), we get that

|c_{n}| ∥x^{n}∥_{A} ≤ ∑_{j=0}^{n}(|a_{j}| ∥x^{j}∥_{A}) (|b_{n−j}| ∥x^{n−j}∥_{A}) (9.13.12)

for each n ≥ 0.

The absolute convergence of (9.13.6) and (9.13.9) with respect to ∥ · ∥_{A} implies that (9.13.11) converges absolutely with respect to ∥ · ∥_{A} too, with

∑_{n=0}^{∞} |c_{n}| ∥x^{n}∥_{A} ≤ ( ∑_{j=0}^{∞} |a_{j}| ∥x^{j}∥_{A}) ( ∑_{l=0}^{∞} |b_{l}| ∥x^{l}∥_{A}). (9.13.13)

We also have that

∑_{n=0}^{∞} c_{n} x^{n} = ( ∑_{j=0}^{∞} a_{j} x^{j}) ( ∑_{l=0}^{∞} b_{l} x^{l}) (9.13.14)

under these conditions. Of course, if x is nilpotent in A, then (9.13.6), (9.13.9), and (9.13.11) reduce to finite sums, and (9.13.14) holds automatically. More precisely, one does not need a norm on A in this case.

9.14 Power series and Banach algebras

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers, with a multiplicative identity element e_{A}, and let ∑_{j=0}^{∞} a_{j} z^{j} be a power series with real or complex coefficients, as appropriate. Also let r be a nonnegative real number in the set E_{1} defined in Section 9.12, so that ∑_{j=0}^{∞} |a_{j}| r^{j} converges. If x ∈ A and ∥x∥_{A} ≤ r, then (9.13.7) converges, by the comparison test. This implies that (9.13.6) converges absolutely with respect to ∥ · ∥_{A}, so that

f̃(x) = ∑_{j=0}^{∞} a_{j} x^{j} (9.14.1)

defines an A-valued function on

{x ∈ A : ∥x∥_{A} ≤ r}. (9.14.2)

Note that (9.14.1) is bounded on (9.14.2), with

∥f̃(x)∥_{A} ≤ ∑_{j=0}^{∞} |a_{j}| r^{j} (9.14.3)

when ∥x∥_{A} ≤ r and ∥e_{A}∥_{A} = 1.

As in Section 9.12, the sequence of partial sums

∑_{j=0}^{n} a_{j} x^{j} (9.14.4)

converges uniformly to (9.14.1) on (9.14.2), essentially as in Weierstrass’ criterion for uniform convergence. These partial sums are continuous as A-valued functions of x on A, with respect to the metric associated to ∥·∥_{A}. In fact, these partial sums are uniformly continuous on bounded subsets of A, and on (9.14.2) in particular. It follows that (9.14.1) is uniformly continuous on (9.14.2).

Let ρ be the radius of convergence of ∑_{j=0}^{∞} a_{j} z^{j}, as in Section 9.12, and suppose that ρ > 0. If x ∈ A and

∥x∥_{A} < ρ, (9.14.5)

then ∥x∥_{A} ∈ E_{1}, which is to say that (9.13.7) converges. This implies that (9.13.6) converges absolutely with respect to ∥ · ∥_{A}, as before, so that (9.14.1) defines an A-valued function on

{x ∈ A : ∥x∥_{A} < ρ}. (9.14.6)

We also have that (9.14.1) is continuous on (9.14.6), with respect to the metric associated to ∥ · ∥_{A}, and its restriction to (9.14.6). This follows from the fact that the restriction of (9.14.1) to (9.14.2) is continuous when 0 ≤ r < ρ, as in the preceding paragraph.

Suppose now that x ∈ A satisfies

r_{A}(x) = lim_{j→∞} ∥x^{j}∥_{A}^{1/j} < ρ. (9.14.7)

This implies that there is an r ∈ R such that

lim_{j→∞} ∥x^{j}∥_{A}^{1/j} < r < ρ. (9.14.8)

It follows that ∥x^{j}∥_{A}^{1/j} < r (9.14.9)

for all but finitely many j, so that

∥x^{j}∥_{A} < r^{j} (9.14.10)

for all but finitely many j. Note that ∑_{j=0}^{∞} |a_{j}| r^{j} converges, because r < ρ, as in Section 9.12. This implies that (9.13.5) converges, by the comparison test, so that (9.13.6) converges absolutely with respect to ∥ · ∥_{A}, and we can define f̃(x) ∈ A as in (9.14.1).

In the complex case, if f is a holomorphic function on an open subset U of C that contains the spectrum of x ∈ A. then f̃(x) ∈ A can be defined in a natural way, as mentioned in Section 8.13. If U is an open disk centered at 0 and r_{A}(x) is less than the radius of U, then this corresponds to the remarks in the preceding paragraph.

9.15. MORE ON F̃(X) 219

9.15 More on f̃(x)

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A}, and let ∑_{j=0}^{∞} a_{j}z^{j} be a power series with real or complex coefficients, as appropriate, again. Suppose that ∑_{j=0}^{∞} |a_{j}| r^{j} converges for some r ≥ 0, and put f(z) = ∑_{j=0}^{∞} a_{j} z^{j} when z ∈ R or C, as appropriate, and |z| ≤ r. Similarly, if x ∈ A and ∥x∥_{A} ≤ r, then f̃(x) ∈ A may be defined as in (9.14.1).

Put

p_{n}(z) = ∑_{j=0}^{n} a_{j} z^{n} (9.15.1)

for each nonnegative integer n, so that p_{n}(z) → f(z) as n → ∞ uniformly for z ∈ R or C, as appropriate, with |z| ≤ r, as in Section 9.12. If x ∈ A, then

p̃_{n}(x) = ∑_{j=0}^{n} a_{j} x^{j} (9.15.2)

defines an element of A for each n, as in Section 8.13. If ∥x∥_{A} ≤ r, then

p̃_{n}(x) → f̃(x) as n → ∞, (9.15.3)

with respect to the metric on A associated to ∥ · ∥_{A}. Similarly, if w ∈ R or C, as appropriate, and |w| ≤ r, then

p̃_{n}(x) − p_{n}(w) e_{A} → f̃(x) − f(w) e_{A} as n → ∞. (9.15.4)

If w ∈ σ_{A}(x), then p̃_{n}(x) − p_{n}(w) e_{A} is not invertible in A for any n, as in Section 8.13. This implies that f̃(x) − f(w) e_{A} is not invertible, because of (9.15.4), and the fact that the invertible elements of A form an open set. This means that f(w) ∈ σ_{A}(f̃(x)). (9.15.5)

It follows that f(σ_{A}(x)) ⊆ σ_{A}(f̃(x)). (9.15.6)

More precisely, if w ∈ σ_{A}(x), then |w| ≤ r, as in Section 6.8, so that (9.15.5) holds.

Similarly, if ∑_{j=0}^{∞} a_{j} z^{j} has radius of convergence ρ > 0, and x ∈ A satisfies (9.14.7), then f̃(x) ∈ A may be defined as in (9.14.1), and satisfies (9.15.3). If w ∈ R or C and |w| < ρ, then f(w) can be defined in the usual way, and

p_{n}(w) → f(w) as n → ∞, (9.15.7)

so that (9.15.4) holds. If w ∈ σ_{A}(x), then

|w| ≤ r_{A}(x) < ρ, (9.15.8) where the first step is as in Section 6.13. In this case, (9.15.5) holds, for the same reasons as before. This implies that (9.15.6) holds, as before.

Suppose now that (V, ∥ · ∥_{V}) is a Banach space over the real or complex numbers with V ≠ {0}, and that A = BL(V), with the corresponding operator norm ∥ · ∥_{op}. Suppose again that ∑_{j=0}^{∞} |a_{j}| r^{j} converges, so that f̃(T) ∈ BL(V) may be defined as before when T ∈ BL(V) and

∥T ∥_{op} ≤ r. (9.15.9)

Let λ ∈ R or C, as appropriate, be an eigenvalue of T with eigenvector v, and remember that |λ| ≤ ∥T ∥_{op}, as in Section 9.2. Observe that

(f̃(T))(v) = f(λ) v, (9.15.10)

because of the analogous statement for polynomials. This means that

f(λ) ∈ σ_{p}(f̃(T)), (9.15.11)

so that f(σ_{p}(T)) ⊆ σ_{p}(f̃(T)). (9.15.12)

Now let λ ∈ R or C, as appropriate, be an approximate eigenvalue of T, and let {v_{j}}_{j=1}^{∞} be a corresponding sequence of unit vectors in V, as in Section 9.3. Thus |λ| ≤ ∥T ∥_{op}, as before. One can check that

∥(f̃(T))(v_{j}) − f(λ) v_{j}∥_{V} → 0 as j → ∞, (9.15.13)

using the analogous statement for T^{l} for each l ≥ 1. This means that f(λ) is an approximate eigenvalue of f̃(T), so that

f(λ) ∈ σ_{ap}(f̃(T)). (9.15.14)

It follows that f(σ_{ap}(T)) ⊆ σ_{ap}(f̃(T)). (9.15.15)

If ∑_{j=0}^{∞} a_{j} z^{j} has radius of convergence ρ > 0 and T ∈ BL(V) satisfies

r_{BL(V)}(T) < ρ, (9.15.16)

then f̃(T) ∈ BL(V) may be defined as before. Suppose that λ ∈ R or C is an eigenvalue of T with eigenvector v, so that

|λ| ≤ r_{BL(V)}(T) < ρ, (9.15.17)

where the first step is as in Section 9.2. In this case, (9.15.10), (9.15.11), and (9.15.12) hold, for essentially the same reasons as before. Similarly, if λ is an approximate eigenvalue of T, then the first inequality in (9.15.17) holds, as in Section 9.3. If {v_{j}}_{j=1}^{∞} is a corresponding sequence of unit vectors in V, then (9.15.13), (9.15.14), and (9.15.15) hold, for essentially the same reasons as before.

Some more algebras, power series

10.1 Power series and Lipschitz conditions

Let A be an associative algebra over the real or complex numbers. If x, y ∈ A and n is a positive integer, then

x^{n} − y^{n} = ∑_{j=1}^{n}(x^{j} y^{n−j} − x^{j−1} y^{n−j+1}) = ∑_{j=1}^{n} x^{j−1} (x − y) y^{n−j}. (10.1.1)

More precisely, one can simply drop the factors of x^{j−1} when j = 1, as well as the factors of y^{n−j} when j = n. If ∥ · ∥_{A} is a submultiplicative norm on A, then we get that

(10.1.2)  ∥x^{n} − y^{n}∥_{A} ≤ ∑_{j=1}^{n} ∥x^{j−1}∥_{A} ∥x − y∥_{A} ∥y^{n−j}∥_{A}
≤ n max(∥x∥_{A}, ∥y∥_{A})^{n−1} ∥x − y∥_{A}.

If r is a nonnegative real number and ∥x∥_{A}, ∥y∥_{A} ≤ r, then we get that

∥x^{n} − y^{n}∥_{A} ≤ n r^{n−1} ∥x − y∥_{A}. (10.1.3)

This means that x ↦ x^{n} (10.1.4)

is Lipschitz with constant n r^{n−1}, as an A-valued function on

{x ∈ A : ∥x∥_{A} ≤ r}. (10.1.5)

Of course, this uses the metric on A associated to ∥ · ∥_{A}, and its restriction to (10.1.5). This Lipschitz condition could also be obtained from some of the remarks in Section 7.10.

Suppose now that A is a Banach algebra with a multiplicative identity element e_{A}, and let ∑_{j=0}^{∞} a_{j} z^{j} be a power series with real or complex coefficients, as appropriate. Suppose also that

∑_{j=1}^{∞} j |a_{j}| r^{j−1} (10.1.6)

converges, which implies in particular that ∑_{j=0}^{∞} |a_{j}| r^{j} converges. Thus

f̃(x) = ∑_{j=0}^{∞} a_{j} x^{j} (10.1.7)

defines an A-valued function on (10.1.5), as in Section 9.14. If x, y ∈ A and ∥x∥_{A}, ∥y∥_{A} ≤ r, then

f̃(x) − f̃(y) = ∑_{j=1}^{∞} a_{j} (x^{j} − y^{j}), (10.1.8)

because the j = 0 terms cancel. This implies that

∥f̃(x) − f̃(y)∥_{A} ≤ ∑_{j=1}^{∞} |a_{j}| ∥x^{j} − y^{j}∥_{A}. (10.1.9)

It follows that

∥f̃(x) − f̃(y)∥_{A} ≤ ∑_{j=1}^{∞} j |a_{j}| r^{j−1} ∥x − y∥_{A}, (10.1.10)

because of (10.1.3). This means that (10.1.7) is Lipschitz on (10.1.5), with constant (10.1.6). Of course, if a_{j} = 0 for all but finitely many j, then the completeness of A is not needed here. If ∑_{j=0}^{∞} a_{j} z^{j} has radius of convergence ρ, as in Section 9.12, and if 0 ≤ r < ρ, then it is well known that (10.1.6) converges.

Note that

∑_{j=1}^{∞} j a_{j} z^{j−1} = ∑_{j=0}^{∞}(j + 1) a_{j+1} z^{j} (10.1.11)

is the power series obtained by differentiating ∑_{j=0}^{∞} a_{j} z^{j} formally term-by-term. If (10.1.6) converges, then (10.1.11) converges absolutely for every z ∈ R or C, as appropriate, with |z| ≤ r, by the comparison test. If r > 0, then it is well known that (10.1.11) is in fact the derivative of ∑_{j=0}^{∞} a_{j} z^{j} when |z| ≤ r, considered as a one-sided derivative in the real case when |z| = r. In the complex case, one should use an appropriate complex derivative here, and a suitable interpretation for |z| = r. In both cases, one can use the same type of Lipschitz conditions as before to show that the derivative of the sum is equal to the sum of the derivatives.

10.2 Another property of power series

Let ∑_{j=0}^{∞} a_{j} z^{j} be a power series with real or complex coefficients, and suppose that (10.1.6) converges for some r > 0, so that ∑_{j=0}^{∞} |a_{j}| r^{j} converges, as before. If w, z ∈ R or C, as appropriate, and |z|, |w| ≤ r, then ∑_{j=0}^{∞} a_{j} z^{j} and ∑_{j=0}^{∞} a_{j} w^{j} converge absolutely, and

(10.2.1) ∑_{j=0}^{∞} a_{j} z^{j} − ∑_{j=0}^{∞} a_{j} w^{j} = ∑_{j=1}^{∞} a_{j} (z^{j} − w^{j})

= ∑_{j=1}^{∞} a_{j} ( ∑_{l=0}^{j−1} z^{l} (z − w) w^{j−l−1}),

where the second step is as in (10.1.1). We would like to reexpress this as

∑_{j=0}^{∞} a_{j} z^{j} − ∑_{j=0}^{∞} a_{j} w^{j} = (z − w) ∑_{j=1}^{∞} a_{j}( ∑_{l=0}^{j−1} z^{l} w^{j−l−1}). (10.2.2)

If z ≠ w, then this follows from (10.2.1), including the convergence of the series on the right. If z = w, then the convergence of the series on the right follows from the convergence of (10.1.6).

Put

b_{l}(w) = ∑_{j=l+1}^{∞} a_{j} w^{j−l−1} (10.2.3)

for each l ≥ 0, and note that the series on the right converges absolutely. It is easy to see that

∑_{l=0}^{∞} b_{l}(w) z^{l} = ∑_{j=1}^{∞} a_{j} ( ∑_{l=0}^{j−1} z^{l} w^{j−l−1}), (10.2.4)

at least formally, by interchanging the order of summation. This means that

∑_{j=0}^{∞} a_{j} z^{j} − ∑_{j=0}^{∞} a_{j} w^{j} = (z − w) ∑_{l=0}^{∞} b_{l}(w) z^{l}, (10.2.5)

at least formally, by (10.2.2).

Let c_{j,l} be defined for j, l ≥ 0 by

(10.2.6)  c_{j,l} = a_{j} when 0 ≤ l ≤ j − 1
= 0 otherwise.

Observe that

∑_{j=0}^{∞} ( ∑_{l=0}^{∞} |c_{j,l}| r^{j−1}) = ∑_{j=1}^{∞} ( ∑_{l=0}^{j−1} |a_{j}| r^{j−1}) = ∑_{j=1}^{∞} j |a_{j}| r^{j−1}. (10.2.7)

This means that

∑_{(j,l)∈(Z_{+}∪{0})^{2}} |c_{j,l}| r^{j−1} = ∑_{j=1}^{∞} j |a_{j}| r^{j−1}. (10.2.8)

More precisely, the left side of (10.2.8), as defined in Section 2.4, is equal to the left side of (10.2.7), as in Section 9.8.

Similarly,

∑_{(j,l)∈(Z_{+}∪{0})^{2}} |c_{j,l}| r^{j−1} = ∑_{l=0}^{∞} ( ∑_{j=0}^{∞} |c_{j,l}| r^{j−1}) = ∑_{l=0}^{∞} ( ∑_{j=l+1}^{∞} |a_{j}| r^{j−1}). (10.2.9)

Thus

∑_{l=0}^{∞} ( ∑_{j=l+1}^{∞} |a_{j}| r^{j−1}) = ∑_{j=1}^{∞} j |a_{j}| r^{j−1}, (10.2.10)

by (10.2.8). Note that

|b_{l}(w)| ≤ ∑_{j=l+1}^{∞} |a_{j}| r^{j−l−1} (10.2.11)

for each l ≥ 0, so that

∑_{l=0}^{∞} |b_{l}(w)| r^{l} ≤ ∑_{j=1}^{∞} j |a_{j}| r^{j−1}. (10.2.12)

Clearly

∑_{(j,l)∈(Z_{+}∪{0})^{2}} |c_{j,l}| |z|^{l} |w|^{j−l−1} ≤ ∑_{(j,l)∈(Z_{+}∪{0})^{2}} |c_{j,l}| r^{j−1}, (10.2.13)

where the summand on the left is interpreted as being equal to 0 unless l ≤ j−1, even when w = 0. Of course, the right side is finite, by (10.2.8). It follows that

∑_{(j,l)∈(Z_{+}∪{0})^{2}} c_{j,l} z^{l} w^{j−l−1} (10.2.14)

converges in the generalized sense, as in Section 2.8, where again the summand is interpreted as being equal to 0 unless l ≤ j − 1. Both sides of (10.2.4) are equal to (10.2.14), by summing in one index at a time, as in Section 9.9. Thus (10.2.4) holds, which implies that (10.2.5) holds.

10.3 Taking w to be fixed

Let us continue with the same notation and hypotheses as in the previous section, and let us now take w to be fixed. Put f(z) = ∑_{j=0}^{∞} a_{j} z^{j} and

g(z) = ∑_{l=0}^{∞} b_{l}(w) z^{l}, (10.3.1)

10.3. TAKING W TO BE FIXED 225

so that f(z) − f(w) = (z − w) g(z), (10.3.2)

by (10.2.5).

If ∑_{j=0}^{∞} a_{j} z^{j} has radius of convergence ρ > 0, then it is well known that (10.1.6) converges when 0 < r < ρ. This implies that ∑_{l=0}^{∞} b_{l}(w) z^{l} has radius of convergence at least ρ, because of (10.2.12). More precisely, this works for any w ∈ R or C, as appropriate, such that |w| < ρ.

Note that the radius of convergence of ∑_{j=0}^{∞} a_{j} z^{j} is less than or equal to the radius of convergence of ∑_{l=0}^{∞} b_{l}(w) z^{l}, because of (10.2.5). Thus the radius of convergence of ∑_{l=0}^{∞} b_{l}(w) z^{l} is equal to ρ too,

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers, as appropriate, with a multiplicative identity element e_{A}. If x ∈ A and ∥x∥_{A} ≤ r, then f̃(x) = ∑_{j=0}^{∞} a_{j} x^{j} defines an element of A, as in Section 9.14. Similarly,

g̃(x) = ∑_{l=0}^{∞} b_{l}(w) x^{l} (10.3.3)

defines an element of A under these conditions. We also have that

f̃(x) − f(w) e_{A} = (x − w e_{A}) g̃(x). (10.3.4)

This could be shown in the same way as before, or obtained from (10.3.2).

If ∑_{j=0}^{∞} a_{j} z^{j} has radius of convergence ρ, then f̃(x) and g̃(x) can be defined in the same way when ∥x∥_{A} < ρ, and (10.3.4) holds. Similarly, if r_{A}(x) < ρ, then f̃(x) and g̃(x) can be defined in the same way, as in Section 9.14. We also have that (10.3.4) holds in this case. As before, we can take w to be any real or complex number with |w| < ρ here.

Of course, g̃(x) and x − w e_{A} commute with each other. If f̃(x) − f(w) e_{A} is invertible in A, then it follows that x − w e_{A} is invertible in A, as in Section 6.13. This implies that f(w) ∈ σ_{A}(f̃(x)) (10.3.5)

when w ∈ σ_{A}(x).

It follows that f(σ_{A}(x)) ⊆ σ_{A}(f̃(x)). (10.3.6)

More precisely, if ∥x∥_{A} ≤ r and w ∈ σ_{A}(x), then |w| ≤ r, so that (10.3.5) holds. Remember that the same conclusions were obtained in Section 9.15, under the hypothesis that ∑_{j=0}^{∞} |a_{j}| r^{j} converges, without asking that (10.1.6) converge.

Similarly, if ∑_{j=0}^{∞} a_{j} z^{j} has radius of convergence ρ, r_{A}(x) < ρ, and w is an element of σ_{A}(x), then |w| ≤ r_{A}(x) < ρ, (10.3.7)

as in Section 6.13. In this case, (10.3.5) holds, for essentially the same reasons as before. This implies that (10.3.6) holds, as before. This is another approach to the analogous statements in Section 9.15.

Let (V, ∥ · ∥_{V}) be a Banach space over the real or complex numbers, as appropriate, with V ≠ {0}, and suppose now that A = BL(V), with the corresponding operator norm. If T ∈ BL(V) and ∥T ∥_{op} ≤ r, and λ ∈ R or C is an eigenvalue or approximate eigenvalue of T, then we saw in Section 9.15 that f(λ) is an eigenvalue or approximate eigenvector of f̃(T), as appropriate. These statements could also be obtained from (10.3.2) when (10.1.6) converges. Similarly, if ∑_{j=0}^{∞} a_{j} z^{j} has radius of convergence ρ > 0 and r_{BL(V)}(T) < ρ, then the analogous statements could be obtained from (10.3.4).

10.4 The exponential function

Let (A, ∥·∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. If x ∈ A, then the exponential of x in A may be defined by the usual power series,

exp x = exp_{A} x = ∑_{j=0}^{∞}(1/j!) x^{j}. (10.4.1)

More precisely, the series on the right converges absolutely with respect to ∥·∥_{A}, because

∑_{j=0}^{∞}(1/j!) ∥x^{j}∥_{A} ≤ ∑_{j=0}^{∞}(1/j!)∥x∥_{A}^{j} = exp ∥x∥_{A}, (10.4.2)

using the ordinary exponential function on the right side. This also shows that

∥ exp x∥_{A} ≤ exp ∥x∥_{A}. (10.4.3)

Note that exp_{A} 0 = e_{A}.

If x, y ∈ A commute with each other, then

(x + y)^{n} = ∑_{j=0}^{n} \binom{n}{j} x^{j} y^{n−j} (10.4.4)

for each nonnegative integer n, with suitable interpretations when n = 0, by the binomial theorem. This implies that

(10.4.5)  exp(x + y) = ∑_{n=0}^{∞}(1/n!) (x + y)^{n}

= ∑_{n=0}^{∞} ( ∑_{j=0}^{n}(1/j!) (1/(n − j)!) x^{j} y^{n−j}).

The right side corresponds to the Cauchy product of the series defining exp x and exp y, as in Section 9.13. It follows that

(10.4.6) exp(x + y) = (exp x) (exp y) under these conditions, because these series converge absolutely, as before.

In particular,

(10.4.7) (exp x) (exp(−x)) = (exp(−x)) (exp x) = e_{A}

for every x ∈ A. This means that exp x is invertible in A, with

(exp x)^{−1} = exp(−x). (10.4.8)

Let (B, ∥ · ∥_{B}) be another Banach algebra over the real or complex numbers, as appropriate, with a multiplicative identity element e_{B}. Also let ϕ be an algebra homomorphism from A into B that is bounded as a linear mapping and satisfies ϕ(e_{A}) = e_{B}. If x ∈ A, then it is easy to see that

ϕ(exp_{A} x) = exp_{B} ϕ(x). (10.4.9)

This works as well when ϕ is an opposite algebra homomorphism, as in Section 6.4. In the complex case, one may wish to consider conjugate-linear mappings too, which may be considered as real-linear in particular.

Let x ↦ x^{∗} be an algebra involution on A, as in Section 6.4, which may be conjugate-linear in the complex case. Suppose that this involution is bounded as a linear or conjugate-linear mapping from A into itself, as appropriate, so that ∥x^{∗}∥_{A} ≤ C ∥x∥_{A} (10.4.10)

for some nonnegative real number C and every x ∈ A. Thus

(exp x)^{∗} = exp(x^{∗}) (10.4.11)

for every x ∈ A, as in (10.4.9). If x is anti-self-adjoint with respect to this involution, in the sense that x^{∗} = −x, then

(10.4.12) (exp x)^{∗} = exp(x^{∗}) = exp(−x) = (exp x)^{−1}.

This means that exp x is an element of the subgroup U(A) of the group G(A) of invertible elements of A defined in Section 7.5, using this involution.

10.5 Algebra derivations

Let A be an algebra in the strict sense over R or C. A linear mapping δ from A into itself is said to be a derivation if

δ(x y) = δ(x) y + x δ(y) (10.5.1)

for every x, y ∈ A, as on p4 of [158], p7 of [300], and p2 of [301]. The derivations of A form a linear subspace of the space L(A) of all linear mappings from A into itself. If δ_{1}, δ_{2} are derivations on A, then one can check that

δ_{1} ∘ δ_{2} − δ_{2} ∘ δ_{1} (10.5.2) is a derivation on A as well, as in [158, 300, 301].

Let δ be a derivation on A, and let n be a nonnegative integer. If x, y ∈ A, then it is well known that

δ^{n}(x y) = ∑_{j=0}^{n} \binom{n}{j} δ^{j}(x) δ^{n−j}(y). (10.5.3)

This is a version of the Leibniz rule.

Let ∥ · ∥_{A} be a norm on A, and suppose that multiplication on A is bounded as a bilinear mapping from A × A into A with respect to ∥ · ∥_{A}, so that

∥x y∥_{A} ≤ C ∥x∥_{A} ∥y∥_{A} (10.5.4)

for some nonnegative real number C and all x, y ∈ A. Suppose also that A is complete with respect to the metric associated to ∥·∥_{A}, so that the space BL(A) of bounded linear mappings from A into itself is a Banach algebra with respect to the corresponding operator norm ∥ · ∥_{op}. If δ is bounded as a linear mapping from A into itself, then exp δ may be defined as a bounded linear mapping from A into itself too, as in the previous section.

If x, y ∈ A, then

(10.5.5) (exp δ)(x y) = ∑_{n=0}^{∞}(1/n!) δ^{n}(x y)

= ∑_{n=0}^{∞} ( ∑_{j=0}^{n}(1/j!) (1/(n − j)!) δ^{j}(x) δ^{n−j}(y)),

using (10.5.3) in the second step. The right side corresponds to the Cauchy product of

(10.5.6) (exp δ)(x) = ∑_{j=0}^{∞}(1/j!) δ^{j}(x) and (exp δ)(y) = ∑_{l=0}^{∞}(1/l!) δ^{l}(y),

as in Section 9.11. Note that these series converge absolutely with respect to ∥ · ∥_{A}, because δ is bounded on A. It follows that

(10.5.7) (exp δ)(x y) = ((exp δ)(x)) ((exp δ)(y)),

as before. This shows that exp δ is an automorphism of A, as an algebra in the strict sense.

10.6 Commutators and conjugations

Let A be an associative algebra over the real or complex numbers. If a ∈ A, then put δ_{a}(x) = a x − x a (10.6.1) for every x ∈ A. This defines a linear mapping from A into itself. One can check that (10.6.2) δ_{a} is a derivation on A.

This type of derivation in this case is called an inner derivation of A.

Remember that L_{a}, R_{a} are the left and right multiplication operators on A associated to a as in Sections 6.3 and 6.4. Note that

δ_{a} = L_{a} − R_{a}. (10.6.3)

We also have that L_{a} ∘ R_{a} = R_{a} ∘ L_{a}, (10.6.4)

because A is associative.

Suppose that (A, ∥ · ∥_{A}) is a Banach algebra with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Remember that L_{a} and R_{a} are bounded linear mappings on A, with ∥L_{a}∥_{op} = ∥R_{a}∥_{op} = ∥a∥_{A}, (10.6.5)

as before. This implies that δ_{a} is bounded as a linear mapping from A into itself, with ∥δ_{a}∥_{op} ≤ 2 ∥a∥_{A}. (10.6.6)

Of course, the space BL(A) of bounded linear mappings from A into itself is a Banach algebra with respect to the operator norm, because A is complete, by hypothesis. Thus the exponentials of L_{a}, R_{a}, and δ_{a} may be defined as elements of BL(A) as in Section 10.4. In fact, we have that

(10.6.7)  exp δ_{a} = exp(L_{a} − R_{a}) = (exp L_{a}) ∘ (exp(−R_{a}))
= (exp L_{a}) ∘ (exp R_{a})^{−1},

because of (10.6.4).

Remember that a ↦ L_{a} is an algebra homomorphism from A into BL(A), with L_{e_{A}} equal to the identity mapping I_{A} on A, as in Section 6.3. Similarly, a ↦ R_{a} is an opposite algebra homomorphism from A into BL(A), with R_{e_{A}} = I_{A}, as in Section 6.4. This implies that

(10.6.8) exp L_{a} = exp_{BL(A)} L_{a} = L_{exp_{A} a}

and (10.6.9) exp R_{a} = exp_{BL(A)} R_{a} = R_{exp_{A} a},

as in Section 10.4. If x ∈ A, then we get that

(10.6.10) (exp δ_{a})(x) = (exp_{BL(A)} δ_{a})(x) = (exp_{A} a) x (exp_{A} a)^{−1},

by (10.6.7). This corresponds to Exercise 15 on p260 of [286].

230 CHAPTER 10. SOME MORE ALGEBRAS, POWER SERIES

10.7 A criterion for local invertibility

Let (V, ∥ · ∥_{V}) be a Banach space over the real or complex numbers, let v_{0} ∈ V and r_{0} > 0 be given, and remember that \overline{B}(v_{0}, r_{0}) is the closed ball in V centered at v_{0} with radius r_{0} with respect to the metric associated to ∥ · ∥_{V}, as in Section 1.2. Also let g be a mapping from \overline{B}(v_{0}, r_{0}) into V, and put ϕ(v) = v − g(v) (10.7.1) on \overline{B}(v_{0}, r_{0}). Suppose that (10.7.2) ϕ is Lipschitz with constant c_{0} ≥ 0 on \overline{B}(v_{0}, r_{0}), with respect to the metric on V associated to ∥ · ∥_{V} and its restriction to \overline{B}(v_{0}, r_{0}). This implies that (10.7.3) g is Lipschitz with constant 1 + c_{0} on \overline{B}(v_{0}, r_{0}), as in Section 2.1. Suppose too that c_{0} < 1, and note that (1 − c_{0}) ∥v − w∥_{V} ≤ ∥g(v) − g(w)∥_{V} (10.7.4) for every v, w ∈ \overline{B}(v_{0}, r_{0}), as in Section 7.11.

If y ∈ V, then put ϕ_{y}(v) = ϕ(v) + y = v − g(v) + y (10.7.5) on \overline{B}(v_{0}, r_{0}). Clearly (10.7.6) ϕ_{y} is Lipschitz with constant c_{0} on \overline{B}(v_{0}, r_{0}), because of (10.7.2). Observe that

(10.7.7) ϕ_{y}(v) − v_{0} = ϕ(v) + y − v_{0} = ϕ(v) − ϕ(v_{0}) + ϕ(v_{0}) + y − v_{0}
= ϕ(v) − ϕ(v_{0}) + y − g(v_{0})

on \overline{B}(v_{0}, r_{0}). This implies that

∥ϕ_{y}(v) − v_{0}∥_{V} ≤ ∥ϕ(v) − ϕ(v_{0})∥_{V} + ∥y − g(v_{0})∥_{V}
(10.7.8)  ≤ c_{0} ∥v − v_{0}∥_{V} + ∥y − g(v_{0})∥_{V} ≤ c_{0} r_{0} + ∥y − g(v_{0})∥_{V}

on \overline{B}(v_{0}, r_{0}).

Suppose that ∥y − g(v_{0})∥_{V} ≤ (1 − c_{0}) r_{0}, (10.7.9) so that ∥ϕ_{y}(v) − v_{0}∥_{V} ≤ r_{0} (10.7.10) on \overline{B}(v_{0}, r_{0}), by (10.7.8). Equivalently, this means that ϕ_{y}(\overline{B}(v_{0}, r_{0})) ⊆ \overline{B}(v_{0}, r_{0}). (10.7.11)

10.7. A CRITERION FOR LOCAL INVERTIBILITY 231 Note that \overline{B}(v_{0}, r_{0}) is complete as a metric space with respect to the restriction of the metric on V associated to ∥·∥_{V}, because V is complete, by hypothesis, and \overline{B}(v_{0}, r_{0}) is a closed set in V, as in Section 1.6. Thus the contraction mapping theorem implies that there is a unique point v_{y} ∈ \overline{B}(v_{0}, r_{0}) such that ϕ_{y}(v_{y}) = v_{y}, (10.7.12) as in Section 7.12.

Of course, (10.7.12) is the same as saying that g(v_{y}) = y, (10.7.13) by the definition (10.7.5) of ϕ_{y}. This shows that \overline{B}(v_{0}, (1 − c_{0}) r_{0}) ⊆ g(\overline{B}(v_{0}, r_{0})) (10.7.14) under these conditions. We also have that (1 − c_{0}) ∥v_{y} − v_{0}∥_{V} ≤ ∥g(v_{y}) − g(v_{0})∥_{V} = ∥y − g(v_{0})∥_{V} , (10.7.15) by (10.7.4). This implies that \overline{B}(v_{0}, (1 − c_{0}) r) ⊆ g(\overline{B}(v_{0}, r)) (10.7.16) when 0 < r ≤ r_{0}, which could be obtained from the same argument as before too. Similarly, B(v_{0}, (1 − c_{0}) r) ⊆ g(B(v_{0}, r)) (10.7.17) when 0 < r ≤ r_{0}, where B(v_{0}, r) is the open ball in V centered at v_{0} with radius r with respect to the metric associated to ∥ · ∥_{V}.

This basically corresponds to part of the inverse function theorem for functions defined on subsets of Banach spaces, as in Theorem 10.39 on p252 of [286]. More precisely, let T be a bounded linear mapping from V into itself with bounded inverse. Suppose that f is a mapping from \overline{B}(v_{0}, r_{0}) into V such that (10.7.18) T − f is Lipschitz with constant c_{1} on \overline{B}(v_{0}, r_{0}), for some c_{1} ≥ 0. This implies that I − T^{−1} ∘ f = T^{−1} ∘ (T − f) (10.7.19) is Lipschitz with constant c_{1} ∥T^{−1}∥_{op} on \overline{B}(v_{0}, r_{0}). If c_{1} ∥T^{−1}∥_{op} < 1, (10.7.20) then one can apply the previous remarks to T^{−1} ∘f, to get analogous properties of f.

10.8 More on local invertibility

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A}, and let r_{0} > 0 be given. We would like to consider some of the remarks in the previous section with V = A, v_{0} = 0, and where g is given by a power series

g(x) = ∑_{j=0}^{∞} a_{j} x^{j}, (10.8.1)

with real or complex coefficients, as appropriate. Suppose that

∑_{j=0}^{∞} |a_{j}| r_{0}^{j} (10.8.2)

converges as an infinite series of nonnegative real numbers, so that the right side of (10.8.1) converges absolutely with respect to ∥ · ∥_{A} when x ∈ A satisfies ∥x∥_{A} ≤ r_{0}, as in Section 9.14. More precisely, this defines g as an A-valued function on the closed ball \overline{B}(0, r_{0}) in A centered at 0 with radius r_{0} with respect to the metric associated to ∥ · ∥_{A}. We have also seen that g is bounded and uniformly continuous on \overline{B}(0, r_{0}), with respect to the metric on A associated to ∥ · ∥_{A} and its restriction to \overline{B}(0, r_{0}).

In fact, we shall ask that

∑_{j=1}^{∞} j |a_{j}| r_{0}^{j−1} (10.8.3)

converges, which implies that (10.8.1) converges. In this case, g is Lipschitz on \overline{B}(0, r_{0}), with constant (10.8.3), as in Section 10.1. Put

ϕ(x) = x − g(x) = −a_{0} e_{A} + (1 − a_{1}) x − ∑_{j=2}^{∞} a_{j} x^{j} (10.8.4)

on \overline{B}(0, r_{0}), as in (10.7.1). Observe that ϕ is Lipschitz with constant

|a_{1} − 1| + ∑_{j=2}^{∞} j |a_{j}| r_{0}^{j−1} (10.8.5)

on \overline{B}(0, r_{0}), as in Section 10.1. Thus we would like to have that

|a_{1} − 1| + ∑_{j=2}^{∞} j |a_{j}| r_{0}^{j−1} < 1, (10.8.6)

as before.

If y ∈ A, then we put

ϕ_{y}(x) = ϕ(x) + y = x − g(x) + y (10.8.7)

10.8. MORE ON LOCAL INVERTIBILITY 233 on \overline{B}(0, r_{0}), as in (10.7.5). Note that ϕ_{y} is Lipschitz with constant (10.8.5) on \overline{B}(0, r_{0}) too, as before. Equivalently,

ϕ_{y}(x) = y − a_{0} e_{A} + (1 − a_{1}) x − ∑_{j=2}^{∞} a_{j} x^{j} (10.8.8)

on \overline{B}(0, r_{0}), so that

∥ϕ_{y}(x)∥_{A} ≤ ∥y − a_{0} e_{A}∥_{A} + |1 − a_{1}| r_{0} + ∑_{j=2}^{∞} |a_{j}| r_{0}^{j} (10.8.9)

on \overline{B}(0, r_{0}). If

∥y − a_{0} e_{A}∥_{A} + |1 − a_{1}| r_{0} + ∑_{j=2}^{∞} |a_{j}| r_{0}^{j} ≤ r_{0}, (10.8.10)

then we get that ϕ_{y}(\overline{B}(0, r_{0})) ⊆ \overline{B}(0, r_{0}). (10.8.11) If (10.8.6) holds as well, then the contraction mapping theorem implies that ϕ_{y} has a unique fixed point in \overline{B}(0, r_{0}), as in Section 7.12.

We may often be interested in having a_{1} = 1, (10.8.12) so that (10.8.6) reduces to

∑_{j=2}^{∞} j |a_{j}| r_{0}^{j−1} < 1. (10.8.13)

Similarly, (10.8.10) reduces to

∥y − a_{0} e_{A}∥_{A} + ∑_{j=2}^{∞} |a_{j}| r_{0}^{j} ≤ r_{0} (10.8.14)

in this case. If

∑_{j=2}^{∞} |a_{j}| r_{0}^{j−1} < 1, (10.8.15)

then (10.8.14) holds when

∥y − a_{0} e_{A}∥_{R} ≤ (1 − ∑_{j=2}^{∞} |a_{j}| r_{0}^{j−1}) r_{0}. (10.8.16)

Note that (10.8.15) holds when (10.8.13) holds. Under these conditions, we can get (10.8.13) by using a smaller positive real number in place of r_{0}, if necessary.

10.9 More on the exponential function

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1, and let r be a positive real number. The restriction of the exponential function to the closed ball \overline{B}(0, r) in A centered at 0 with radius r is Lipschitz with constant

∑_{j=1}^{∞} j (1/j!) r^{j−1}, (10.9.1)

as in Section 10.1. Note that (10.9.1) is equal to

∑_{j=1}^{∞}(1/(j − 1)!) r^{j−1} = ∑_{j=0}^{∞}(1/j!) r^{j} = exp r. (10.9.2)

Similarly, the restriction of (exp x) − x (10.9.3) to \overline{B}(0, r) is Lipschitz with constant

∑_{j=2}^{∞} j (1/j!) r^{j−1} = ∑_{j=2}^{∞}(1/(j − 1)!) r^{j−1} = ∑_{j=1}^{∞}(1/j!) r^{j} = (exp r) − 1. (10.9.4)

Of course, this tends to 0 as r → 0. It follows that (10.9.5) exp x is open at 0, as a mapping from A into itself, as in Section 10.7.

Remember that G(A) is the group of invertible elements of A, as in Section 6.6. Thus exp A = {exp x : x ∈ A} (10.9.6) is a subset of G(A). Using (10.9.5), we get that (10.9.7) e_{A} is an element of the interior of exp A.

Let Γ(A) be the subgroup of G(A) generated by exp A, as in the proof of Theorem 10.44 on p258 of [286]. Clearly e_{A} is an element of the interior of Γ(A), because exp A ⊆ Γ(A). This implies that (10.9.8) Γ(A) is an open set in A, as in [286].

If A is commutative, then it is easy to see that (10.9.9) exp x is open at every point in A, using (10.9.5). This means that (10.9.10) exp x is an open mapping on A, as in Section 7.13. This is related to a remark on p257 of [286]. In this case, exp A is a subgroup of G(A), so that Γ(A) = exp A. (10.9.11)

10.10. THE SUBGROUP G_{1}(A) OF G(A) 235

10.10 The subgroup G_{1}(A) of G(A)

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Remember that the group G(A) of invertible elements of A is an open set in A, as in Section 6.6. Let G_{1}(A) be the connected component of G(A) that contains e_{A}, as on p257 of [286]. Equivalently, G_{1}(A) is the union of all of the connected subsets of G(A) that contain e_{A}.

Remember that open balls in A are convex, as in Section 1.2. This means that open balls in A are path connected. It follows that G(A) is locally path connected, because G(A) is an open set in A. This implies that the connected components of G(A) are the same as the path-connected components of G(A), by standard arguments. We also get that the connected components of G(A) are open sets.

Thus G_{1}(A) is the same as the path-connected component of G(A) that contains e_{A}. This is the set of invertible elements of A that can be connected to e_{A} by a continuous path in G(A). Note that

(10.10.1) G_{1}(A) is an open set in A,

as in part (a) of Theorem 10.44 on p258 of [286].

One can check that

(10.10.2) G_{1}(A) is a subgroup of G(A),

using the characterization of G_{1}(A) in terms of continuous paths in G(A) starting at e_{A}. This is another part of part (a) of Theorem 10.44 on p258 of [286].

In fact, (10.10.3) G_{1}(A) is a normal subgroup of G(A),

which is the third part of part (a) of Theorem 10.44 on p258 of [286]. This can also be obtained from the characterization of G_{1}(A) in terms of continuous paths in G(A) starting at e_{A}.

If x ∈ A, then exp(t x), 0 ≤ t ≤ 1, (10.10.4)

defines a continuous path from e_{A} to exp x in G(A). This implies that exp x is an element of G_{1}(A), so that

exp A ⊆ G_{1}(A), (10.10.5)

as in part (b) of Theorem 10.44 on p258 of [286].

Let Γ(A) be the subgroup of G(A) generated by exp A, as in the previous section. Observe that Γ(A) ⊆ G_{1}(A), (10.10.6)

because of (10.10.2) and (10.10.5). Part (b) of Theorem 10.44 on p258 of [286] states that Γ(A) = G_{1}(A). (10.10.7)

Remember that Γ(A) is an open set in A, as in (10.9.8). This implies that all of the cosets of Γ(A) in G(A) are open sets. It follows that G(A) \ Γ(A) is an open set, because it is the union of all of the cosets of Γ(A) in G(A) other than Γ(A). One can use this to get that

G_{1}(A) ⊆ Γ(A), (10.10.8)

because G_{1}(A) is a connected subset of G(A) that contains e_{A}.

If A is commutative, then

exp A = G_{1}(A), (10.10.9)

by (10.9.11) and (10.10.7). This is part (c) of Theorem 10.44 on p258 of [286].

10.11 Some induced homomorphisms

Let A, B be associative algebras, both real or both complex, with multiplicative identity elements e_{A}, e_{B}, respectively. Also let ϕ be an algebra homomrohism from A into B such that ϕ(e_{A}) = e_{B}. If x ∈ A is invertible, then ϕ(x) is invertible in B, with ϕ(x^{−1}) = ϕ(x)^{−1}. (10.11.1)

Thus ϕ(G(A)) ⊆ G(B), (10.11.2)

and the restriction of ϕ to G(A) defines a group homomorphism into G(B).

Suppose now that (A, ∥ · ∥_{A}), (B, ∥ · ∥_{B}) are Banach algebras, and that ϕ is bounded as a linear mapping from A into B. This means that ϕ is continuous, and one can use that to get that

ϕ(G_{1}(A)) ⊆ G_{1}(B). (10.11.3)

Remember that G_{1}(A), G_{1}(B) are normal subgroups of G(A), G(B), as in the previous section. Thus the corresponding quotient groups G(A)/G_{1}(A) and G(B)/G_{1}(B) may be defined in the usual way. It follows from (10.11.3) that

(10.11.4) ϕ induces a group homomormphism from G(A)/G_{1}(A) into G(B)/G_{1}(B).

Let V be a vector space over the real or complex numbers, and let X, Y be nonempty sets. If ψ is a mapping from X into Y, then

Ψ(f) = f ∘ ψ (10.11.5)

defines a mapping from the space of all V -valued functions on Y into the space of all V -valued functions on X. More precisely, the spaces of V -valued functions on X and Y are vector spaces over the real or complex numbers, as appropriate, with respect to pointwise addition and scalar multiplication, and Ψ is a linear mapping. It is easy to see that

(10.11.6) Ψ is one-to-one when ψ(X) = Y.

Let ∥ · ∥_{V} be a norm on V. If f is a bounded V -valued function on Y, then (10.11.5) is a bounded V -valued function on X, so that the restriction of Ψ to ℓ^{∞}(Y, V) defines a linear mapping into ℓ^{∞}(X, V). Observe that

∥Ψ(f)∥_{ℓ^{∞}(X,V)} ≤ ∥f∥_{ℓ^{∞}(Y,V)} (10.11.7)

for every f ∈ ℓ^{∞}(Y, V). More precisely, the operator norm of the restriction of Ψ to ℓ^{∞}(Y, V) is equal to 1, with respect to the corresponding supremum norms. If ψ(X) = Y, then

∥Ψ(f)∥_{ℓ^{∞}(X,V)} = ∥Ψ(f)∥_{ℓ^{∞}(Y,V)} (10.11.8)

for every f ∈ ℓ^{∞}(Y, V).

Suppose now that X, Y are metric or topological spaces, and that ψ is continuous. If f is a continuous mapping from Y into V, then (10.11.5) is a continuous mapping from X into V. Thus the restriction of Ψ to C(Y, V) defines a linear mapping into C(X, V). Note that

(10.11.9) Ψ is one-to-one on C(Y, V) when ψ(X) is dense in Y.

Similarly, if f is bounded and continuous on Y, then (10.11.5) is bounded and continuous on X, so that the restriction of Ψ to C_{b}(Y, V) is a bounded linear mapping into C_{b}(X, V) with respect to the supremum norm. If ψ(X) is dense in Y, then (10.11.8) holds for every f ∈ C_{b}(Y, V).

Let A_{0} be an algebra in the strict sense over the real or complex numbers, so that the spaces of A_{0}-valued functions on X and Y are algebras in the strict sense over R or C, as appropriate, with respect to pointwise multiplication of functions, as in Section 7.9. In this case, (10.11.5) defines an algebra homomorphism from the space of A_{0}-valued functions on Y into the space of A_{0}-valued functions on X. Suppose that A_{0} has a multiplicative identity element e_{A_{0}}, so that the functions on X and Y equal to e_{A_{0}} at every point are the multiplicative identity elements in the algebras of A_{0}-valued functions on X and Y, respectively, as before. Of course, Ψ sends the function on Y equal to e_{A_{0}} at every point to the function on X equal to e_{A_{0}} at every point.

10.12 Some continuity conditions

Let X, Y be nonempty metric or topological spaces, and let (Z, d_{Z}) be a metric space. Also let F be a mapping from X × Y into Z. If y ∈ Y, then

F_{y}(x) = F (x, y) (10.12.1) defines a mapping from X into Z. Thus

y ↦ F_{y} (10.12.2)

defines a mapping from Y into the space of mappings from X into Z. Conversely, every mapping from Y into the space of mappings from X into Z corresponds to a mapping from X × Y into Z in this way.

Similarly, if x ∈ X, then F (x, y) may be considered as a mapping from Y into Z. Consider the collection

E = {F (x, ·) : x ∈ X} (10.12.3)

of all of the mappings from Y into Z that correspond to some x ∈ X in this way. This collection is said to be equicontinuous at a point y_{0} ∈ Y if for every ϵ > 0 there is an open set V ⊆ Y such that y_{0} ∈ V and

d_{Z}(F (x, y), F (x, y_{0})) < ϵ (10.12.4)

for every x ∈ X and y ∈ V. This condition clearly implies that F (x, y) is continuous as a function of y on Y at y_{0} for every x ∈ X. If X has only finitely many elements, and if F (x, y) is continuous in y at y_{0} for every x ∈ X, then E is equicontinuous at y_{0}.

Suppose for the moment that

(10.12.5) F_{y} is bounded as a mapping from X into Z

for each y ∈ Y. Remember that the space B(X, Z) of all mappings from X into Z is a metric space with respect to the supremum metric associated to d_{Z}. One can check that (10.12.6) E is equicontinuous at y_{0}

if and only if (10.12.2) is continuous at y_{0} as a mapping from Y into B(X, Z), with respect to the supremum metric on B(X, Z).

Suppose for the moment again that (10.12.6) holds. Let x_{0} ∈ X be given, and suppose also that

(10.12.7) F_{y_{0}}(x) = F (x, y_{0}) is continuous in x at x_{0},

as a mapping from X into Z. Under these conditions, one can verify that

(10.12.8) F (x, y) is continuous at (x_{0}, y_{0}),

as a mapping from X × Y into Z, with respect to the product topology on X × Y. If X and Y are both metric spaces, then one can use a suitable product metric on X × Y, as in Section 5.11.

Let ϵ > 0 be given. If (10.12.8) holds, then there are open sets U(x_{0}) ⊆ X and V (x_{0}) ∈ Y such that x_{0} ∈ U(x_{0}), y_{0} ∈ V (x_{0}), and

d_{Z}(F (x, y), F (x_{0}, y_{0})) < ϵ/2 (10.12.9) for every x ∈ U(x_{0}) and y ∈ V (x_{0}). This implies that

d_{Z}(F (x, y), F (x, y_{0})) ≤ d_{Z}(F (x, y), F (x_{0}, y_{0})) + d_{Z}(F (x_{0}, y_{0}), F (x, y_{0}))
(10.12.10)  < ϵ/2 + ϵ/2 = ϵ

for every x ∈ U(x_{0}) and y ∈ V (x_{0}).

Suppose that (10.12.8) holds for every x_{0} ∈ X. If X is compact, then X can be covered by finitely many open sets U(x_{0}) as in the preceding paragraph. If we take V to be the intersection of the finitely many corresponding open sets V (x_{0}) in Y, then we get that (10.12.6) holds.

If X and Y are both metric spaces, then one can use a suitable product metric on X × Y, as before. If X and Y are both compact, then X × Y is compact, by Tychonoff’s theorem. Under these conditions, the continuity of F on X × Y implies that F is uniformly continuous. In particular, this implies that E is equicontinuous at every y_{0} ∈ Y.

10.13 Some remarks about homotopies

Let X and Z be nonempty metric or topological spaces, and let f, g be continuous mappings from X into Z. A homotopy between f and g is a continuous mapping from X × [0, 1] into Z that corresponds to f on X × {0} and to g on X × {1}. This uses the product topology on X × [0, 1] associated to the topology induced on [0, 1] by the standard topology on the real line. If X is a metric space, then one can use a suitable product metric on X × [0, 1], as in Section 5.11.

In particular, a homotopy between f and g can be used to define a mapping from [0, 1] into the space C(X, Z) of continuous mappings from X into Z that is equal to f at 0 and to g at 1. The continuity of the homotopy as a mapping from X × [0, 1] into Z may be considered as a continuity condition on this mapping from [0, 1] into C(X, Z).

Suppose that (Z, d_{Z}) is a metric space, so that the space C_{b}(X, Z) of bounded continuous mappings from X into Z is a metric space with respect to the supremum metric. A mapping from [0, 1] into C_{b}(X, Z) determines a mapping from X × [0, 1] into Z, as before. The continuity of this mapping from [0, 1] into C_{b}(X, Y) with respect to the supremum metric on C_{b}(X, Z) is equivalent to the equicontinuity condition for the corresponding mapping from X × [0, 1] into Z at every point in Y = [0, 1] discussed in the previous section. It follows that a continuous mapping from [0, 1] into C_{b}(X, Y) determines a continuous mapping from X × [0, 1] into Z, with respect to the usual product topology on X × [0, 1], as before.

Suppose that X is compact, so that every continuous mapping from X into Z is bounded. In this case, a continuous mapping from X×[0, 1] into Z corresponds to a continuous mapping from [0, 1] into C_{b}(X, Z). More precisely, continuous mappings from X × [0, 1] into Z satisfy the equicontinuity condition discussed in the previous section at every point in [0, 1], because X is compact. This implies that the corresponding mapping from [0, 1] into C_{b}(X, Z) is continuous, as before.

Let (A_{0}, ∥ · ∥_{A_{0}}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A_{0}}. If X is any metric or topological space, then C(X, A_{0}) is an associative algebra over the real or complex numbers, as appropriate, as in Section 7.9. Remember that f ∈ C(X, A_{0}) is invertible if and only if f(x) is invertible in A_{0} for every x ∈ X. Thus

G(C(X, A_{0})) = C(X, G(A_{0})), (10.13.1)

where G(A_{0}) is considered as a metric space, using the resptriction of the metric on A_{0} associated to ∥ · ∥_{A_{0}} to G(A_{0}).

The space C_{b}(X, A_{0}) of all continuous A_{0}-valued functions on X is a Banach algebra with respect to the supremum norm, as in Sections 7.8 and 7.9. Suppose that X is compact again, so that this is the same as C(X, A_{0}). In this case, a continuous path in (10.13.1) corresponds to a homotopy between continuous mappings from X into G(A_{0}), as before.

Thus the path connected components of (10.13.1) are the same as the homotopy classes of continuous mappings from X into G(A_{0}). In particular,

G_{1}(C(X, A_{0})) (10.13.2)

consists of continuous mappings from X into G(A_{0}) that are homotopic to the constant function on X equal to e_{A_{0}} at every point, as continuous mappings from X into G(A_{0}). The quotient group

G(C(X, A_{0}))/G_{1}(C(X, A_{0})) (10.13.3)

corresponds to the set of homotopy classes of continuous mappings from X into A_{0}. This is related to Exercise 16 on p261 of [286].

10.14 More on G_{1}(A)

Let A be an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. If x ∈ A, then r_{A}(x) can be defined in the usual way, as in Sections 6.13 and 7.2. If t ∈ R or C, as appropriate, then one can check that r_{A}(t x) = |t| r_{A}(x). (10.14.1)

Suppose now that (A, ∥·∥_{A}) is a Banach algebra with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Suppose also that x ∈ A is quasinilpotent, so that r_{A}(x) = 0, as in Section 7.2. This implies that e_{A} − x is invertible in A, as in Section 6.13.

If t ∈ R or C, as appropriate, then r_{A}(t x) = 0, by (10.14.1), so that t x is quasinilpotent as well. This implies that

e_{A} − t x ∈ G(A), (10.14.2)

10.14. MORE ON G_{1}(A) 241

as in the preceding paragraph. It follows that e_{A} − t x ∈ G_{1}(A), (10.14.3) by the definition of G_{1}(A), as in Section 10.10.

Now let x ∈ G(A) be given, and put f(λ) = λ x − (λ − 1) e_{A} = λ x + (1 − λ) e_{A}. (10.14.4) for every real or complex number λ, as appropriate. Note that f(0) = e_{A}, f(1) = x. (10.14.5) Let E be the set of λ ∈ R or C, as appropriate, such that f(λ) ∈ G(A). (10.14.6) Thus 0, 1 ∈ E, by (10.14.5). We also have that E is an open set in R or C, as appropriate, with respect to the standard Euclidean metric, because G(A) is an open set in A with respect to the metric associated to ∥ · ∥_{A}, as in Section 6.6.

If λ ≠ 0, then (10.14.6) is the same as saying that x − λ^{−1} (λ − 1) e_{A} ∈ G(A). (10.14.7) Equivalently, this means that λ^{−1} (λ − 1) = 1 − λ^{−1} (10.14.8) is not in the spectrum σ_{A}(x) of x with respect to A. If (10.14.9) 0, 1 are in the same connected component of E, then it follows that x ∈ G_{1}(A). (10.14.10) This corresponds to part of the proof of part (d) of Theorem 10.44 on p258 of [286]. This is also related to Exercise 26 on p262 of [286].

Of course, (10.14.9) holds when {λ ∈ R : 0 ≤ λ ≤ 1} ⊆ E. (10.14.11) One can verify that this happens exactly when σ_{A}(x) ∩ {λ ∈ R : λ ≤ 0} = ∅, (10.14.12) because 0 ∈ E automatically. In the real case, (10.14.9) is equivalent to (10.14.11), and (10.14.12) is the same as saying that σ_{A}(x) ⊆ (0, +∞). (10.14.13)

In the complex case, there is exactly one unbounded connected component of C \ σ_{A}(x), because σ_{A}(x) is bounded, as in Section 6.8. In this case, one can check that (10.14.9) holds if and only if (10.14.14) 0 is in the unbounded connected component of C \ σ_{A}(x).

242 CHAPTER 10. SOME MORE ALGEBRAS, POWER SERIES

10.15 Some additional properties of G_{1}(A)

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}. Suppose that x ∈ A satisifes x^{n} = e_{A} (10.15.1) for some positive integer n. Consider the polynomial p(t) = t^{n}. Under these conditions, we have that p(σ_{A}(x)) ⊆ σ_{A}(p̃(x)) = {1}, (10.15.2) where the first step is as in Section 8.13.

In the real case, (10.15.2) implies that σ_{A}(x) ⊆ {1} (10.15.3) when n is odd. In the complex case, (10.15.2) implies that (10.15.4) C \ σ_{A}(x) is connected for all n ≥ 1, because σ_{A}(x) has only finitely many elements.

Suppose now that (A, ∥ · ∥_{A}) is a Banach algebra, with ∥e_{A}∥_{A} = 1. In the real case, we get that (10.14.10) holds when n is odd, because (10.15.3) implies (10.14.13). In the complex case, (10.14.10) holds for every n ≥ 1, because (10.15.4) implies (10.14.14). This corresponds to part of the proof of part (d) of Theorem 10.44 on p258 of [286], and to Exercise 26 on p262 of [286].

Suppose from now on in this section that A is commutative. Suppose also that w ∈ G(A) satisfies w^{n} ∈ G_{1}(A) (10.15.5) for some positive integer n. Remember that G_{1}(A) = exp A when A is commutative, as in Section 10.10. Thus w^{n} = exp a (10.15.6) for some a ∈ A.

Put y = exp(n^{−1} a) (10.15.7) and z = w y^{−1}. (10.15.8) Observe that z^{n} = w^{n} y^{−n} = (exp a) (exp(−a)) = e_{A}. (10.15.9) In the real case, this implies that z ∈ G_{1}(A) (10.15.10)

10.15. SOME ADDITIONAL PROPERTIES OF G_{1}(A) 243

when n is odd, as before. In the complex case, we get that (10.15.10) holds for any n ≥ 1.

Remember that y ∈ G_{1}(A), as in Section 10.10. If (10.15.10) holds, then it follows that w = z y ∈ G_{1}(A), (10.15.11)

because G_{1}(A) is a subgroup of G(A).

Let [w] be the image of w under the natural quotient homomorphism from G(A) onto the quotient group G(A)/G_{1}(A). Thus (10.15.5) is the same as saying that

(10.15.12) [w]^{n} = [w^{n}] is the identity element in G(A)/G_{1}(A).

Similarly, (10.15.11) is the same as saying that

(10.15.13) [w] is the identity element in G(A)/G_{1}(A).

This corresponds to part (d) of Theorem 10.44 on p258 of [286].

Part III

Algebras, norms, and operators, 2 Algebras, norms, and square roots

11.1 Exponentials of nilpotent elements

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}. Suppose that x ∈ A is nilpotent, so that x^{l} = 0 for some positive integer l. Under these conditions, the exponential of x in A may be defined by

exp x = exp_{A} x = ∑_{j=0}^{∞}(1/j!) x^{j} = ∑_{j=0}^{l−1}(1/j!) x^{j}. (11.1.1)

Similarly, if t ∈ R or C, as appropriate, then (t x)^{l} = t^{l} x^{l} = 0, and

exp(t x) = ∑_{j=0}^{l−1}(1/j!) t^{j} x^{j}. (11.1.2)

Let y be another nilpotent element of A, so that y^{m} = 0 for some m ≥ 1. Suppose that x and y commute, and let us check that x + y is nilpotent as well. More precisely, (x + y)^{l+m−1} = 0. (11.1.3)

Indeed, (x + y)^{l+m−1} can be expanded into a sum of products of powers of x and y, where the sum of the powers of x and y is equal to l+m−1. This implies that in each term, the power of x is at least l, or the power of y is at least m.

This implies that the exponential of x + y may be defined as before. We also have that (11.1.4) exp(x + y) = (exp x) (exp y),

as in Section 10.4. In particular, we can take y = −x, to get that exp x is invertible in A, with (exp x)^{−1} = exp(−x). (11.1.5)

Let B be another associative algebra over the real or complex numbers, as appropriate, with a multiplicative identity element e_{B}, and let ϕ be an algebra homomorphism from A into B with ϕ(e_{A}) = e_{B}. If x ∈ A is nilpotent, then ϕ(x) is nilpotent in B, and

ϕ(exp_{A} x) = exp_{B} ϕ(x). (11.1.6)

This also works when ϕ is an oppositve algebra homomorphism, which may by conjugate-linear in the complex case.

Suppose for the moment that A is an algebra in the strict sense over R or C, and let δ be a derivation on A, as in Section 10.5. If δ is nilpotent as a linear mapping from A into itself, then exp δ may be defined as a linear mapping from A into itself, as before. In fact,

(11.1.7) exp δ is an automorphism of A,

as an algebra in the strict sense, as in Section 10.5. This corresponds to some remarks on p8f of [158] when A is a Lie algebra.

Let A be an associative algebra over R or C with a multiplicative identity element e_{A} again, and let L_{a}, R_{a} be the left and right multiplication operators on A associated to a ∈ A. If a is nilpotent, then it is easy to see that L_{a} and R_{a} are nilpotent as elements of the algebra L(A) of linear mappings from A into itself, so that their exponentials are defined as elements of L(V) too. In this case, we have that (11.1.8) exp L_{a} = exp_{L(A)} L_{a} = L_{exp_{A} a}

and (11.1.9) exp R_{a} = exp_{L(A)} R_{a} = R_{exp_{A} a},

as in (11.1.6).

Remember that δ_{a} = L_{a} − R_{a} is a derivation on A, and that L_{a} and R_{a} commute on A, as in Section 10.6. If a is nilpotent, then it follows that δ_{a} is nilpotent in L(A), as in (11.1.3). We also get that

(11.1.10) exp δ_{a} = (exp L_{a}) ∘ (exp R_{a})^{−1},

as in (11.1.4). This means that

(11.1.11) (exp δ_{a})(x) = (exp_{L(A)} δ_{a})(x) = (exp_{A} a) x (exp_{A} a)^{−1}

for every x ∈ A, because of (11.1.8) and (11.1.9). This is related to some remarks on p9 of [158].

11.2 One-sided inverses

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}, and let x be an element of A. We say that w ∈ A is a left inverse of x if w x = e_{A}. (11.2.1)

Similarly, z ∈ A is a right inverse of x if

x z = e_{A}. (11.2.2)

If w and z are left and right inverses of x in A, respectively, then

w = w (x z) = (w x) z = z. (11.2.3)

This means that x is invertible in A, with

x^{−1} = w = z. (11.2.4)

Let V, W be vector spaces, both real or both complex, and let T be a linear mapping from V into W. A linear mapping L from W into V is said to be a left inverse of T if L ∘ T = I_{V} , (11.2.5)

the identity mapping on V. This means that

L(T (v)) = v (11.2.6)

for every v ∈ V, which implies that T is one-to-one on V. If V = W, then this is the same as a left inverse of T in the algebra L(V) of linear mappings from V into itself.

Similarly, a linear mapping R from W into V is said to be a right inverse of T if T ∘ R = I_{W} . (11.2.7)

This is the same as saying that

T (R(w)) = w (11.2.8)

for every w ∈ W, which implies that T maps V onto W. If V = W, then this is the same as a right inverse of T in L(V).

If V and W have the same finite dimension, then it is well known that T is one-to-one on V if and only if T (V) = W. In this case, we get that T is invertbile as a linear mapping from V into W when T has a left or right inverse.

Let ∥ · ∥_{V}, ∥ · ∥_{W} be norms on V, W, respectively, and suppose that T is a bounded linear mapping from V into W, with respect to these norms. A bounded linear mapping L from W into V is a left inverse of T as a bounded linear mapping from V into W if (11.2.5) holds. This implies that

∥v∥_{V} ≤ ∥L∥_{op,W V} ∥T (v)∥_{W} (11.2.9)

for every v ∈ V. If V = W, and the two norms are the same, then L is the same as a left inverse of T in the algebra BL(V) of bounded linear mappings from V into itself.

Similarly, a bounded linear mapping R from W into V is said to be a right inverse of T as a bounded linear mapping from V into W if (11.2.7) holds. One can check that this implies that

(11.2.10) T is an open mapping from V onto W.

If V = W, and the two norms are the same, then R is the same as a right inverse of T in BL(V).

Let A be any associative algebra over R or C with a multiplicative identity element e_{A} again, and let x, y ∈ A be given. If x y is invertible in A, then

x (y (x y)^{−1}) = (x y) (x y)^{−1} = e_{A} (11.2.11)

and ((x y)^{−1} x) y = (x y)^{−1} (x y) = e_{A}, (11.2.12)

so that y (x y)^{−1} is a right inverse of x in A, and (x y)^{−1} x is a left inverse of y. Similarly, if y x is invertible in A, then

y (x (y x)^{−1}) = (y x) (y x)^{−1} = e_{A} (11.2.13)

and ((y x)^{−1} y) x = (y x)^{−1} (y x) = e_{A}, (11.2.14)

so that x (y x)^{−1} is a right inverse of y in A, and (y x)^{−1} y is a left inverse of x in A. If x y and y x are both invertible in A, then it follows that x and y are invertible in A, as in part (b) of Exercise 1 on p259 of [286]. If x and y commute, then this corresponds to a remark in Section 6.13.

11.3 Topological divisors of zero

Let A be an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}, and let x be an element of A. Let us say that x is a left topological divisor of zero in A if there is a sequence {z_{j}}_{j=1}^{∞} of elements of A such that ∥z_{j}∥_{A} = 1 (11.3.1)

for each j and

lim_{j→∞} x z_{j} = 0, (11.3.2)

as in Exercise 4 on p300 of [291]. Similarly, let us say that x is a right topological divisor of zero in A if there is a sequence {w_{j}}_{j=1}^{∞} of elements of A such that

∥w_{j}∥_{A} = 1 (11.3.3)

for each j and

lim_{j→∞} w_{j} x = 0, (11.3.4)

as in [291]. If there is a sequence {y_{j}}_{j=1}^{∞} of elements of A such that

∥y_{j}∥_{A} = 1 (11.3.5)

for each j and

lim_{j→∞} x y_{j} = lim_{j→∞} y_{j} x = 0, (11.3.6)

then we may say that x is a topological divisor of zero in A, as in Exercise 5 on p259 of [286]. In this case, x is both a left and right topological divisor of zero, with w_{j} = z_{j} = y_{j} (11.3.7)

for each j.

Suppose that A has a multiplicative identity element e_{A}. If x is a left topological divisor of zero in A, then it is easy to see that

(11.3.8) x does not have a left inverse in A.

Similarly, if x is a right topological divisor of zero in A, then

(11.3.9) x does not have a right inverse in A.

Let B be another associative algebra over the real or complex numbers, as appropriate, with a submultiplicative norm ∥ · ∥_{B}. Suppose that ϕ is an algebra homomorphism from A into B that is bounded as a linear mapping. If x ∈ A is a left or right topological divisor of zero in A, or simply a topological divisor of zero in A, then ϕ(x) has the analogous property in B.

Suppose now that A is a Banach algebra, with a multiplicative identity element e_{A}. Let x ∈ A be an element of the boundary of the set G(A) of invertible elements of A. This implies that x is not invertible in A, because G(A) is an open set in A, as in Section 6.6, and that there is a sequence {u_{j}}_{j=1}^{∞} of elements of G(A) that converges to x with respect to the metric on A associated to ∥ · ∥_{A}.

We also have that ∥u_{j}^{−1}∥_{A} → ∞ as j → ∞ (11.3.10)

under these conditions. More precisely, the norms of the u_{j}^{−1}’s are not bounded, because x is not invertible, as in Section 6.7. This means that (11.3.10) holds after passing to a subsequence, which would suffice for the remarks in the next paragraph. The same argument shows that the norms of the u_{j}^{−1}’s are not bounded along any subsequence, which implies (11.3.10). This corresponds to Lemma 10.17 on p238 of [286].

Put y_{j} = u_{j}^{−1}/∥u_{j}^{−1}∥_{A} (11.3.11)

for each j, which automatically satisfies (11.3.5). One can check that (11.3.6) holds under these conditions, so that

(11.3.12) x is a topological divisor of zero in A.

This corresponds to part (a) of Exercise 5 on p259 of [286].

Let a ∈ A be given, and remember that σ_{A}(a) is the spectrum of a with respect to A, as in Section 6.8. Suppose that λ ∈ R or C, as appropriate, is an element of the boundary of σ_{A}(a), with respect to the standard Euclidean metric on R or C, as appropriate. This implies that λ e_{A} − a is an element of the boundary of G(A), so that

(11.3.13) λ e_{A} − a is a topological divisor of zero in A,

as in the preceding paragraph. This corresponds to part (a) of Exercise 4 on p300 of [291]. Note that this could be used as another way to obtain some of the remarks in Section 7.3.

11.4 Another condition on a norm

Let A be an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. Consider the condition that there be a nonnegative real number C such that

∥x∥_{A} ∥y∥_{A} ≤ C ∥x y∥_{A} (11.4.1)

for every x, y ∈ A. It is easy to see that this implies that

(11.4.2) 0 is the only left or right topological divisor of zero in A.

Of course, (11.4.2) implies that

(11.4.3) 0 is the only topological divisor of zero in A.

Suppose that A is a Banach algebra, with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Note that (11.4.2) holds when every nonzero element of A is invertible in A, because of (11.3.8) and (11.3.9). Let a ∈ A be given, and suppose that λ ∈ R or C, as appropriate, is an element of the boundary of the spectrum σ_{A}(a) of a with respect to A. If (11.4.3) holds, then we get that

a = λ e_{A}, (11.4.4)

because of (11.3.13).

Remember that σ_{A}(a) is a bounded subset of R or C, as appropriate, as in Section 6.8. If σ_{A}(a) ≠ ∅, (11.4.5)

then it follows that ∂σ_{A}(a) ≠ ∅, (11.4.6)

where ∂σ_{A}(a) is the boundary of σ_{A} in R or C, as appropriate. This uses the well-known fact that R, C are connected with respect to their standard Euclidean metrics. If (11.4.3) holds, then we get that (11.4.4) holds for some λ ∈ R or C, as appropriate, as in the preceding paragraph.

In the complex case, (11.4.5) holds for every a ∈ A, as in Section 6.8. This means that A = {λ e_{A} : λ ∈ C} (11.4.7)

when (11.4.3) holds. This corresponds to Theorem 10.19 on p239 of [286] when (11.4.1) holds. The analogous statement when (11.4.3) holds corresponds to part (b) of Exercise 5 on p259 of [286].

11.5. SOME PROPERTIES OF THE SPECTRUM 251

11.5 Some properties of the spectrum

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}. Also let ϕ be an algebra automorphism of A, which is to say an algebra isomorphism of A onto itself, or an opposite algebra isomorphism from A onto itself. As usual, we allow ϕ to be conjugate-linear in the complex case. Remember that ϕ(e_{A}) = e_{A}, (11.5.1) as in Sections 6.3 and 6.4.

If a ∈ A and λ ∈ R or C, as appropriate, then ϕ(λ e_{A} − a) = λ e_{A} − ϕ(a), (11.5.2) except in the complex case when ϕ is conjugate-linear, for which we have ϕ(λ e_{A} − a) = \overline{λ} e_{A} − ϕ(a). (11.5.3) It follows that σ_{A}(ϕ(a)) = σ_{A}(a) (11.5.4) in the first case, and σ_{A}(ϕ(a)) = \{\overline{λ} : λ ∈ σ_{A}(a)\} (11.5.5) in the second case. The right side may also be denoted \overline{σ_{A}(a)} sometimes, although this notation may be used for the closure of σ_{A}(a) in R or C, as appropriate, as well.

In particular, if x is an invertible element of A, then conjugation by x defines an algebra automorphism of A, and we get that σ_{A}(x a x^{−1}) = σ_{A}(a), (11.5.6) as in (11.5.4). If y ∈ A, then we can take a = y x to get that σ_{A}(x y) = σ_{A}(y x). (11.5.7) This corresponds to part (c) of Exercise 2 on p259 of [286].

Let x, y ∈ A be given, and suppose that e_{A} − x y ∈ G(A). (11.5.8) Under these conditions, it is well known that e_{A} − y x ∈ G(A). (11.5.9) This corresponds to part (a) of Exercise 2 on p259 of [286], and to the hint in part (b) of Exercise 1 on p299 of [291]. This also corresponds to Exercise (3) on p7 of [11] in the case of linear mappings. More precisely, one can check that (e_{A} − y x)^{−1} = e + y (e_{A} − x y)^{−1} x, (11.5.10) as in [286, 291]. In [11], the hint is given to look at how the formal Neumann series for (e_{A} − x y)^{−1} and (e_{A} − y x)^{−1} are related, and to use that to give a rigorous proof of (11.5.9). Indeed, these formal Neumann series are related as in (11.5.10).

One can use this to get that

σ_{A}(x y) \ {0} = σ_{A}(y x) \ {0}. (11.5.11)

This corresponds to Exercise (4) on p7 of [11] in the case of linear operators, the first part of part (b) of Exercise 2 on p259 of [286], and part (b) of Exercise 1 on p299 of [291].

Now let A be an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. Thus r_{A}(x) may be defined for x ∈ A as in Sections 6.13 and 7.2. If x, y ∈ A, then

r_{A}(x y) = r_{A}(y x), (11.5.12)

as in Exercise 24 on p262 of [286]. If A is a complex Banach algebra with a multiplicative identity element, then this can be obtained from (11.5.11), using the relationship between r_{A}(·) and the spectrum mentioned in Section 6.14. This can also be shown more directly using the hint mentioned in [286], as follows.

Observe that (x y)^{j} = x (y x)^{j−1} y (11.5.13)

for every positive integer j. This implies that

∥(x y)^{j}∥_{A} ≤ ∥x∥_{A} ∥(y x)^{j−1}∥_{A} ∥y∥_{A} (11.5.14)

for each j. Let r be a real number such that

r_{A}(y x) < r, (11.5.15)

so that ∥(y x)^{j}∥_{A} < r^{j} (11.5.16)

when j is sufficiently large. It follows that

∥(x y)^{j}∥_{A} ≤ ∥x∥_{A} ∥y∥_{A} r^{j−1} (11.5.17)

when j is sufficiently large, and thus

∥(x y)^{j}∥_{A}^{1/j} ≤ ∥x∥_{A}^{1/j} ∥y∥_{A}^{1/j} r^{1−(1/j)}. (11.5.18)

Using this, we get that r_{A}(x y) ≤ r. (11.5.19)

This implies that r_{A}(x y) ≤ r_{A}(y x). (11.5.20)

Of course, the opposite inequality can be obtained in the same way, to get (11.5.12).

11.6 Commutativity and norms

Let (A, ∥·∥_{A}) be a Banach algebra over the complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Suppose that there is a nonnegative real number C such that ∥x y∥_{A} ≤ C ∥y x∥_{A} (11.6.1)

for all x, y ∈ A. We would like to show that A is commutative, as in part (a) of Exercise 25 on p262 of [286].

If z ∈ A is invertible, then one can check that

∥z y z^{−1}∥_{A} ≤ C∥y∥_{A} (11.6.2)

for every y ∈ A, using (11.6.1). This implies that

∥(exp(λ x)) y (exp(−λ x))∥_{A} ≤ C∥y∥_{A} (11.6.3)

for all x, y ∈ A and λ ∈ C.

Let x, y ∈ A, and consider

f(λ) = f_{x,y}(λ) = (exp(λ x)) y (exp(−λ x)) (11.6.4)

as a function of λ ∈ C with values in A. More precisely, we would like to consider this as a holomorphic function of λ ∈ C with values in A. The right side may be expressed as a power series in λ with coefficients in A that converges absolutely with respect to ∥ · ∥_{A} for every λ ∈ C, using Cauchy products, for instance.

In particular, if μ is a bounded linear functional on A, then

μ(f(λ)) = μ((exp(λ x)) y (exp(−λ x))) (11.6.5)

may be expressed as a power series in λ with complex coefficients that converges absolutely for every λ ∈ C. Thus (11.6.5) is holomorphic as a complex-valued function of λ ∈ C in the usual sense. This function is also bounded on C, because of (11.6.3). Liouville’s theorem implies that this function is constant on C, so that μ(f(λ)) = μ(f(0)) (11.6.6)

for every λ ∈ C. One can use this and the Hahn–Banach theorem to get that

f(λ) = f(0) (11.6.7)

for every λ ∈ C.

It follows that (exp(λ x)) y = y (exp(λ x)) (11.6.8)

for all x, y ∈ A and λ ∈ C. One can use this to get that A is commutative.

11.7 More on commutativity and norms

Let A be an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. Of course,

∥x^{2}∥_{A} ≤ ∥x∥_{A}^{2} (11.7.1)

for every x ∈ A. Suppose that there is a nonnegative real number C such that

∥x∥_{A}^{2} ≤ C ∥x^{2}∥_{A} (11.7.2)

for every x ∈ A. If x ∈ A, then one can check that

∥x∥_{A}^{2^{l}} ≤ C^{2^{l}−1} ∥x^{2^{l}}∥_{A} (11.7.3)

for every positive integer l. This corresponds to (3) in the proof of Lemma 11.11 on p270 of [286].

Thus ∥x∥_{A} ≤ C^{1−2^{−l}} ∥x^{2^{l}}∥_{A}^{2^{−1}} (11.7.4)

for every l ≥ 1. This implies that

∥x∥_{A} ≤ C r_{A}(x), (11.7.5)

where r_{A}(x) is as in Sections 6.13 and 7.2, as in (4) on p270 of [286]. If x, y ∈ A, then it follows that

∥x y∥_{A} ≤ C r_{A}(x y) = C r_{A}(y x) ≤ C ∥y x∥_{A}, (11.7.6)

using (11.5.12) in the second step.

Suppose for the moment that A is a complex Banach algebra with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. In this case, (11.7.6) implies that A is commutative, as in the previous section. This shows that A is commutative when (11.7.2) holds. This corresponds to part (b) of Exercise 25 on p262 of [286] when C = 1.

Let A be any associative algebra over R or C with a submultiplicative norm again. Also let m ≥ 2 be an integer, and suppose that there is a nonnegative real number C(m) such that

∥x∥_{A}^{m} ≤ C(m) ∥x^{m}∥_{A} (11.7.7)

for every x ∈ A. If n ≥ 2 is an integer, then there is a nonnegative real number C_{n}(m) such that ∥x∥_{A}^{n} ≤ C_{n}(m) ∥x^{n}∥_{A} (11.7.8)

for every x ∈ A. This corresponds to Exercise 16 on p291 of [286]. In particular, we can take n = 2, to get a condition like (11.7.2). One could use the same type of arguments as before in this case as well. If A is a complex Banach algebra with a multiplicative identity element, then it follows that A is commutative, as in the preceding paragraph.

If x ∈ A, then

r_{A}(x) = lim_{j→∞} ∥x^{j}∥_{A}^{1/j} = lim_{l→∞} ∥x^{l m}∥_{A}^{1/(l m)} = r_{A}(x^{m})^{1/m}. (11.7.9)

If (11.7.5) holds for some C ≥ 0 and all x ∈ A, then

∥x∥_{A}^{m} ≤ C^{m} r_{A}(x)^{m} = C^{m} r_{A}(x^{m}) ≤ C^{m} ∥x^{m}∥_{A} (11.7.10)

for every x ∈ A. This corresponds to another part of Lemma 11.11 on p270 of [286] when m = 2.

11.8 A continuity property

Let (V, ⟨·, ·⟩_{V}) be a real or complex Hilbert space, with associated norm ∥ · ∥_{V}, and V ≠ {0}. Also let f be a continuous real-valued function on the real line. If T is a bounded self-adjoint linear mapping from V into itself, then f̃(T) may be defined as a bounded self-adjoint linear mapping from V into itself as in Section 8.14. More precisely, f̃(T) only depends on the restriction of f to the spectrum σ(T) = σ_{BL(V)}(T) of T with respect to the algebra BL(V) of bounded linear mappings from V into itself. In fact,

∥f̃(T)∥_{op} = max{|f(λ)| : λ ∈ σ(T)}, (11.8.1)

as before.

The space of bounded self-adjoint linear mappings from V into itself is a real-linear subspace of BL(V). Let us check that

T ↦ f̃(T) (11.8.2)

is continuous on this space, with respect to the metric associated to the operator norm. This is a version of Exercise (3) on p51 of [11]. More precisely, if r is a positive real number, then (11.8.2) is uniformly continuous on

(11.8.3) {T ∈ BL(V) : T is self-adjoint, and ∥T ∥_{op} ≤ r}.

If p is a polynomial with real coefficients, then

T ↦ p̃(T) (11.8.4)

is uniformly continuous on

{T ∈ BL(V) : ∥T ∥_{op} ≤ r} (11.8.5)

with respect to the metric associated to the operator norm. This corresponds to a remark in Section 9.14. We also have that (11.8.4) is Lipschitz on (11.8.5) with respect to the metric associated to the operator norm, where the constant can be estimated as in Section 10.1. In particular, (11.8.4) has these properties on (11.8.3).

Otherwise, one can approximate f by polynomials uniformly on [−r, r], by Weierstrass’ approximation theorem. Of course, if T is a bounded self-adjoint linear mapping from V into itself with ∥T ∥_{op} ≤ r, then

σ(T) ⊆ [−r, r]. (11.8.6)

Indeed, if λ ∈ σ(T), then |λ| ≤ ∥T ∥_{op} ≤ r, (11.8.7)

where the first step is as in Section 6.8. In the complex case, we are also using the fact that the spectrum of T is contained in the real line when T is self-adjoint, as in Section 8.11.

If T is a bounded self-adjoint linear mapping from V into itself and p is a polynomial with real coefficients, then

∥f̃(T) − p̃(T)∥_{op} = max{|f(λ) − p(λ)) : λ ∈ σ(T)}, (11.8.8)

as in (11.8.1). This implies that

∥f̃(T) − p̃(T)∥_{op} ≤ max{|f(λ) − p(λ)| : λ ∈ [−r, r]} (11.8.9)

when ∥T ∥_{op} ≤ r, by (11.8.6). Weierstrass’ theorem implies that for each ϵ > 0 there is a polynomial p with real coefficients such that

max{|f(λ) − p(λ)| : λ ∈ [−r, r]} < ϵ. (11.8.10)

In this case, we get that ∥f̃(T) − p̃(T)∥_{op} < ϵ (11.8.11)

when ∥T ∥_{op} ≤ r, as in (11.8.9). One can use this to get the uniform continuity of (11.8.2) on (11.8.3), because of the analogous statement for polynomials.

11.9 More on self-adjoint operators

Let (V, ⟨·, ·⟩_{V}) be a real or complex Hilbert space again, with associated norm ∥ · ∥_{V}, and V ≠ {0}. Also let T be a bounded self-adjoint linear mapping from V into itself, with spectrum σ(T) = σ_{BL(V)} with respect to BL(V). Remember that T is said to be nonnegative on V with respect to ⟨·, ·⟩_{V} when ⟨T (v), v⟩_{V} ≥ 0 for every v ∈ V, as in Section 8.10. This may be expressed by saying that

T ≥ 0 (11.9.1)

on V. This is equivalent to the condition that α(T) ≥ 0, in the notation of Section 8.9.

Remember that σ(T) ⊆ R in the complex case, because T is self-adjoint, as in Section 8.11. It is well known that (11.9.1) holds if and only if

σ(T) ⊆ {λ ∈ R : λ ≥ 0}. (11.9.2)

This follows from the characterization of α(T) in Section 8.12.

Let f be a continuous real-valued function on σ(T), and let f̃(T) be the corresponding bounded self-adjoint linear mapping from V into itself, as in Section 8.14. Remember that

f(σ(T)) = σ(f̃(T)), (11.9.3)

as in Section 9.5. It follows that

f̃(T) ≥ 0 (11.9.4)

on V if and only if f ≥ 0 (11.9.5)

on σ(T), as in the preceding paragraph.

Remember that we may consider BL(V) as an associative algebra over the real numbers, even when V is complex, as in Section 8.14. Let A_{0}(T) be the subalgebra of BL(V), as an algebra over the real numbers, generated by T and the identity mapping I = I_{V} on V. Equivalently,

(11.9.6) A_{0}(T) = {p̃(T) : p is a polynomial with real coefficients}.

Note that A_{0}(T) is commutative, and that the elements of A_{0}(T) are selfadjoint.

Consider the closure A(T) = \overline{A_{0}(T)} (11.9.7)

of A_{0}(T) in BL(V) with respect to the metric associated to the operator norm. This is also a commutative subalgebra of BL(V), as an algebra over the real numbers. The elements of A(T) are self-adjoint, as before.

Remember that f ↦ f̃(T) (11.9.8)

is a homomorphism from C(σ(T), R) into BL(V), as algebras over the real numbers, as in Section 8.14. This mapping is also an isometry with respect to the supremum norm on C(σ(T), R) and the operator norm on BL(V), as in (11.8.1). It is easy to see that (11.9.8) maps C(σ(T), R) into A(T), because the continuous functions on σ(T) defined by polynomials with real coefficients are dense in C(σ(T), R) with respect to the supremum metric, by the Stone– Weierstrass theorem.

Of course, C(σ(T), R) is complete as a metric space, with respect to the supremum metric. This implies that the image

{f̃(T) : f ∈ C(σ(T), R)} (11.9.9)

of C(σ(T), R) under (11.9.8) is complete with respect to the metric associated to the operator norm, because (11.9.8) is an isometry, as before. It follows that (11.9.9) is a closed set in BL(V), with respect to the metric associated to the operator norm, as in Section 1.6. Clearly (11.9.9) contains A_{0}(T), and (11.9.9) is contained in A(T), as in the preceding paragraph. This means that (11.9.9) is equal to A(T).

11.10 Nonnegative square roots

Let us continue with the same notation and hypotheses as in the previous section. If f is a continuous nonnegative real-valued function on σ(T), then f has a unique nonnegative square root in C(σ(T), R). This implies that every nonnegative element of A(T) has a unique nonnegative square root in A(T).

Suppose now that T is nonnegative on V. It follows that there is a unique R ∈ A(T) such that R ≥ 0 on V and

R ∘ R = T. (11.10.1)

We would like to show that R is the unique square root of T among all nonnegative self-adjoint bounded linear mappings from V into itself, as in Theorem 12.33 on p3.14 of [286]. Let B be a nonnegative self-adjoint bounded linear mapping from V into itself such that

B ∘ B = T. (11.10.2)

The same remarks as in the previous section and at the beginning of this section can be used for B in place of T. Thus we let A_{0}(B) be the subalgebra of BL(V), as an algebra over the real numbers, generated by B and I, and we let A(B) be its closure in BL(V).

Of course, T ∈ A_{0}(B), by (11.10.2). Note that B is the unique nonnegative square root of T in A(B), by the remarks at the beginning of the section. We also have that A_{0}(T) ⊆ A_{0}(B), (11.10.3)

by (11.10.2). This implies that

A(T) ⊆ A(B). (11.10.4)

In particular, R ∈ A(B), so that B = R.

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}. If x, y ∈ A satisfy

x = y^{2}, (11.10.5)

and if x is invertible in A, then y is invertible in A too, as in Section 6.13.

11.11 Polar decompositions

Let (V, ⟨·, ·⟩_{V}) be a real or complex Hilbert space, with associated norm ∥ · ∥_{V}, and V ≠ {0}. If T is any bounded linear mapping from V into itself, then T^{∗} ∘ T is a bounded self-ajoint linear mapping from V into itself that is also nonnegative, as in Section 8.10. This implies that there is a unique bounded self-adjoint linear mapping P from V into itself that is nonnegative on V and satisfies P ∘ P = T^{∗} ∘ T, (11.11.1) as in the previous section.

Of course, (11.11.1) is equivalent to saying that

⟨(P ∘ P)(v), w⟩_{V} = ⟨(T^{∗} ∘ T)(v), w⟩_{V} (11.11.2)

for all v, w ∈ V. This implies that

⟨(P ∘ P)(v), v⟩_{V} = ⟨(T^{∗} ∘ T)(v), v⟩_{V} (11.11.3)

for every v ∈ V. Conversely, (11.11.3) implies (11.11.2), using suitable polarization identities, as in Section 8.5. This also uses the fact that P ∘ P and T^{∗} ∘ T are self-adjoint in the real case, to get that both sides of (11.11.2) are symmetric bilinear forms on V.

Observe that (11.11.3) is the same as saying that

⟨P (v), P (v)⟩_{V} = ⟨T (v), T (v)⟩_{V} (11.11.4)

for every v ∈ V. Equivalently, this means that

∥P (v)∥_{V} = ∥T (v)∥_{V} (11.11.5)

for every v ∈ V. It follows that P is the unique bounded self-adjoint linear mapping from V into itself that is nonnegative and satisfies (11.11.5). This corresponds to Theorem 12.34 on p314 of [286].

Suppose now that T is invertible on V, so that T^{∗} is invertible, and thus T^{∗} ∘ T is invertible too. This implies that P is invertible on V as well, as in the previous section. Put U = T ∘ P^{−1}, (11.11.6)

so that T = U ∘ P. (11.11.7)

Note that U is invertible on V.

Observe that U^{∗} = P^{−1} ∘ T^{∗}, so that

U^{∗} ∘ U = P^{−1} ∘ T^{∗} ∘ T ∘ P^{−1} = I. (11.11.8)

This means that U^{−1} = U^{∗}, (11.11.9)

because U is invertible. Thus U is a unitary mapping from V onto itself, or equivalently an orthogonal transformation on V in the real case. One could also use (11.11.4) to get that ⟨·, ·⟩_{V} is invariant under U.

Conversely, suppose that (11.11.7) holds for some unitary mapping U on V and bounded self-adjoint linear mapping P from V into itself that is nonnegative on V. One can check directly that (11.11.1) holds, or equivalently that (11.11.5) holds. This implies that P is uniquely determined by T, as before. It follows that U is uniquely determined by T as well, because P is invertible. This corresponds to part (a) of Theorem 12.35 on p315 of [286].

The expression (11.11.7) for T is called the polar decomposition of T.

Suppose that T is normal on V, in the sense that T commutes with T^{∗}, so that T^{∗} ∘ T commutes with T and T^{∗}. If P is the nonnegative self-adjoint square root of T^{∗} ∘ T, as before, then P is in the closure of the subalgebra of BL(V), as an algebra over the real numbers, generated by T^{∗} ∘ T and I, as in the previous section. This means that P commutes with T and T^{∗} too, so that the corresponding unitary operator U commutes with T and T^{∗} as well. In particular, U commutes with T^{∗} ∘ T, and thus also with P. This corresponds to part of part (b) of Theorem 12.35 on p315 of [286], which will be discussed further in the next section.

11.12 More on polar decompositions

Let (V, ⟨·, ·⟩_{V}) be a real or complex Hilbert space again, with associated norm ∥ · ∥_{V}, and V ≠ {0}, and let T be a bounded linear mapping from V into itself. An expression for T as in (11.11.7), where P is a bounded nonnegative selfadjoint linear mapping from V into itself, and U is a unitary mapping from V onto itself, is called a polar decomposition of T, as on p315 of [286]. Of course, (11.11.7) is the same as saying that

T (v) = U(P (v)) (11.12.1)

for every v ∈ V.

In this case, P satisfies (11.11.5), or equivalently

⟨P (v), P (w)⟩_{V} = ⟨T (v), T (w)⟩_{V} (11.12.2)

for every v, w ∈ V, because U is unitary. This means that (11.11.2) holds, so that (11.11.1) holds. It follows that P is uniquely determined by T, as in Section 11.10, and as mentioned in [286].

This implies that the restriction of U to P (V) is uniquely determined by T as well. More precisely, the restriction of U to the closure \overline{P (V)} of P (V) in V is uniquely determined by T.

Note that ker P = ker T, (11.12.3)

by (11.11.5). We also have that

ker P = P (V)^{⊥}, (11.12.4)

as in Section 8.11, because P is self-adjoint, and thus normal. Thus P (V) is dense in V exactly when ker P = {0}, which is the same as saying that

ker T = {0}, (11.12.5)

because of (11.12.3). In this case, it follows that U is uniquely determined by P. This was mentioned in the previous section when T is invertible on V.

In order to try to get a polar decomposition for T, we can start with the unique bounded nonnegative self-adjoint linear mapping P from V into itself that satisfies (11.11.1). We would like to define U initially as a mapping from P (V) onto T (V) by (11.12.1). More precisely, U is well-defined on P (V) in this way because of (11.12.3). Remember that (11.11.1) is equivalent to (11.11.2), which is equivalent to (11.12.2). This implies that U is unitary as a linear mapping from P (V) onto T (V), with respect to the restrictions of the inner product to these linear subspaces.

Equivalently, U is an isometric linear mapping from P (V) onto T (V) with respect to the restriction of the norm to these linear subspaces. This implies that U has a unique extension to an isometric linear mapping from \overline{P (V)} onto \overline{T (V)}, as in Section 2.2. This uses the fact that V is complete with respect to the metric associated to ∥ · ∥_{V}, so that \overline{P (V)} and \overline{T (V)} are complete with respect to the restrictions of the metric to these linear subspaces, as in Section 1.6. Let us also use U to denote this extension, which is unitary with respect to the restrictions of the inner product on V to \overline{P (V)} and \overline{T (V)}.

Thus a unitary mapping corresponds exactly to an extension of U to a unitary mapping from V onto itself. Such an extension of U should map P (V)^{⊥} onto T (V)^{⊥}, and is uniquely determined by this mapping. Remember that

ker T^{∗} = T (V)^{⊥}, (11.12.6)

as in Section 8.11. This means that extensions of U to unitary mappings from V onto itself correspond exactly to unitary mappings from ker T onto ker T^{∗}, if there are any, because of (11.12.3) and (11.12.4).

Suppose for the moment that T is normal, so that

ker T = ker T^{∗}, (11.12.7)

as in Section 8.11. In this case, extensions of U to unitary mappings from V onto itself correspond exactly to unitary mappings from ker T onto itself, as in the preceding paragraph. Let us simply take the extension corresponding to the identity mapping on ker T, which we shall denote U as well. This corresponds to part (b) of Theorem 12.35 on p315 of [286].

Remember that P commutes with T and T^{∗} when T is normal, as mentioend at the end of the previous section. One can check directly that U commutes with P and T under these conditions, which is another part of part (b) of Theorem 12.35 on p315 of [286]. This implies that U commutes with T^{∗} too, because U^{∗} = U^{−1}.

If V has finite dimension, then P (V) and T (V) have the same codimension in V, because they have the same dimension. This implies that U can be extended to a unitary mapping from V onto itself, as mentioned on p316 of [286].

Of course, any extension of U to a bounded linear mapping from V into itself satisfies (11.11.7), because that only involves the restriction of U to P (V). Such an extension corresponds exactly to a bounded linear mapping from P (V)^{⊥} into V. The extension defined by taking U = 0 on P (V)^{⊥} is called a partial isometry on V, as on p316 of [286].

262 CHAPTER 11. ALGEBRAS, NORMS, AND SQUARE ROOTS

11.13 Commuting with adjoints

Let A be an associative algebra over the real or complex numbers with an involution x ↦ x^{∗} that may be conjugate-linear in the complex case. If x, y ∈ A commute, then x^{∗} y^{∗} = y^{∗} x^{∗}. (11.13.1) In some case one might also like to have that x y^{∗} = y^{∗} x, (11.13.2) which is the same as saying that x^{∗} y = y x^{∗}. (11.13.3) Of course, x automatically commutes with itself, but it may not commute with x^{∗}.

Suppose that x commutes with y again. If x is self-adjoint or anti-self-adjoint, then (11.13.3) holds, so that (11.13.2) holds. If A has a multiplicative identity element e_{A} and x is invertible in A, then x^{−1} commutes with y. If x^{∗} = x^{−1}, then it follows that (11.13.3) holds, so that (11.13.2) holds again.

Let (V, ⟨·, ·⟩_{V}) be a complex Hilbert space with associated norm ∥ · ∥_{V} and V ≠ {0}, and let A, B, and T be bounded linear mappings from V into itself. Suppose that A, B are normal, and that A ∘ T = T ∘ B. (11.13.4) Under these conditions, A^{∗} ∘ T = T ∘ B^{∗}, (11.13.5) as in Theorem 12.16 on p300 of [286]. This was initially shown by Fuglede when A = B, and extended to this formulation by Putnam. The proof that follows was found by Rosenblum, as in [286].

Using (11.13.4), we get that A^{l} ∘ T = T ∘ B^{l} (11.13.6) for all positive integer l. This implies that (exp(\overline{λ} A)) ∘ T = T ∘ (exp(\overline{λ} B)) (11.13.7) for all λ ∈ C. Equivalently, this means that T = (exp(−\overline{λ} A)) ∘ T ∘ (exp(\overline{λ} B)) (11.13.8) for every λ ∈ C.

Put f(λ) = (exp(λ A^{∗})) ∘ T ∘ (exp(−λ B^{∗})) (11.13.9) for every λ ∈ C. Observe that

f(λ) = (exp(λ A^{∗} − \overline{λ} A)) ∘ T ∘ (exp(−λ B^{∗} + \overline{λ} B)) (11.13.10)

for every λ ∈ C, because of (11.13.8), and the normality of A, B. We also have that exp(λ A^{∗} − \overline{λ} A), exp(−λ B^{∗} + \overline{λ} B) (11.13.11)

are unitary operators on V for every λ ∈ C, because

λ A^{∗} − \overline{λ} A, −λ B^{∗} + \overline{λ} B (11.13.12)

are anti-self-adjoint on V. This implies that

∥f(λ)∥_{op} = ∥T ∥_{op} (11.13.13)

for every λ ∈ C.

We may consider f(λ) as a holomorphic function of λ ∈ C with values in the algebra BL(V) of bounded linear mappings from V into itself. In fact, f(λ) may be expressed as an absolutely convergent power series in λ with coefficients in BL(V), using Cauchy products. In particular, if v, w ∈ V, then

f_{v,w}(λ) = ⟨(f(λ))(v), w⟩_{V} (11.13.14)

is a holomorphic complex-valued function of λ ∈ C. Note that

|f_{v,w}(λ)| ≤ ∥T ∥_{op} ∥v∥_{V} ∥w∥_{V} (11.13.15)

for every λ ∈ C, because of (11.13.13).

Thus Liouville’s theorem implies that f_{v,w}(λ) is constant as a function of λ ∈ C, so that f_{v,w}(λ) = f_{v,w}(0) (11.13.16)

for every λ ∈ V. This means that

(exp(λ A^{∗})) ∘ T = T ∘ (exp(λ B^{∗})) (11.13.17)

for every λ ∈ C. One can get (11.13.5) by considering the derivative of both sides of (11.13.17) in λ at 0.

The analogous statement for real Hilbert spaces can be reduced to the complex case using complexification. This will be discussed further in the next chapter.

11.14 Similar normal operators

Let (V, ⟨·, ·⟩_{V}) be a complex Hilbert space with associated norm ∥ · ∥_{V} and V ≠ {0} again. Also let A, B, and T be bounded linear mappings from V into itself such that A and B are normal, T is invertible, and

A = T ∘ B ∘ T^{−1}. (11.14.1)

264 CHAPTER 11. ALGEBRAS, NORMS, AND SQUARE ROOTS

If T = U ∘ P is the polar decomposition of T, as in Section 11.11, then A = U ∘ B ∘ U^{−1}. (11.14.2) This is Theorem 12.36 on p316 of [286], which is due to Putnam.

Of course, (11.14.1) is the same as saying that A ∘ T = T ∘ B. This implies that A^{∗} ∘ T = T ∘ B^{∗}, as in the previous section. It follows that T^{∗} ∘ A = (A^{∗} ∘ T)^{∗} = (T ∘ B^{∗})^{∗} = B ∘ T^{∗}. (11.14.3) Remember that P ∘ P = T^{∗} ∘ T, so that B ∘ P ∘ P = B ∘ T^{∗} ∘ T = T^{∗} ∘ A ∘ T = T^{∗} ∘ T ∘ B = P ∘ P ∘ B. (11.14.4)

We also have that P is in the closure of the subalgebra of BL(V), as an algebra over the real numbers, generated by T^{∗} ∘ T and I, as in Section 11.11. This means that P is in the closure of the subalgebra of BL(V), as an algebra over the real numbers, generated by P ∘ P and I. It follows that B ∘ P = P ∘ B, (11.14.5) because of (11.14.4). It is easy to obtain (11.14.2) from (11.14.1) and (11.14.5).

The arguments in this section also work for real Hilbert spaces, as long as we have (11.13.5). That will be discussed in the next chapter, as mentioned in the previous section.

11.15 More on square roots

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A} and a submultiplicative norm ∥ · ∥_{A}. Put f(x) = (e_{A} + x)^{2} = e_{A} + 2 x + x^{2} (11.15.1) for each x ∈ A. Thus

(11.15.2)  f(x) − f(y) = 2 (x − y) + x^{2} − y^{2}
= 2 (x − y) + x (x − y) + (x − y) y

for every x, y ∈ A.

It follows that 2 ∥x − y∥_{A} ≤ ∥f(x) − f(y)∥_{A} + (∥x∥_{A} + ∥y∥_{A}) ∥x − y∥_{A} (11.15.3) for every x, y ∈ A. Equivalently, this means that (2 − ∥x∥_{A} − ∥y∥_{A}) ∥x − y∥_{A} ≤ ∥f(x) − f(y)∥_{A} (11.15.4) for every x, y ∈ A. In particular, if ∥x∥_{A} + ∥y∥_{A} < 2 (11.15.5)

11.15. MORE ON SQUARE ROOTS 265

and f(x) = f(y), (11.15.6) then we get that x = y. (11.15.7)

If x ∈ A, then 2 ∥x∥_{A} ≤ ∥f(x) − e_{A}∥_{A} + ∥x∥_{A}^{2}, (11.15.8) by (11.15.3), with y = 0. In particular, ∥x∥_{A} ≤ ∥f(x) − e_{A}∥_{A} (11.15.9) when ∥x∥_{A} ≤ 1. More precisely, (11.15.8) implies that 1 − ∥f(x) − e_{A}∥_{A} ≤ 1 − 2 ∥x∥_{A} + ∥x∥_{A}^{2} = (1 − ∥x∥_{A})^{2}. (11.15.10) If ∥x∥_{A}, ∥f(x) − e_{A}∥_{A} ≤ 1, then it follows that (1 − ∥f(x) − e_{A}∥_{A})^{1/2} ≤ 1 − ∥x∥_{A}. (11.15.11) This means that ∥x∥_{A} ≤ 1 − (1 − ∥f(x) − e_{A}∥_{A})^{1/2}. (11.15.12)

Let a ∈ A be given, and suppose that we want to find x ∈ A such that f(x) = e_{A} + a. (11.15.13) Put g(x) = x + (1/2) x^{2} (11.15.14) for each x ∈ A, so that (11.15.13) is the same as saying that g(x) = (1/2) a. (11.15.15) Similarly, put ϕ_{a}(x) = (1/2) a − (1/2) x^{2} (11.15.16) for each x ∈ A, so that (11.15.15) is equivalent to ϕ_{a}(x) = x. (11.15.17)

If x, y ∈ A, then

(11.15.18) ϕ_{a}(x) − ϕ_{a}(y) = (1/2) (y^{2} − x^{2})
= (1/2) (y − x) y + (1/2) x (y − x).

This implies that ∥ϕ_{a}(x) − ϕ_{a}(y)∥_{A} ≤ (1/2) (∥x∥_{A} + ∥y∥_{A}) ∥x − y∥_{A}. (11.15.19) Note that ∥ϕ_{a}(x)∥_{A} ≤ (1/2) ∥a∥_{A} + (1/2) ∥x∥_{A}^{2} (11.15.20) for every x ∈ A.

Let \overline{B}(0, r) be the closed ball in A centered at 0 with radius r ≥ 0 with respect to the metric associated to ∥ · ∥_{A}. If x, y ∈ \overline{B}(0, r), then

∥ϕ_{a}(x) − ϕ_{a}(y)∥_{A} ≤ r ∥x − y∥_{A}, (11.15.21)

by (11.15.19). Similarly, if a, x ∈ \overline{B}(0, r), then

∥ϕ_{a}(x)∥_{A} ≤ (1/2) r + (1/2) r^{2}, (11.15.22)

by (11.15.20). If r ≤ 1, then (11.15.22) implies that

∥ϕ_{a}(x)∥ ≤ r. (11.15.23)

This means that ϕ_{a}(\overline{B}(0, r)) ⊆ \overline{B}(0, r) (11.15.24)

when ∥a∥_{A} ≤ r and r ≤ 1.

Suppose that A is complete with respect to the metric associated to ∥·∥_{A}, so that \overline{B}(0, r) with respect to the restriction of this metric to \overline{B}(0, r), as in Section 1.6. If ∥a∥_{A} < 1, then we take take r = ∥a∥_{A} in (11.15.21) and (11.15.24) to get that ϕ_{a} has a unique fixed point x_{a} in \overline{B}(0, ∥a∥_{A}), by the contraction mapping theorem. This means that x_{a} is the unique element of \overline{B}(0, ∥a∥_{A}) that satisfies (11.15.15), or equivalently (11.15.13).

Complexifications, nets, and C^{∗} algebras

12.1 Complexifying real vector spaces

Let V be a vector space over the real numbers. We would like to define the complexification of V, which is a vector space V_{C} over the complex numbers. We start by defining V_{C} as a vector space over the real numbers to be the Cartesian product V × V with itself, with respect to coordinatewise addition and scalar multiplication. This is the same as the direct sum of V with itself, as a vector space over R, as in Section 5.12.

Let us define multiplication by i on V_{C} by putting

i (v_{1}, v_{2}) = (−v_{2}, v_{1}) (12.1.1)

for every v_{1}, v_{2} ∈ V. Note that

i (i (v_{1}, v_{2})) = i (−v_{2}, v_{1})) = (−v_{1}, −v_{2}) (12.1.2)

for every v_{1}, v_{2} ∈ V. One can use (12.1.1) to define multiplication by any complex number on V_{C} in an obvious way, using also scalar multiplication by real numbers. One can check that this makes V_{C} into a vector space over the complex numbers.

We may identify v ∈ V with (v, 0) ∈ V_{C}, so that V corresponds to V × {0}, as a real-linear subspace of V_{C}. Thus (v_{1}, v_{2}) ∈ V_{C} may be expressed as v_{1}+i v_{2}. Observe that C_{V} (v_{1} + i v_{2}) = v_{1} − i v_{2} (12.1.3)

defines a one-to-one conjugate-linear mapping from V C onto itself.

268 CHAPTER 12. COMPLEXIFICATIONS, NETS, AND C^{∗} ALGEBRAS

Suppose for the moment that V is a linear subspace of the space of realvalued functions on a nonempty set X. In this case, V_{C} may be identified with the linear subspace of the space of all complex-valued functions on X whose real and imaginary parts are elements of V. Using this identification, (12.1.3) corresponds to taking the complex conjugate of a complex-valued function on X.

Let W be a vector space over the complex numbers, which may also be considered as a vector space over the real numbers. If T is a real-linear mapping from V into W, then

T_{C}(v_{1} + i v_{2}) = T (v_{1}) + i T (v_{2}) (12.1.4)

define s complex-linear mapping from V_{C} into W. This is the unique complexlinear mapping from V_{C} into W that agrees with T on V.

Let Y be another vector space over the real numbers, and let Y_{C} be its complexification, as before. Suppose that R is a linear mapping from V into Y, which may be considered as a real-linear maping from V into Y_{C}, by identifying Y with a real-linear subspace of Y_{C}, as before. This leads to a complex-linear mapping R_{C} from V_{C} into Y_{C}, as in the preceding paragraph. This may be called the complexification of R. Observe that

R_{C} ∘ C_{V} = C_{Y} ∘ R_{C}, (12.1.5)

where C_{Y} is defined on Y_{C} as in (12.1.3).

Conversely, let A be a complex-linear mapping from V_{C} into Y_{C} such that

A ∘ C_{V} = C_{Y} ∘ A. (12.1.6)

This implies that A maps V into Y, as real-linear subspaces of V_{C} and Y_{C}, respectively. Let A_{0} be the restriction of A to V, considered as a real-linear mapping into Y. It is easy to see that

A = (A_{0})_{C}. (12.1.7)

Let Z be another vector space over the real numbers, with complexification Z_{C}, and let B be a linear mapping from Y into Z. This leads to a complex-linear mapping B_{C} from Y_{C} into Z_{C}, as before. Similarly, B ∘ R is a linear mapping from V into Z, which leads to a complex-linear mapping (B ∘ R)_{C} from V_{C} into Z_{C}. One can check that

(B ∘ R)_{C} = B_{C} ∘ R_{C}. (12.1.8)

12.2 Complexifying inner products

Let V be a vector space over the real numbers with an inner product ⟨·, ·⟩_{V}, and let V_{C} be the complexification of V, as in the previous sections. The complexification of ⟨·, ·⟩_{V} may be defined on V_{C} by

(12.2.1)  ⟨v_{1} + i v_{2}, w_{1} + i w_{2}⟩_{V_{C}} = ⟨v_{1}, v_{2}⟩_{V} + i ⟨v_{2}, w_{1}⟩_{V}
−i ⟨v_{1}, w_{2}⟩_{V} + ⟨v_{2}, w_{2}⟩_{V} .

It is easy to see that this defines a sesquilinear form on V_{C}. In fact, it is an inner product on V_{C}, because

⟨v_{1} + i v_{2}, v_{1} + i v_{2}⟩_{V_{C}} = ⟨v_{1}, v_{1}⟩_{V} + ⟨v_{2}, v_{2}⟩_{V} (12.2.2)

for every v_{1}, v_{2} ∈ V.

Note that

Re⟨v_{1} + i v_{2}, w_{1} + i w_{2}⟩_{V_{C}} = ⟨v_{1}, w_{1}⟩_{V} + ⟨v_{2}, w_{2}⟩_{V} (12.2.3)

for all v_{1}, v_{2}, w_{1}, w_{2} ∈ V. This is the same as the inner product on _{C}, as a vector space over the real numbers, and considered as the direct sum of two copies of V, obtained from ⟨·, ·⟩_{V} on both copies of V as in Section 5.12. If ∥ · ∥_{V}, ∥ · ∥_{V_{C}} are the norms on V, V_{C}, respectively, associated to their inner products, then

∥v_{1} + i v_{2}∥_{V_{C}}^{2} = ∥v_{1}∥_{V}^{2} + ∥v_{2}∥_{V}^{2} (12.2.4)

for all v_{1}, v_{2} ∈ V. If V is complete with respect to the metric associated to ∥ · ∥_{V}, then V_{C} is complete with respect to the metric associated to ∥ · ∥_{V_{C}}, as before.

Let (W, ⟨·, ·⟩_{W}) be another real inner product space, with complexification (W_{C}, ⟨·, ·⟩_{W_{C}}), and associated norms ∥·∥_{W}, ∥·∥_{W_{C}}, respectively. Also let T be a linear mapping from V into W, and let T_{C} be the corresponding complex-linear mapping from V_{C} into W_{C}, as in the previous section. If v_{1}, v_{2} ∈ V, then

∥T (v_{1} + i v_{2})∥_{W_{C}}^{2} = ∥T (v_{1}) + i T (v_{2})∥_{W_{C}}^{2} = ∥T (v_{1})∥_{W}^{2} + ∥T (v_{2})∥_{W}^{2} , (12.2.5)

where the second step is as in (12.2.4). If T is a bounded linear mapping from V into W, then one can use this to check that T_{C} is bounded as a linear mapping from V_{C} into W_{C}, with

∥T_{C}∥_{op,V_{C}W_{C}} = ∥T ∥_{op,V W} . (12.2.6)

Suppose that V, W are real Hilbert spaces, so that V_{C}, W_{C} are complex Hilbert spaces, as before. If T is a bounded linear mapping from V into W, then the adjoint T^{∗} of T is a bounded linear mapping from W into V, as in Section 3.5. This leads to a bounded linear mapping (T^{∗})_{C} from W_{C} into V_{C}, as in the preceding paragraph. One can verify that

(T^{∗})_{C} = (T_{C})^{∗}, (12.2.7)

the adjoint of T_{C}. Thus one may denote this operator simply as T_{C}^{∗}.

One can use this to get the analogue of the statement in Section 11.13 for real Hilbert spaces from the previous version for complex Hilbert spaces, as mentioned earlier. This permits one to use the same arguments as in Section 11.14 for real Hilbert spaces, as before.

12.3 Convergence of nets

A binary relation ⪯ on a set A is said to be a partial ordering if it is reflexive, transitive, and satisfies

a = b when a ⪯ b and b ⪯ a (12.3.1)

for all a, b ∈ A. Sometimes (12.3.1) is not included in the definition of a partial ordering, which may not lead to any additional complications for the present purposes. The term pre-order may also be used for binary relations that are reflexive and transitive, but may not satisfy (12.3.1). If a partial ordering ⪯ on A has the additional property that for every a, b ∈ A,

a ⪯ b or b ⪯ a, (12.3.2)

then ⪯ is said to be a linear ordering or total ordering on A.

A partially-ordered set (A, ⪯) is said to be a directed system if

(12.3.3) for every a, b ∈ A there is a c ∈ A such that a, b ⪯ c.

Note that linearly-ordered sets are directed systems. Similarly, let us say that a pre-ordered set (A, ⪯) is a pre-directed system if it satisfies (12.3.3). As before, the term directed system is sometimes used for this, which may not lead to additional complications for the present purposes.

Let (A, ⪯) be a nonempty directed system or pre-directed system, and let Z be a set. A net of elements of Z indexed by A is a family {z_{a}}_{a∈A} that associates to each a ∈ A an element z_{a} of Z. This is the same as a function on A with values in Z, but we shall normally use this notation and terminology in this situation. If A is the set of positive integers with the standard ordering, then a net of elements of Z indexed by A is the same as a sequence of elements of Z.

Suppose now that Z is a metric or topological space. A net {z_{a}}_{a∈A} of elements of Z indexed by A is said to converge to a point z ∈ Z if for every open set V ⊆ Z with z ∈ V there is an a ∈ A such that

z_{b} ∈ V (12.3.4)

for every b ∈ A with a ⪯ b. If (Z, d_{Z}) is a metric space, then this is equivalent to asking that for each ϵ > 0 there be an a ∈ A such that

d_{Z}(z, z_{b}) < ϵ (12.3.5)

for every b ∈ A with a ⪯ b. If A is the set of positive integers with the standard ordering, then convergence of a net of elements of Z indexed by A is the same as convergence of a sequence of elements of Z.

Let Y be a topological space, and let B(y) be a local base for the topology of Y at a point y ∈ Y. If U, V ∈ B(y), then put

U ⪯ V when V ⊆ U. (12.3.6)

One can check that B(y) is a directed system with respect to ⪯. Let {y_{U}}_{U∈B(y)} be a net of elements of Y indexed by B(y). If

y_{U} ∈ U (12.3.7)

for every U ∈ B(y), then it is easy to see that

(12.3.8) {y_{U}}_{U∈B(y)} converges to y

as a net of elements of Y.

One can check that the limit of a convergent net in a metric space is unique. Similarly, the limit of a convergent net in a Hausdorff topological space is unique.

Let (A, ⪯) be a nonempty directed system or pre-directed system again, and let (Z, d_{Z}) be a metric space. A net {z_{a}}_{a∈A} of elements of Z is said to be a Cauchy net if for every ϵ > 0 there is an a ∈ A such that

d_{Z}(z_{b}, z_{c}) < ϵ (12.3.9)

for every b, c ∈ A with a ⪯ b, c. If A is the set of positive integrs with the standard ordering, then a Cauchy net of elements of Z indexed by A is the same as a Cauchy sequence in Z. It is easy to see that a convergence net of elements of Z is a Cauchy net.

If Z is complete with respect to d_{Z}, and {z_{a}}_{a∈A} is a Cauchy net of elements of Z indexed by A, then it is well known that {z_{a}}_{a∈A} converges to an element z of Z. Indeed, for each positive integer j there is an a_{j} ∈ A such that

d_{Z}(z_{b}, z_{c}) < 1/j (12.3.10)

for every b, c ∈ A with a_{j} ⪯ b_{j}, c_{j}. We can also choose α_{j} ∈ A recursively for each j ≥ 1 in such a way that a_{j} ⪯ α_{j} (12.3.11)

and α_{j−1} ⪯ α_{j} (12.3.12)

when j ≥ 2, because A is a (pre-)directed system. It is easy to see that {z_{α_{j}}}_{j=1}^{∞} is a Cauchy sequence in Z under these conditions. If Z is complete, then {z_{α_{j}}}_{j=1}^{∞} converges to an element z of Z, and one can verify that {z_{a}}_{a∈A} converges to z as well.

12.4 More on convergence of nets

Let (A, ⪯) be a nonempty directed or pre-directed system again, and let Y, Z be metric or topological spaces. Suppose that {y_{a}}_{a∈A} is a net of elements of Y indexed by A that converges to y ∈ Y. If a mapping ϕ from Y into Z is continuous at y, then it is easy to see that

(12.4.1) {ϕ(y_{a})}_{a∈A} converges to ϕ(y), as a net of elements of Z.

Conversely, suppose that ϕ is not continuous at y. This means that there is an open set W ⊆ Z such that ϕ(y) ∈ W and

ϕ(U) ⊈ W (12.4.2)

for any open set U ⊆ Y with y ∈ U. Let B(y) be a local base for the topology of Y at y, which may be considered as a directed system with as in the previous section. If U ∈ B(y), then let y_{U} be an element of U such that

ϕ(y_{U}) ∉ W. (12.4.3)

Under these conditions, {y_{U}}_{U∈B(x)} converges to y in Y, as in (12.3.8), but {ϕ(y_{U})}_{U∈B(x)} does not converge to ϕ(y) in Z, because of (12.4.3).

Now le Y be a nonempty set, let {f_{a}}_{a∈A} be a net of functions on Y with values in Z indexed by A, and let f be another Z-valued function on Y. We say that {f_{a}}_{a∈A} converges pointwise to f on Y if

(12.4.4) {f_{a}(y)}_{a∈A} converges to f(y)

in Z for every y ∈ Y. Similarly, if (Z, d_{Z}) is a metric space, then {f_{a}}_{a∈A} converges to f uniformly on Y if for every ϵ > 0 there is an a ∈ A such that

d_{Z}(f(y), f_{b}(y)) < ϵ (12.4.5)

for every b ∈ A with a ⪯ b and y ∈ Y. These definitions reduce to the usual ones for sequences when A is the set of positive integers with the standard ordering. Of course, uniform convergence automatically implies pointwise convergence.

Suppose that (Z, d_{Z}) is a metric space, and let B(Y, Z) be the space of all bounded mappings from Y into Z, as in Section 7.8. As in the case of sequences, convergence of a net in B(Y, Z) with respect to the supremum metric is equivalent to uniform convergence. Let {f_{a}}_{a∈A} be any net of Z-valued function on Y indexed by A again, and let f be another Z-valued function on Y. If Y is a metric or topological space, y ∈ Y, f_{a} is continuous at y for each a ∈ A, and {f_{a}}_{a∈A} converges to f uniformly on Y, then

(12.4.6) f is continuous at y

as well. If Y is a metric space, f_{a} is uniformly continuous on Y for each a ∈ A, and {f_{a}}_{a∈A} converges to f uniformly on Y, then

(12.4.7) f is uniformly continuous on Y

too.

Suppose now that Z is a vector space over the real or complex numbers with a norm ∥ · ∥_{Z}, let {z_{a}}_{a∈A} be a net of elements of Z indexed by A, and let z be another element of Z. Suppose for the moment that {z_{a}}_{a∈A} converges to z with respect to the metric on Z associated to ∥ · ∥_{Z}. If t ∈ R or C, as appropriate, then one can check that

(12.4.8) {t z_{a}}_{a∈A} converges to t z in Z.

Similarly, if {w_{a}}_{a∈A} is a net of elements of Z that converges to w ∈ Z, then

(12.4.9) {w_{a} + z_{a}}_{a∈A} converges to w + z in Z.

We say that {z_{a}}_{a∈A} converges weakly to z ∈ Z if for every bounded linear functional λ on Z, (12.4.10) {λ(z_{a})}_{a∈A} converges to λ(z)

in R or C, as appropriate. If A is the set of positive integers with the standard ordering, then this is the same weak convergence of sequences, as in Section 4.9. Otherwise, this is the same as the convergence of the net with respect to the weak topology on Z. Weak convergence is implied by convergence with respect to the metric associated to the norm, as before. The weak limit of a net in Z is unique when it exists, because the dual space Z′ of bounded linear functionals on Z separates points in Z, by the Hahn–Banach theorem. Note that weak convergence of nets satisfies the same type of properties for sums and scalar multiples of convergent nets as in the preceding paragraph. This follows from the analogous statements for convergent nets in R and C, which may be considered as particular cases of the previous statements.

Suppose that Y is another vector space over the real or complex numbers, as appropriate, with a norm ∥ · ∥_{Y}. Also let {T_{a}}_{a∈A} be a net of bounded linear mappings from Y into Z indexed by A, and let T be another bounded linear mapping from Y into Z. Under these conditions, {T_{a}}_{a∈A} converges to T with respect to the strong operator topology on the space BL(Y, Z) of bounded linear mappings from Y into Z if and only if {T_{a}}_{a∈A} converges to T pointwise on Y. If A is the set of positive integers with the standard ordering, then this corresponds to the type of convergence discussed in Section 4.4. If {T_{a}}_{a∈A} converges to T with respect to the metric on BL(Y, Z) associated to the operator norm, then it is easy to see that {T_{a}}_{a∈A} converges to T pointwise on Y.

In particular, we can take Z = R or C, with the standard absolute value function as the norm. Let {μ_{a}}_{a∈A} be a net of bounded linear functionals on Y indexed by A, and let μ be another bounded linear functional on Y. Pointwise convergence of {μ_{a}}_{a∈A} to μ on Y is equivalent to the convergence of {μ_{a}}_{a∈A} to μ with respect to the weak^{∗} topology on the dual space Y′ of bounded linear functionals on Y with respect to ∥ · ∥_{Y}. If A is the set of positive integers with the standard ordering, then this corresponds to the type of convergence mentioned in Section 4.5.

12.5 Nets and approximate eigenvalues

Let V be a vector space over the real or complex numbers, and let v be a nonzero element of V. Consider the collection A_{v} of linear mappings T from V into itself such that v is an eigenvector of T with eigenvalue λ_{v}(T). Note that λ_{v}(T) is unique in this case, because v ≠ 0. It is easy to see that A_{v} is a subalgebra of the algebra L(V) of all linear mappings from V into itself, and of course A_{v} contains the identity mapping I = I_{V} on V. We also have that λ_{v} is an algebra homomorphism from A_{v} into R or C, as appropriate, with λ_{v}(I) = 1.

Suppose now that ∥ · ∥_{V} is a norm on V, and that (A, ⪯) is a nonempty directed or pre-directed system. Let {v_{a}}_{a∈A} be a net of elements of V such that ∥v_{a}∥_{V} = 1 (12.5.1)

for each a ∈ A. Consider the collection A of bounded linear mappings T from V into itself for which there is a real or complex number λ(T), as appropriate, such that (12.5.2) {∥T (v_{a}) − λ(T) v_{a}∥_{V} }_{a∈A} converges to 0,

as a net of real numbers. One can check that λ(T) is uniquely determined by (12.5.2), and satisfies |λ(T)| ≤ ∥T ∥_{op}. (12.5.3)

We also have that λ(T) is an approximate eigenvalue of T, as in Section 9.3.

It is easy to see that A is a linear subspace of the space BL(V) of all bounded linear mappings from V into itself, and that λ defines a linear functional on A. In fact, one can verify that A is a subalgebra of BL(V), and that λ is an algebra homomorphism from A into R or C, as appropriate. Clearly I ∈ A, with λ(I) = 1.

Let us check that A is a closed set in BL(V), with respect to the metric associated to the operator norm. Let \overline{A} be the closure of A in BL(V) with respect to this metric. There is a unique extension of λ to a bounded linear functional on \overline{A} with respect to the operator norm, as in Section 2.2, and we shall also use λ to denote this extension. If T ∈ \overline{A}, then one can verify that (12.5.2) holds, by approximating T by elements of A. This implies that T ∈ A, so that \overline{A} = A.

Now let (V, ⟨·, ·⟩_{V}) be a real or complex Hilbert space, and let T_{0} be a bounded linear mapping from V into itself that is normal, in the sense that it commutes with its adjoint. Also let μ_{0} be a real or complex number, as appropriate, in the spectrum of T_{0}. This implies that μ_{0} is an approximate eigenvalue of T_{0}, as in Section 9.5.

It follows that there is a sequence {v_{j}}_{j=1}^{∞} of unit vectors in V such that

lim_{j→∞} ∥T_{0}(v_{j}) − μ_{0} v_{j}∥_{V} = 0, (12.5.4)

as in Section 9.3. We also get that

lim_{j→∞} ∥T_{0}^{∗}(v_{j}) − μ_{0} v_{j}∥_{V} = 0 (12.5.5)

in the real case, and

lim_{j→∞} ∥T_{0}^{∗}(v_{j}) − \overline{μ_{0}} v_{j}∥_{V} = 0 (12.5.6)

12.6. C^{∗} ALGEBRAS 275

in the complex case, as in Section 9.5.

Consider the collection A_{0} of bounded linear mappings T from V into itself for which there is a real or complex number λ_{0}(T), as appropriate, such that

lim_{j→∞} ∥T (v_{j}) − λ_{0}(T) v_{j}∥_{V} = 0. (12.5.7)

This is a closed subalgebra of BL(V) that contains I, as before, and which contains T_{0}, T_{0}^{∗} in this case.

12.6 C^{∗} Algebras

Let (A, ∥·∥_{A}) be a Banach algebra over the complex numbers with a conjugatelinear algebra involution x ↦ x^{∗}. Suppose that ∥ · ∥_{A} satisfies the C^{∗} identity

∥x^{∗} x∥_{A} = ∥x∥_{A}^{2} (12.6.1)

for every x ∈ A, as in Section 7.7. Under these conditions, A is called a C^{∗} algebra with respect to the norm and involution. Remember that (12.6.1) implies that ∥x^{∗}∥_{A} = ∥x∥_{A} (12.6.2)

for every x ∈ A, as in Section 7.7.

This corresponds to Definition 2.2.1 on p46 of [11], and is mentioned on p260 of [291]. If A has a nonzero multiplicative identity element e_{A}, then we have that ∥e_{A}∥_{A} = 1, as in Section 7.7. This may sometimes be included in the definition of a C^{∗} algebra.

Sometimes the term B^{∗} algebra is used for what is called a C^{∗} algebra here, as in Definition 11.17 on p276 of [286]. However, this term is also sometimes used for a complex Banach algebra with a norm-preserving conjugate-linear involution, as on p260 of [291]. The latter may be called a Banach ∗-algebra as well, as in Definition 2.5.1 on p57 of [11]. A conjugate-linear algebra involution on a complex associative algebra may be denoted x ↦ x , as in Definition 21.6 on p313 of [151]. Similarly, a complex Banach algebra with a conjugate-linear involution that preserves the norm may be caller a Banach -algebra, as in [151].

If X is a nonempty set, then the space ℓ^{∞}(X, C) of bounded complex-valued functions on X is a commutative complex Banach algebra with respect to the supreum norm. It is easy to see that

f ↦ \overline{f} (12.6.3)

is a conjugate-linear algebra involution on ℓ^{∞}(X, C). If f ∈ ℓ^{∞}(X, C), then

∥\overline{f} f∥_{∞} = ∥|f|^{2}∥_{∞} = ∥f∥_{∞}^{2}, (12.6.4)

so that ℓ^{∞}(X, C) is a C^{∗} algebra with respect to this involution.

Let (V, ⟨·, ·⟩_{V}) be a complex Hilbert space, with the associated norm ∥ · ∥_{V}, and remember that the algebra BL(V) of bounded linear mappings from V into itself is a complex Banach algebra with respect to the corresponding operator norm. We have seen that the Hilbert space adjoint defines a conjugate-linear involution on BL(V) that satisfies the C^{∗} identity, as in Section 3.5, so that BL(V) is a C^{∗} algebra.

Let (A, ∥ · ∥_{A}) be a complex Banach algebra again. If A_{1} is a subalgebra of A that is also a closed set with respect to the metric associated to ∥ · ∥_{A}, then A_{1} may be considered as a complex Banach algebra with respect to the restriction of ∥ · ∥_{A} to A_{1}. If x ↦ x^{∗} is a conjugate-linear involution on A, and if x^{∗} ∈ A_{1} for each x ∈ A_{1}, then the restriction of x ↦ x^{∗} to x ∈ A_{1} defines an involution on A_{1}. If A is a C^{∗} algebra, and A_{1} has all of the properties just mentioned, then A_{1} is a C^{∗} algebra with respect to the restrictions of the norm and involution to A_{1}.

Let V be a complex Hilbert space again, and let A_{1} be a closed subalgebra of BL(V) that contains the adjoints of all of its elements. Thus A_{1} is a C^{∗} algebra with respect to the restrictions of the operator norm and the involution defined by taking the adjoint to A_{1}, as in the preceding paragraph. In this case, A_{1} may be called a C^{∗} algebra of operators, as in Definition 2.1.2 on p42 of [11].

Let X be a nonempty metric or topological space. The algebra C_{b}(X, C) of bounded continuous complex-valued functions on X is a closed subalgebra of ℓ^{∞}(X, C) that is invariant under complex conjugation, and thus a C^{∗} algebra with respect to the supremum norm and (12.6.3).

12.7 Some remarks about involutions

Let A_{1}, A_{2}, and A_{3} be algebras in the strict sense, all real or all complex. Also let ϕ be a mapping from A_{1} into A_{2}, and let ψ be a mapping from A_{2} into A_{3}. Of course, if ϕ and ψ are linear mappings, then their composition ψ ∘ ϕ is a linear mapping from A_{1} into A_{3}. In the complex case, we may be concerned with situations where one of ϕ and ψ is complex-linear and the other is conjugate-linear, which implies that

(12.7.1) ψ ∘ ϕ is conjugate-linear

as well. If ϕ and ψ are both conjugate-linear, then

(12.7.2) ψ ∘ ϕ is complex-linear.

If ϕ and ψ are both algebra homomorphisms, then ψ ∘ ϕ is an algebra homomorphism too, as in Section 6.3. If one of ϕ and ψ is an algebra homomorphism and the other is an opposite algebra homomorphism, then

(12.7.3) ψ ∘ ϕ is an opposite algebra homomorphism.

If ϕ and ψ are both opposite algebra homomorphisms, then

(12.7.4) ψ ∘ ϕ is an algebra homomorphism.

Let a_{1} ↦ a_{1}^{∗_{1}} and a_{2} ↦ a_{2}^{∗_{2}} be involutions on A_{1} and A_{2}, respectively. In the complex case, these involutions should be both complex-linear or both conjugate-linear. An algebra homomorphism ϕ from A_{1} into A_{2} is said to be a ∗-homomorphism if ϕ(a_{1}^{∗_{1}}) = ϕ(a_{1})^{∗_{2}} (12.7.5)

for every a_{1} ∈ A_{1}. One may also consider ϕ to be a homomorphism from A_{1} into A_{2} as algebras in the strict sense with involutions in this case. Similarly, we may refer to ∗-isomorphisms and ∗-automorphisms for algebra isomorphisms and automorphisms that are ∗-homomorphisms, which may be considered as isomorphisms or automorphisms of algebras in the strict sense with involutions, as appropriate, as well.

Note that the definition of a homomorphism between C^{∗} algebras on p262 of [291] includes (12.7.5). Here we shall always use the term ∗-homomorphism for an algebra homomorphism that satisfies (12.7.5), so that there is no ambiguity with ordinary homomorphisms between algebras.

Now let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}, and let x ↦ x^{∗} be an algebra involution on A, which may be conjugate-linear in the complex case. Suppose that a is an invertible element of A, so that

ϕ_{a}(x) = a x a^{−1} (12.7.6)

defines an algebra automorphism of A. Thus

x^{#} = a x^{∗} a^{−1} (12.7.7)

defines an opposite algebra isomorphism from A onto itself, as before. In the complex case, if x ↦ x^{∗} is conjugate-linear, then (12.7.7) is conjugate-linear in x too.

If x ∈ A, then

(x^{#})^{#} = a (x^{#})^{∗} a^{−1} = a (a x^{∗} a^{−1})^{∗} a^{−1} = a (a^{−1})^{∗} x a^{∗} a^{−1}. (12.7.8)

Suppose for the moment that a^{∗} = t a (12.7.9)

for some nonzero real or complex number t, as appropriate. This implies that

(a^{−1})^{∗} = (a^{∗})^{−1} = t^{−1} a^{−1}. (12.7.10)

It follows that (x^{#})^{#} = x. (12.7.11)

This means that x ↦ x^{#} defines another algebra involution on A when (12.7.9) holds, as in Exercise 27 on p326 of [286].

Note that ϕ_{a}(x)^{∗} = (a^{−1})^{∗} x^{∗} a^{∗} (12.7.12)

278 CHAPTER 12. COMPLEXIFICATIONS, NETS, AND C^{∗} ALGEBRAS for every x ∈ A. Suppose for the moment again that a^{∗} = τa^{−1} (12.7.13) for some nonzero real or complex number τ, as appropriate. This means that (a^{−1})^{∗} = (a^{∗})^{−1} = τ^{−1} a. (12.7.14) Using this, we get that ϕ_{a}(x)^{∗} = ϕ_{a}(x^{∗}) (12.7.15) for every x ∈ A, so that ϕ_{a} is a ∗-automorphism of A. This is related to Exercises (7) and (8) on p50 of [11].

Let ∥ · ∥_{A} be a submultiplicative norm on A with ∥e_{A}∥_{A} = 1, and note that 1 ≤ ∥a∥_{A} ∥a^{−1}∥_{A}. Suppose that ∥a∥_{A} ∥a^{−1}∥_{A} = 1. (12.7.16) If y ∈ A, then ∥y∥_{A} ≤ ∥a^{−1}∥_{A} ∥a y∥_{A} = ∥a∥_{A}^{−1} ∥a y∥_{A}. (12.7.17) This implies that ∥a y∥_{A} = ∥a∥_{A} ∥y∥_{A}, (12.7.18) because of submultiplicativity of ∥ · ∥_{A}. Similarly, ∥y a∥_{A} = ∥a∥_{A} ∥y∥_{A} (12.7.19) and ∥a^{−1} y∥_{A} = ∥y a^{−1}∥_{A} = ∥a∥_{A}^{−1} ∥y∥_{A}. (12.7.20)

It follows that ∥ϕ_{a}(y)∥_{A} = ∥a y a^{−1}∥_{A} = ∥y∥_{A} (12.7.21) under these conditions. This implies that ∥x^{#}∥_{A} = ∥x^{∗}∥_{A} (12.7.22) for every x ∈ A. In particular, if x ↦ x^{∗} preserves the norm, then x ↦ x^{#} has the same property.

12.8 Self-adjoint elements

Let A be an associative algebra over the complex numbers with a nonzero multiplicative identity element e_{A} and a conjugate-linear algebra involution x ↦ x^{∗}. Suppose that a ∈ A is self-adjoint with respect to this involution, so that a^{∗} = a. It would be nice if σ_{A}(a) ⊆ R. (12.8.1)

Remember that this holds when A is the algebra of bounded linear mappings from a complex Hilbert space into itself, with the involution defined by the adjoint, as in Section 8.11.

Similarly, let h be a nonzero algebra homomorphism from A into C, so that h(e_{A}) = 1. It would be nice if h(a) ∈ R. (12.8.2)

Remember that h(a) ∈ σ_{A}(a), as in Section 6.9, so that (12.8.1) implies (12.8.2). If A is a commutative Banach algebra, then every element of σ_{A}(a) is of the form h(a) for some nonzero homomorphism h from A into C, as in Section 6.12. This means that (12.8.1) would follow from (12.8.2) in this case.

Let ∥ · ∥_{A} be a submultiplicative norm on A that satisfies the C^{∗} identity ∥x^{∗} x∥_{A} = ∥x∥_{A}^{2} for every x ∈ A. Let us show that (12.8.2) holds in this case, using the argument on p277 of [286]. Let α, β be the real and imaginary parts of h(a), so that we would like to get that β = 0. Also let t ∈ R be given, and put z = a + i t e_{A}. (12.8.3)

Thus h(z) = h(a) + i t = α + i (β + t) (12.8.4)

and z^{∗} z = a^{∗} a + t^{2} e_{A} = a^{2} + t^{2} e_{A}. (12.8.5)

Observe that α^{2} + (β + t)^{2} = |h(z)|^{2} ≤ ∥z∥_{A}^{2}, (12.8.6)

where the second step is as in Section 6.9. Using the C^{∗} identity, we get that

∥z∥_{A}^{2} = ∥z^{∗} z∥_{A} ≤ ∥a^{∗} a∥_{A} + t^{2} = ∥a∥_{A}^{2} + t^{2}. (12.8.7)

Combining (12.8.6) and (12.8.7), we obtain that

α^{2} + β^{2} + 2 β t + t^{2} ≤ ∥a∥_{A}^{2} + t^{2}, (12.8.8)

so that α^{2} + 2 β t ≤ ∥a∥_{A}^{2}. (12.8.9)

This implies that β = 0, because t ∈ R is arbitrary.

Alternatively, suppose that A is a C^{∗} algebra, and let t ∈ R be given again. Remember that exp_{A}(i t a) may be defined as in Section 10.4, and satisfies

(12.8.10) (exp_{A}(i t a))^{∗} = exp_{A}(−i t a^{∗}) = exp_{A}(−i t a) = (exp_{A}(i t a))^{−1}.

This implies that ∥ exp_{A}(i t a)∥_{A} = 1, (12.8.11)

by the C^{∗} identity. We also have that

h(exp_{A}(i t a)) = exp(i t h(a)), (12.8.12) where the right side uses the usual complex exponential function. This is because h is a bounded linear functional on A, as well as an algebra homomorphism, as in Section 6.9.

Using the boundedness of h as a linear functional on A again, we get that

| exp(i t h(a))| = |h(exp_{A}(i t a))| ≤ 1. (12.8.13)

This implies (12.8.2), because t ∈ R is arbitrary. This is the argument used on p48 of [11].

If A is a commutative C^{∗} algebra, then it follows that (12.8.1) holds. If A is a C^{∗} algebra that is not necessarily commutative, then observe that the subalgebra A_{0} of A generated by a and e_{A} is commutative and invariant under the involution. This means that the closure \overline{A_{0}} of A_{0} in A with respect to the metric associated to ∥ · ∥_{A} is a commutative C^{∗} algebra. It follows that

σ\overline{A_{0}}(a) ⊆ R, (12.8.14)

as before. Remember that σ_{A}(a) is contained in σ\overline{A_{0}}(a), because \overline{A_{0}} is a subalgebra of A, as in Section 7.3. Thus (12.8.1) follows from (12.8.14). This corresponds to Corollary 1 on p48 of [11], and to part (a) of Theorem 11.28 on p282 of [286].

As another approach, let t ∈ R be given again, and put f_{t}(z) = exp(i t z) on C, which can be defined by a convergent power series using the power series for the exponential function on C. Note that

f_{t}(σ_{A}(a)) ⊆ σ_{A}(exp_{A}(i t a)), (12.8.15)

as in Section 10.3. The right side is contained in the closed unit disk in C, because of (12.8.11), as in Section 6.8. This implies (12.8.1), because t ∈ R is arbitrary. This is a variant of the proof of the first part of part (iv) of 2.1 on p262 of [291]. In [291], one takes t = 1, and observes more precisely that the spectrum of exp_{A}(i a) is contained in the unit circle. This permits one to obtain (12.8.1) from (12.8.15) with t = 1.

12.9 Some maximal ideals

Let X be a nonempty metric or topological space, and note that the spaces C_{b}(X, R), C_{b}(X, C) of bounded continuous real and complex-valued functions on X, respectively, are Banach algebras with respect to the supremum norm. If x ∈ X, then h_{x}(f) = f(x) (12.9.1)

defines an algebra homomorphism from C_{b}(X, R), C_{b}(X, C) onto R, C, respectively, as in Section 6.9. Thus the kernel of h_{x} is a maximal proper ideal in each of C_{b}(X, R) and C_{b}(X, C), as in Section 6.12.

Remember that X is said to be a Urysohn space if continuous real-valued functions on X separate points in X, as in Section 5.5. More precisely, this implies that bounded continuous real-valued functions on X separate points in X, as before. This means that if x, y ∈ X and x ≠ y, then

h_{x}(f) ≠ h_{y}(f) (12.9.2)

for some f ∈ C_{b}(X, R). Of course, metric spaces are Urysohn spaces, as before.

Suppose that X is compact, so that C_{b}(X, R), C_{b}(X, C) are the same as the algebras C(X, R), C(X, C) of continuous real and complex-valued functions on X, respectively. If I is a maximal proper ideal in C(X, R) or C(X, C), then it is well known that I = ker h_{x} (12.9.3)

for some x ∈ X. To see this, it suffices to show that

I ⊆ ker h_{x} (12.9.4)

for some x ∈ X, because I is supposed to be a maximal proper ideal.

Suppose for the sake of a contradiction that for each x ∈ X, (12.9.4) does not hold. This means that for every x ∈ X there is an f_{x} ∈ I such that

f_{x}(x) ≠ 0. (12.9.5)

We may as well suppose that f_{x} is a nonnegative real-valued continuous function on X for each x ∈ X, by replacing f_{x} with f_{x}^{2} in the real case, and with |f_{x}|^{2} = f_{x} \overline{f_{x}} in the complex case, if necessary.

If x ∈ X, then f_{x}^{−1}((0, +∞)) is an open subset of X that contains x. If X is compact, then there are finitely many elements x_{1}, …, x_{n} of X such that

X ⊆ ⋃_{j=1}^{n} f_{x_{j}}^{−1}((0, +∞)). (12.9.6)

This means that

f = ∑_{j=1}^{n} f_{x_{j}} (12.9.7)

is strictly positive at every point in X. Note that f ∈ I, because f_{x_{j}} ∈ I for each j. This is a contradiction, because I is supposed to be a proper ideal in C(X, R) or C(X, C), and 1/f is continuous on X.

If h is any nonzero algebra homomorphism from C(X, R) or C(X, C) into R or C, as appropriate, then the kernel of h is a maximal proper ideal in C_{b}(X, R) or C(X, C), as appropriate, as in Section 6.12. This implies that

ker h = ker h_{x} (12.9.8)

for some x ∈ X, as before. One can use this to get that

h = h_{x}, (12.9.9)

because h(1_{X}) = 1, as in Section 6.9.

This is essentially the same argument as used in Example 11.13 (a) on p271 of [286]. Another argument is used in the proof of Theorem 1.10.4 on p28 of [11].

If X is a compact Hausdorff topological space, then it is well known that X is normal. In particular, this means that X is a Urysohn space, by Urysohn’s lemma. This implies that there is only one x ∈ X such that (12.9.9) holds in this case.

12.10 Commutative Banach algebras

Let (A, ∥ · ∥_{A}) be a commutative Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. The set of all nonzero algebra homomorphisms from A into R or C, as appropriate, may be denoted Sp(A), or Sp_{R}(A) or Sp_{C}(A), to indicate whether A is considered as an algebra over R or C. This may be called the Gelfand spectrum of A, at least in the complex case, as on p25 of [11]. This is also known as the maximal ideal space of A in the complex case, because of the correspondence with maximal proper ideals in A, as in Section 6.12.

If h ∈ Sp(A), then h is a bounded linear functional on A, with dual norm less than or equal to 1, as in Section 6.9. More precisely,

∥h∥_{A′} = 1, (12.10.1)

because h(e_{A}) = 1, as before.

If a ∈ A, then let â be the real or complex-valued function, as appropriate, defined on Sp(A) by â(h) = h(a) (12.10.2)

for every h ∈ Sp(A). This is called the Gelfand transform of a, at least in the complex case, as on p26 of [11], and p268 of [286]. Remember that

â(Sp(A)) ⊆ σ_{A}(a), (12.10.3)

as in Section 6.9. In the complex case, we have that

â(Sp(A)) = σ_{A}(a), (12.10.4)

as in Section 6.12. This corresponds to Theorem 1.9.5 on p26 of [11], and to the first part of part (c) of Theorem 11.9 on p268 of [286].

As mentioned in Section 6.9,

(12.10.5) Sp(A) is compact

as a subset of the dual space A′ of bounded linear functionals on A with respect to the weak^{∗} topology. More precisely, Sp(A) is contained in the closed unit ball \overline{B}_{A′}(0, 1) in A′, and \overline{B}_{A′}(0, 1) is compact with respect to the weak^{∗} topology on A′, by the Banach–Alaoglu theorem. To get (12.10.5), it suffices to check that Sp(A) is a closed set in A′ with respect to the weak^{∗} topology, as before.

Let us consider Sp(A) as a topological space, using the topology induced by the weak^{∗} topology on the dual space A′ of bounded linear functionals on A. Equivalently, this is the weakest topology on Sp(A) such that â is continuous for every a ∈ A. It is easy to see that Sp(A) is Hausdorff with respect to this topology, and it is also compact, by (12.10.5). This corresponds to Proposition 1.9.3 on p25 of [11], and to part (a) of Theorem 11.9 on p268 of [286].

Note that Sp(A) ≠ ∅ (12.10.6)

in the complex case, because of (12.10.4). This corresponds to Exercise (1) on p27 of [11]. If A is a Banach algebra over the real numbers, then let us suppose that (12.10.6) holds for the rest of the section. In both cases,

a ↦ â (12.10.7)

defines an algebra homomorphism from A into C(Sp(A), R) or C(Sp(A), C), as appropriate. This may be called the Gelfand map, at least in the complex case, as on p26 of [11], although the term “Gelfand transform” may also be used for this mapping, as on p268 of [286].

The kernel of the Gelfand map (12.10.7) is equal to

⋂_{h∈Sp(A)} ker h. (12.10.8)

In the complex case, this is the same as the intersection of all of the maximal proper ideals in A, which is the Jacobson radical rad(A) of A, as in Section 6.12. This corresponds to part of part (b) of Theorem 11.9 on p268 of [286]. Thus, in the complex case, the Gelfand map is one-to-one if and only if A is semisimple in the sense that rad(A) = {0}.

Observe that ∥â∥_{sup} = ∥â∥_{sup,Sp(A)} ≤ ∥a∥_{A} (12.10.9)

for every a ∈ A, where the left side is the supremum norm of â on Sp(A). This means that the Gelfand map is bounded as a linear mapping from A into C(Sp(A), R) or C(Sp(A), C), as appropriate, with respect to the supremum norm. More precisely, the corresponding operator norm of the Gelfand map is equal to 1, because it sends e_{A} to the constant function equal to 1 on Sp(A). This corresponds to part of Remark 1.9.4 on p26 of [11].

In fact, we have that ∥â∥_{sup} ≤ r_{A}(a) (12.10.10)

for every a ∈ A, where the right side is as in Section 6.13. This follows from (12.10.3), because |λ| ≤ r_{A}(a) (12.10.11)

for every λ ∈ σ_{A}(a), as before. In the complex case, we have that

∥â∥_{sup} = r_{A}(a) (12.10.12)

for every a ∈ A, as in Section 6.14. This corresponds to part of part (c) of Theorem 11.9 on p268 of [286].

12.11 More on the Gelfand map

Let us continue with the same notation and hypotheses as in the previous section. Note that the image Â = {â : a ∈ A} (12.11.1)

of A under the Gelfand map is a subalgebra of C(Sp(A), R) or C(Sp(A), C), as appropriate, as in Remark 1.9.4 on p26 of [11], and part (b) of Theorem 11.9 on p268 of [286]. We also have that Â separates points in Sp(A), by construction, and that Â contains the constant functions on Sp(A).

Consider the condition that there be a positive real number c_{0} such that

c_{0} ∥a∥_{A} ≤ ∥â∥_{sup} (12.11.2)

for every a ∈ A. Of course, this condition implies that the Gelfand map is injective. This condition also implies that Â is a closed set in C(Sp(A), R) or C(Sp(A), C), as appropriate, with respect to the supremum metric, because A is complete, as in Section 7.11. Conversely, if Â is a closed set in C(Sp(A), R) of C(Sp(A), C), as appropriate, then Â is complete with respect to the supremum metric, as in Section 1.6. If the Gelfand map is injective as well, then the open mapping theorem implies that (12.11.2) holds for some c_{0} > 0.

If a ∈ A, then ∥ ˆ(a^{2})∥_{sup} = ∥(â)^{2}∥_{sup} = ∥â∥_{sup}^{2}. (12.11.3)

If (12.11.2) holds for some c_{0} > 0, then we get that

c_{0}^{2} ∥a∥_{A}^{2} ≤ ∥â∥_{sup}^{2} = ∥ ˆ(a^{2})∥_{sup} ≤ ∥a^{2}∥_{A}, (12.11.4)

using (12.10.9) in the third step. This corresponds to part of Lemma 11.11 on p270 of [286]. This is also related to part of part (b) of Theorem 11.12 on p270 of [286].

If (12.11.2) holds with c_{0} = 1, then

∥a∥_{A} = ∥â∥_{sup} (12.11.5)

for every a ∈ A, because of (12.10.9). This implies that

∥a^{2}∥_{A} = ∥a∥_{A}^{2} (12.11.6)

for every a ∈ A. This corresponds to part of Exercise (3) on p27 of [11], and to part of part (a) of Theorem 11.12 on p270 of [286].

Suppose for the moment that

∥a∥_{A}^{2} ≤ C ∥a^{2}∥_{A} (12.11.7)

for some positive real number C and every a ∈ A. This implies that

∥a∥_{A} ≤ C r_{A}(a) (12.11.8) for every a ∈ A, as in Section 11.7. This means that

∥a∥_{A} ≤ C ∥â∥_{sup} (12.11.9)

in the complex case, by (12.10.12). Of course, this is the same as saying that (12.11.2) holds with c_{0} = C^{−1}. This corresponds to parts of Lemma 11.11 and part (b) of Theorem 11.12 on p270 of [286].

In particular, (12.11.6) implies (12.11.5) in the complex case. This corresponds to parts of Exercise (3) on p27 of [11] and part (a) of Theorem 11.12 on p270 of [286].

Suppose from now on in this section that A is a commutative C^{∗} algebra. This implies that r_{A}(a) = ∥a∥_{A} (12.11.10)

for every a ∈ A, as in Section 7.7. It follows that (12.11.5) holds for every a ∈ A, because of (12.10.12). This corresponds to parts of Theorem 2.2.4 on p47 of [11], and Theorem 11.18 on p276 of [286].

If a ∈ A is self-adjoint, then σ_{A}(a) ⊆ R, as in Section 12.8. This means that â is a real-valued function on Sp(A), because of (12.10.4). One can use this to get that ˆ(a^{∗}) = \overline{(â)} (12.11.11)

for every a ∈ A, by expressing a as a_{1} + ia_{2}, where a_{1}, a_{2} ∈ A are self-adjoint. We also obtain that Â = C(Sp(A), C) (12.11.12)

under these conditions, by the Stone–Weierstrass theorem. These statements correspond to additional parts of Theorem 2.2.4 on p47 of [11], and Theorem 11.18 on p276 of [286].

12.12 Bounded continuous functions

Let X be a nonempty metric or topological space, and let us take A = C_{b}(X, R) or C_{b}(X, C), equipped with the supremum norm. Thus A is a commutative Banach algebra over the real or complex numbers with a multiplicative identity element, so that Sp(A) may be defined as in Section 12.10. If x ∈ X, then h_{x}(f) = f(x) defines an element of Sp(A), as in Section 12.9. This means that

x ↦ h_{x} (12.12.1)

defines a mapping from X into Sp(A). This mapping is injective exactly when X is a Urysohn space, as before.

If f ∈ A, then f̂(h) = h(f) defines a continuous real or complex-valued function on Sp(A), as appropriate, as in Section 12.10. Note that

f̂(h_{x}) = h_{x}(f) = f(x) (12.12.2) is continuous as a real or complex-valued function of x ∈ X. One can use this to check that (12.12.1) is continuous as a mapping from X into Sp(A), with respect to the topology defined on Sp(A) previously.

If X is completely regular in the strict sense, then one can check that the topology on X is the weakest topology with respect to which the elements of A are all continuous. This means that the topology on X is the weakest topology with respect to which (12.12.1) is continuous as a mapping into Sp(A). If X is completely regular in the strong sense, then (12.12.1) is a homeomorphism from X onto its image in Sp(A), with respect to the topology induced on the image by the topology defined on Sp(A) in Section 12.10.

If f ∈ A, then σ_{A}(f) = \overline{f(X)}, (12.12.3)

where the right side is the closure of f(X) in R or C, as appropriate. This is analogous to the corresponding statement for arbitrary bounded functions on X, mentioned in Section 6.8. If h ∈ Sp(A), then it follows that

h(f) ∈ \overline{f(X)}, (12.12.4)

because h(f) ∈ σ_{A}(f), as in Section 6.9.

Suppose for the moment that A = C_{b}(X, C). If f ∈ C_{b}(X, R), then

σ_{A}(f) ⊆ R, (12.12.5)

by (12.12.3). In particular, if h ∈ Sp(A), then

h(f) ∈ R, (12.12.6)

by (12.12.4). This implies that

h(\overline{a}) = \overline{h(a)} (12.12.7)

for every a ∈ C_{b}(X, C).

In both the real and complex cases, we would like to check that the image

{h_{x} : x ∈ X} (12.12.8)

of X in Sp(A) under the mapping (12.12.1) is dense in Sp(A). Suppose for the sake of a contradiction that there is an element h of Sp(A) that is not in the closure of (12.12.8) in Sp(A). This means that there is an open set U ⊆ Sp(A) such that h ∈ U and h_{x} ∉ U (12.12.9)

for every x ∈ X.

Because of the way that the topology on Sp(A) is defined, there are finitely many elements f_{1}, …, f_{n} of A and positive real numbers r_{1}, …, r_{n} such that

(12.12.10) {ϕ ∈ Sp(A) : |ϕ(f_{j}) − h(f_{j})| < r_{j} for each j = 1, …, n} ⊆ U.

12.13. SOME REMARKS ABOUT WEAK TOPOLOGIES 287

We can reduce to the case where h(f_{j}) = 0 (12.12.11) for each j = 1, …, n, by subtracting the constant function on X equal to h(f_{j}) from f_{j} if necessary. Using this, (12.12.10) reduces to (12.12.12) {ϕ ∈ Sp(A) : |ϕ(f_{j})| < r_{j} for each j = 1, …, n} ⊆ U. If x ∈ X, then (12.12.9) implies that |f_{j}(x)| = |h_{x}(f_{j})| ≥ r_{j} (12.12.13) for some j.

Put

g = ∑_{j=1}^{n} f_{j}^{2} (12.12.14)

in the real case, and

g = ∑_{j=1}^{n} |f_{j}|^{2} = ∑_{j=1}^{n} f_{j} \overline{f_{j}} (12.12.15)

in the complex case. In both cases, we have that h(g) = 0, (12.12.16) because of (12.12.11). We also have that

g(x) = ∑_{j=1}^{n} |f_{j}(x)|^{2} ≥ min(r_{1}^{2}, …, r_{n}^{2}) (12.12.17)

for every x ∈ X, because of (12.12.13). This implies that g is invertible in A, contradicting (12.12.16).

12.13 Some remarks about weak topologies

Let X, I be nonempty sets, and suppose that f_{j} is a mapping from X into a metric or topological space Y_{j} for each j ∈ I. Under these conditions, it is well known there is a weakest topology on X such that (12.13.1) f_{j} is continuous for each j ∈ I. This is known as the weak topology on X associated to the family of f_{j}’s, j ∈ I.

Let

Y = ∏_{j∈I} Y_{j} (12.13.2)

be the Cartesian product of the Y_{j}’s, j ∈ I, equipped with the corresponding product topology. Using the f_{j}’s, we get a mapping F from X into Y, whose jth coordinate is equal to f_{j} for each j ∈ I. The weak topology on X associated to the f_{j}’s as in the preceding paragraph is the same as the weakest topology on X such that (12.13.3) F is continuous.

Suppose for the moment that the family of f_{j}’s, j ∈ I, separates points in X, so that (12.13.4) F is one-to-one.

In this case,

(12.13.5) F is a homeomorphism from X onto its image in Y,

with respect to the corresponding weak topology on X, and the topology induced on F (X) by the product topology on Y.

Suppose for the moment again that

(12.13.6) I has only finitely or countably many elements,

and that

(12.13.7) the topology on Y_{j} is determined by a metric for each j ∈ I.

It is well known that

(12.13.8) the product topology on Y is determined by a metric

as well under these conditions. If the family of f_{j}’s, j ∈ I, separates points in X, then it follows that

(12.13.9) the corresponding weak topology on X is determined by a metric

too.

Suppose for the moment that for each j ∈ I, there is a base B_{j} for the topology of Y_{j} with only finitely or countably many elements. If (12.13.6) holds, then it is well known that there is a base B_{Y} for the product topology on Y with only finitely or countably many elements. Similarly, there is a base B_{X} for the weak topology on X associated the family of mappings f_{j}, j ∈ I with only finitely or countably many elements under these conditions.

Let X_{0} be a subset of X. One can check that the topology induced on X_{0} by the weak topology on X associated to the family of mappings f_{j}, j ∈ I, is the same as the weak topology on X_{0} associated to the restrictions of the f_{j}’s to X_{0}, j ∈ I.

Sometimes we may be concerned with the weak topology on X determined by a vector space Z of real or complex-valued functions on X. The same topology is determined by any subset of Z whose linear span is Z. Similarly, if Z is an algebra with respect to pointwise multiplication of functions, then the same topology on X is determined by any subset of Z that generates Z as an algebra over the real or complex numbers, as appropriate.

12.14. SOME REMARKS ABOUT SP(A) 289

Let V be a vector space over the real or complex numbers, with a norm ∥ · ∥_{V}. Remember that the dual space V′ of bounded linear functionals on V with respect to ∥ · ∥_{V} is a vector space over R or C, as appropriate, with a corresponding dual norm ∥ · ∥_{V′}, as in Section 3.1. The weak^{∗} topology on V′ is the weak topology corresponding to the linear functions

λ ↦ λ(v), (12.13.10)

v ∈ V, as in Section 4.5. One can get the same topology on V′ using the linear functions (12.13.10) associated to a subset of V whose linear span is equal to V, as in the preceding paragraph.

Let E be a subset of V′. The topology induced on E by the weak^{∗} topology on V′ is the same as the weak topology on E associated to the restrictions of the functions (12.13.10) to E, as before. If E is bounded with respect to ∥ · ∥_{V′}, then one can check that this is the same as the weak topology associated to the restrictions of the functions (12.13.10) to E, with v in a dense subset of V. More precisely, it suffices to use a set of v’s whose linear span is dense in V.

Suppose that V is separable with respect to the metric associated to ∥ · ∥_{V}. If E is bounded with respect to ∥·∥_{V′}, then it follows that the topology induced on E by the weak^{∗} topology on V′ is the same as the weak topology associated to the restrictions of the functions (12.13.10) to E corresponding to finitely or countably many v ∈ V. This implies that there is a metric on E that determines the same topology, as before.

12.14 Some remarks about Sp(A)

Let A be a commutative associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}. As in Section 12.10, we let Sp(A) be the set of all nonzero algebra homomorphisms α from A into R or C, as appropriate, and we may use Sp_{R}(A) or Sp_{C}(A) to indicate whether A is considered as an elgebra over R or C. Remember that

α(e_{A}) = 1, (12.14.1)

because α ≠ 0, as in Section 6.9.

If a ∈ A, then we let â be the real or complex-valued function defined on Sp(A) by â(α) = α(a), (12.14.2)

as in Section 12.10. Remember that â maps Sp(A) into σ_{A}(a), as in Section 6.9.

We may consider Sp(A) as a topological space, using the weakest topology with respect to which â is continuous for each a ∈ A, as before. Note that the collection Â of the functions â, a ∈ A, on Sp(A) separates points in Sp(A), by construction. This implies that Sp(A) is Hausdorff with respect to this weak topology. If Sp(A) ≠ ∅, then a ↦ â defines an algebra homomorphism from A into C(Sp(A), R) or C(Sp(A), C), as before.

Let B be another commutative associative algebra over the real or complex numbers, as appropriate, with a nonzero multiplicative identity element e_{B}. Also let ϕ be an algebra homomorphism from A into B with ϕ(e_{A}) = e_{B}. If β ∈ Sp(B), then it is easy to see that

ϕ̂(β) = β ∘ ϕ (12.14.3)

is an element of Sp(A). In particular, note that

(ϕ̂(β))(e_{A}) = β(ϕ(e_{A})) = β(e_{B}) = 1. (12.14.4)

Let a ∈ A be given, so that ϕ(a) ∈ B, and ˆ (ϕ(a)) is a real or complex-valued function on Sp(B), as appropriate. If β ∈ Sp(B), then

ˆ (ϕ(a))(β) = β(ϕ(a)) = (ϕ̂(β))(a) = â(ϕ̂(β)). (12.14.5)

This means that

ˆ (ϕ(a)) = â ∘ ϕ̂. (12.14.6)

One can use this to get that

(12.14.7) ϕ̂ is continuous as a mapping from Sp(B) into Sp(A).

This uses the fact that ˆ (ϕ(a)) is continuous on Sp(B), by construction.

Let C be a third commutative associative algebra over the real or complex numbers, as appropriate, with a nonzero multiplicative identity element e_{C}, and let ψ be an algebra homomorphism from B into C with ψ(e_{B}) = e_{C}. Thus ψ ∘ ϕ is an algebra homomorphism from A into C that sends e_{A} to e_{C}. It is easy to see that

ˆ (ψ ∘ ϕ) = ϕ̂ ∘ ψ̂, (12.14.8)

as mappings from Sp(C) into Sp(A).

Of course, the identity mapping on A is an algebra homomorphism, for which the induced mapping on Sp(A) is the identity mapping. If ϕ is an algebra isomorphism from A onto B, then ϕ^{−1} is an algebra homomorphism from B into A, which induces a continuous mapping from Sp(A) into Sp(B), as before. Under these conditions,

(12.14.9)  ϕ̂ is a homeomorphism from Sp(B) onto Sp(A),

with

ϕ̂^{−1} = ˆ (ϕ^{−1}). (12.14.10)

This uses the remarks in the preceding paragraph.

12.15. SOME MORE REMARKS ABOUT SP(A) 291

12.15 Some more remarks about Sp(A)

Let us continue with the same notation and hypotheses as in the previous section. Note that (12.15.1) ker ϕ ⊆ ker β ∘ ϕ = ker ϕ̂(β)

for every β ∈ Sp(B).

If ϕ(A) = B, (12.15.2)

then (12.15.3) ϕ̂ is one-to-one as a mapping from Sp(B) into Sp(A).

More precisely, if β_{1}, β_{2} ∈ Sp(B) and

ϕ̂(β_{1}) = ϕ̂(β_{2}), (12.15.4)

then β_{1} = β_{2} on ϕ(A), and thus on B. In this case,

(12.15.5)  ϕ̂(Sp(B)) = {α ∈ Sp(A) : ker ϕ ⊆ ker α}.

In fact,

(12.15.6) ϕ̂ is a homeomorphism from Sp(B) onto its image in Sp(A),

with respect to the induced topology on ϕ̂(Sp(B)) when (12.15.2) holds. This is because the induced topology on ϕ̂(Sp(B)) is the same as the weak topology associated to the restrictions of the functions â, a ∈ A, to ϕ̂(Sp(B)), as in Section 12.13. The restrictions of these functions to ϕ̂(Sp(B)) correspond exactly to the functions on Sp(B) of the form (12.14.6), with a ∈ A. These are the same as the functions b̂, b ∈ B, used to define the usual topology on Sp(B) when (12.15.2) holds.

Suppose now that (B, ∥ · ∥_{B}) is a Banach algebra, so that every h ∈ Sp(B) is a bounded linear functional on B, as in Section 6.9. Suppose also that

(12.15.7) ϕ(A) is dense in B,

with respect to the metric associated to ∥ · ∥_{B}. One can verify that (12.15.3) holds in this case too. Indeed, if β_{1}, β_{2} ∈ Sp(B) satisfy (12.15.4), then β_{1} = β_{2} on ϕ(A), which implies that they are equal to B, because h_{1}, h_{2} are continuous on B.

One can check that (12.15.7) implies (12.15.6) in this case as well. One can start in the same way as before, to get that ϕ̂ is a homeomorphism from Sp(B) onto its image in Sp(A), when Sp(B) is equipped with the weak topology associated to the family of functions of the form ˆ (ϕ(a)), a ∈ A. Thus one would like to verify that this is the same as the usual topology on Sp(B) under these conditions. This corresponds to a remark in Section 12.13, because Sp(B) is bounded with respect to the dual norm on B′, as in Section 6.9.

If α = ϕ̂(β) = β ∘ ϕ for some β ∈ Sp(B), then

|α(a)| ≤ ∥ϕ(a)∥_{B} (12.15.8)

for every a ∈ A, because the dual norm of β on B is less than or equal to 1, as in Section 6.9. Note that this includes the condition that

ker ϕ ⊆ ker α, (12.15.9)

as in (12.15.1). Of course, (12.15.8) does not use (12.15.7).

If (12.15.7) holds, then

(12.15.10)  ϕ̂(Sp(B)) = {α ∈ Sp(A) : (12.15.8) holds}.

To see this, suppose that α ∈ Sp(A) satisfies (12.15.8), and let us define β initially on ϕ(A) by β(ϕ(a)) = α(a) (12.15.11)

for each a ∈ A. Clearly β is well defined on ϕ(A), because of (12.15.9). There is a unique extension of β to a bounded linear functional on B, because of (12.15.7) and (12.15.8), as in Section 2.2. One can check that this extension defines an element of Sp(B) under these conditions.

Algebras, polynomials, and Sp(A)

13.1 Some remarks about continuous functions

Let X be a nonempty metric or topological space, so that C(X, R), C(X, C) are commutative associative algebras over the real and complex numbers, respectively, with a nonzero multiplicative identity element. If x ∈ X, then h_{x}(f) = f(x) defines an algebra homomorphism from each of C(X, R), C(X, C) onto R, C, respectively, as before. Thus

x ↦ h_{x} (13.1.1)

defines a mapping from X into each of Sp(C(X, R)), Sp(C(X, C)). This mapping is injective exactly when X is a Urysohn space, as in Sections 12.9 and 12.12.

If f ∈ C(X, R) or C(X, C), then f̂(h) = h(f) defines a continuous real or complex-valued function on Sp(C(X, R)) or Sp(C(X, C)), as appropriate, with respect to the topology mentioned in Section 12.14. Of course,

f̂(h_{x}) = h_{x}(f) = f(x) (13.1.2)

is continuous as a real or complex-valued function of x ∈ X, as appropriate. One can use this to get that (13.1.1) is continuous as a mapping from X into each of Sp(C(X, R)), Sp(C(X, C)), as in Section 12.12.

If X is completely regular in the strict sense, then the topology on X is the weakest topology with respect to which the elements of C(X, R) or C(X, C) are continuous, as in Section 12.12. This implies that the topology on X is the weakest with respect to which (13.1.1) is continuous as a mapping from X into Sp(C(X, R)) or Sp(C(X, C)), as before. If X is regular in the strong sense, then (13.1.1) is a homeomorphism from X onto its image in each of Sp(C(X, R)), Sp(C(X, C)), with respect to the induced topology on the image, as before.

294 CHAPTER 13. ALGEBRAS, POLYNOMIALS, AND SP(A)

If A = C(X, R) or C(X, C) and f ∈ A, then

σ_{A}(f) = f(X), (13.1.3)

as in Section 8.15. If h ∈ Sp(A), then we get that

h(f) ∈ f(X), (13.1.4)

because h(f) ∈ σ_{A}(f), as in Section 6.9. If A = C(X, C) and f ∈ C(X, R), then it follows that h(f) ∈ R. (13.1.5)

One can use this to get that h(\overline{a}) = \overline{h(a)} (13.1.6)

for every a ∈ C(X, C).

Suppose that h, h′ are both elements of Sp(C(X, R)) or of Sp(C(X, C)), and that h(f) = h′(f) (13.1.7)

for every f ∈ C_{b}(X, R) or C_{b}(X, C), as appropriate. Let f ∈ C(X, R) or C(X, C) be given, as appropriate, and let us check that (13.1.7) also holds in this case. If ϵ is any positive real number, then

f_{ϵ} = \frac{f}{1 + ϵ |f|^{2}} (13.1.8)

is bounded and continuous on X, so that

h(f_{ϵ}) = h′(f_{ϵ}), (13.1.9)

by hypothesis. It is easy to see that

h(f_{ϵ}) = \frac{h(f)}{1 + ϵ |h(f)|^{2}} , (13.1.10)

and similarly for h′, using (13.1.6) in the complex case. One can get (13.1.7) by taking the limit as ϵ → 0 of both sides of (13.1.9).

Let ϕ be the obvious inclusion mapping from C_{b}(X, R) or C_{b}(X, C) into C(X, R) or C(X, C), as appropriate. In both cases, ϕ is an algebra homomorphism which is compatible with the multiplicative identity elements. This leads to a continuous mapping ϕ̂ from Sp(C(X, R)) or Sp(C(X, C)) into Sp(C_{b}(X, R)) or Sp(C_{b}(X, C)), as appropriate, as in Section 12.14. Equivalently, ϕ̂ sends an element h of Sp(C(X, R)) or Sp(C(X, C)) to its restriction to C_{b}(X, R) or C_{b}(X, C), as appropriate. Note that ϕ̂ is one-to-one in both cases, as in the preceding paragraph.

Let Y be another nonempty metric or topological space, and let θ be a continuous mapping from X into Y. Observe that

Θ(f) = f ∘ θ (13.1.11) defines an algebra homomorphism from each of C(Y, R), C(Y, C) into C(X, R), C(X, C), respectively. Of course, Θ sends constant functions on Y to constant functions on X, with the same constant value. Using Θ, we get a continuous mapping Θ̂ from each of Sp(C(X, R)), Sp(C(X, C)) into Sp(C(Y, R)), Sp(C(Y, C)), respectively, as in Section 12.14. If x ∈ X, then Θ̂ sends the homomorphism associated to evaluation at x to the analogous homomorphism associated to evaluation to θ(x) ∈ Y.

If f ∈ C_{b}(Y, R) or C_{b}(Y, C), then Θ(f) ∈ C_{b}(X, R) or C_{b}(X, C), as appropriate, with ∥Θ(f)∥_{sup,X} ≤ ∥f∥_{sup,Y} . (13.1.12)

If θ(X) is dense in Y, then

∥Θ(f)∥_{sup,X} = ∥f∥_{sup,Y} . (13.1.13)

Let Θ_{b} be the restriction of Θ to C_{b}(Y, R) or C_{b}(Y, C), considered as an algebra homomorphism into C_{b}(X, R) or C_{b}(X, C), as appropriate. This leads to a continuous mapping Θ̂_{b} from Sp(C_{b}(X, R)) or Sp(C_{b}(X, C)) into Sp(C_{b}(Y, R)) or Sp(C_{b}(Y, C)), as appropriate.

Let ψ be the obvious inclusion mapping from C_{b}(Y, R) or C_{b}(Y, C) into C(Y, R) or C(Y, C), as appropriate, and let ϕ be the analogous inclusion mapping for X, as before. Thus

Θ ∘ ϕ = ψ ∘ Θ_{b}, (13.1.14)

by construction. This implies that

ϕ̂ ∘ Θ̂ = Θ̂_{b} ∘ ψ̂, (13.1.15)

as in Section 12.14. Here ϕ̂ is as before, and ψ̂ is the analogue for ψ.

13.2 Some conditions related to compactness

Let X be a topological space. A point p ∈ X is said to be a limit point of a set E ⊆ X if for every open set U ⊆ X with p ∈ U, there is a q ∈ E ∩ U such that q ≠ p. Let us say that p is a strong limit point of E in X if for every open set U ⊆ X with p ∈ U, we have that E ∩ U has infinitely many elements. Thus strong limit points of E in X are automatically limit points of E in X. If X satisfies the first separation condition, then one can check that limit points of E in X are strong limit points of E in X.

We say that E has the limit point property if every infinite subset of E has a limit point in E. Let us say that E has the strong limit point property if every infinite subset of E has a strong limit point in E. This implies that E has the limit point property, and the converse holds when X satisfies the first separation condition. If E is compact, then one can show that E has the strong limit point property, using a standard argument. If E has the limit point property, and the topology on X is determined by a metric, then it is well known that E is compact.

We say that E is countably compact if every covering of E by countably many open subsets of X can be reduced to a finite subcovering. Similarly, we say that E has the Lindelöf propery if every open covering of E in X can be reduced to a subcovering with only finitely or countably many elements. Thus E is compact if and only if E is countably compact and E has the Lindelöf property. It is well known that E is countably compact if and only if E has the strong limit point property.

A collection B of open subsets of X is said to be a base for the topology of X if every open set in X can be expressed as a union of elements of B. If there is a base B for the topology of X with only finitely or countably many elements, then Lindelöf’s theorem implies that every subset of X has the Lindelöf property.

We say that X is separable if there is a dense set in X with only finitely or countably many elements. If there is a base for the topology of X with only finitely or countably many elements, then one can check that X is separable. It is well known that the converse holds when the topology on X is determined by a metric.

Let f be a continuous mapping from X into another topological space Y. If K ⊆ X is compact, then it is well known that f(K) is compact in Y. One can verify that the analogous statements for countable compactness and the Lindelöf property hold as well, using essentially the same argument.

We say that X is pseudocompact if every continuous real-valued function on X is bounded on X. Of course, if X is compact, then X is pseudocompact. More precisely, one can check that X is pseudocompact when X is countably compact. The converse holds when X is normal in the strong sense, as on p20 of [312].

13.3 Polynomials in n variables

Let n be a positive integer. A multi-index is an n-tuple α = (α_{1}, …, α_{n}) of nonnegative integers. In this case, it is convenient to put

|α| = ∑_{j=1}^{n} α_{j}. (13.3.1)

Note that the set of all multi-indices is the same as the set (Z_{+} ∪ {0})^{n} of all n-tuples of elements of Z_{+} ∪ {0}.

Let T_{1}, …, T_{n} be commuting indeterminates. We shall normally try to use upper-case letters for indeterminates, and lower-case letters for elements of R, C, or other associative algebras, as in [62, 131]. A formal polynomial in T_{1}, …, T_{n} with real or complex coefficients may be expressed as

p(T) = p(T_{1}, …, T_{n}) = ∑_{|α|≤N} c_{α} T^{α}, (13.3.2)

13.3. POLYNOMIALS IN N VARIABLES 297

where N is a nonnegative integer, and the sum is taken over all multi-indices α with |α| ≤ N. Of course, c_{α} should be an element of R or C, as appropriate, for each such α. If α is any multi-index, then

T^{α} = T_{1}^{α_{1}} ⋯ T_{n}^{α_{n}} (13.3.3)

is the corresponding formal monomial in T_{1}, …, T_{n}.

The spaces of all formal polynomials in T_{1}, …, T_{n} with real or complex coefficients may be denoted

R[T_{1}, …, T_{n}], C[T_{1}, …, T_{n}], (13.3.4)

respectively. The coefficients c_{α} of a formal polynomial as in (13.3.2) should be considered as being defined for all multi-indices α, with c_{α} = 0 when |α| is strictly larger then N. This means that

α ↦ c_{α} (13.3.5)

defines a real or complex-valued function, as appropriate, on (Z_{+} ∪ {0})^{n}, with finite support. Thus the spaces (13.3.4) of formal polynomials may be defined more precisely as the spaces

c_{00}((Z_{+} ∪ {0})^{n}, R), c_{00}((Z_{+} ∪ {0})^{n}, C) (13.3.6)

of all real or complex-valued functions on (Z_{+}∪{0})^{n}, as appropriate, with finite support. These are vector spaces over the real and complex numbers, respectively, with respect to pointwise addition and scalar multiplication of functions on (Z_{+} ∪ {0})^{n}, which corresponds to termwise addition and scalar multiplication of formal polynomials as in (13.3.2).

If α, β are multi-indices, then α+β is the multi-index defined by coordinatewise addition, as usual. Similarly, multiplication of the corresponding formal monomials is defined by T^{α} T^{β} = T^{α+β}. (13.3.7)

This can be extended to a bilinear operation of multiplication of formal polynomials, which is commutative and associative. This means that the spaces (13.3.4) of formal polynomials are commutative associative algebras over the real and complex numbers, as appropriate, with respect to this definition of multiplication. The “constant” polynomial for which the coefficient of T^{α} is equal to 1 when α = 0 and to 0 otherwise is the multiplicative identity element in each of these algebras.

Of course, formal polynomials like these determine polynomial functions on R^{n} or C^{n}, as appropriate, in the usual way, and an extension of this will be discussed in the next section. Note that the coefficients of such a polynomial are determined by the derivatives of the corresponding polynomial function at

13.4 Polynomials and homomorphisms

Let us continue with the same notation and hypotheses as in the previous section. Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}. Also let a = (a_{1}, …, a_{n}) be an n-tuple of commuting elements of A, so that

a_{j} a_{l} = a_{l} a_{j} (13.4.1)

for all j, l = 1, …, n. If α is a multi-index, then put

a^{α} = a_{1}^{α_{1}} ⋯ a_{n}^{α_{n}}, (13.4.2)

where a_{j}^{α_{j}} is interpreted as being equal to e_{A} when α_{j} = 0, as usual. Note that

a^{α+β} = a^{α} a^{β} (13.4.3)

for all multi-indices α, β.

Let p(T) be a formal polynomial in T_{1}, …, T_{n} with real or complex coefficients, as appropriate, as in (13.3.2). Put

p(a) = p(a_{1}, …, a_{n}) = ∑_{|α|≤N} c_{α} a^{α}, (13.4.4)

where the sum is taken over all multi-indices α with |α| ≤ N, as before. This defines p(a) as an element of A. This is essentially the same as in Section 8.13 when n = 1. We may also use p_{A}(a) = p_{A}(a_{1}, …, a_{n}) for (13.4.4), to indicate the role of A.

It is easy to see that p(T) ↦ p(a) (13.4.5) defines an algebra homomorphism from R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], as appropriate, into A. Note that (13.4.5) sends the multiplicative identity element in R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], as appropriate, to e_{A}. Similarly, (13.4.5) sends T_{j} to a_{j} for each j = 1, …, n. We also have that (13.4.5) is uniquely determined by these properties.

Let B be another associative algebra over the real or complex numbers, as appropriate, and with a multiplicative identity element e_{B}, and let ϕ be an algebra homomorphism from A into B with ϕ(e_{A}) = e_{B}. Put b_{j} = ϕ(a_{j}) for each j = 1, …, n, so that b = (b_{1}, …, b_{n}) is a commuting n-tuple of elements of B. Observe that ϕ(a^{α}) = b^{α} (13.4.6) for every multi-index α. This implies that

ϕ(p_{A}(a_{1}, …, a_{n})) = p_{B}(ϕ(a_{1}), …, ϕ(a_{n})). (13.4.7)

Let X be a nonempty set, and suppose that A is a subalgebra of the algebra of all real or complex-valued functions on X, as appropriate. It is easy to see that (p_{A}(a_{1}, …, a_{n}))(x) = p(a_{1}(x), …, a_{n}(x)) (13.4.8) for every x ∈ X.

13.5. MORE ON POLYNOMIALS, HOMOMORPHISMS 299

13.5 More on polynomials, homomorphisms

Let n be a positive integer, and let T_{1}, ….T_{n} be n commuting indeterminates again. If w ∈ R^{n} or C^{n}, then put h_{w}(p(T)) = p(w) (13.5.1) for every p(T) ∈ R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], as appropriate. This defines an algebra homomorphism from R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}] into R or C, as appropriate, as in the previous section. Note that h_{w}(T_{j}) = w_{j} (13.5.2) for each j = 1, …, n, as before.

Using this, we get a mapping w ↦ h_{w} (13.5.3) from each of R^{n}, C^{n} into Sp(R[T_{1}, …, T_{n}]), Sp(C[T_{1}, …, T_{n}]), respectively. It is easy to see that (13.5.4) (13.5.3) is one-to-one, because of (13.5.2). We also have that (13.5.5) (13.5.3) maps R^{n}, C^{n} onto Sp(R[T_{1}, …, T_{n}]), Sp(C[T_{1}, …, T_{n}]), respectively, because any element of Sp(R[T_{1}, …, T_{n}]) or Sp(C[T_{1}, …, T_{n}]) is uniquely determined by its values at T_{1}, …, T_{n}. More precisely, the inverse mapping is given by h ↦ (h(T_{1}), …, h(T_{n})). (13.5.6)

If p(T) ∈ R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], then we get a real or complexvalued function ˆ (p(T)) on Sp(R[T_{1}, …, T_{n}]) or Sp(C[T_{1}, …, T_{n}]), as appropriate, defined by ˆ (p(T))(h) = h(p(T)), (13.5.7) as in Section 12.14. We take Sp(R[T_{1}, …, T_{n}]) and Sp(C[T_{1}, …, T_{n}]) to be equipped with the weakest topologies with respect to which these functions are continuous, as before. Of course, ˆ (p(T))(h_{w}) = h_{w}(p(T)) = p(w) (13.5.8) for every w ∈ R^{n} or C^{n}, as appropriate. Note that the standard topologies on R^{n}, C^{n} are the weakest topologies with respect to which all polynomial functions are continuous. One can use this to get that (13.5.9) (13.5.3) is a homeomorphism from each of R^{n}, C^{n} onto Sp(R[T_{1}, …, T_{n}]), Sp(C[T_{1}, …, T_{n}]), respectively.

13.6 Polynomials, homomorphisms, and Sp(A)

Let n be a positive integer, and let T_{1}, …, T_{n} be n commuting indetermiinates. Also let A be a commutative associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}, and let a = (a_{1}, …, a_{n}) be an n-tuple of elements of A. Thus

ψ(p(T)) = p_{A}(a) (13.6.1)

defines an algebra homomorphism from R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}] into A, as appropriate, as in Section 13.4. This leads to an induced mapping

ψ̂(α) = α ∘ ψ (13.6.2)

from Sp(A) into Sp(R[T_{1}, …, T_{n}]) or Sp(C[T_{1}, …, T_{n}]), as appropriate, as in Section 12.14. Remember that

ψ̂ is continuous

(13.6.3)

with respect to the usual topology on Sp(A), as before.

Put ϕ(α) = (α(ψ(T_{1})), … α(ψ(T_{n}))) = (α(a_{1}), …, α(a_{n})) (13.6.4)

for each α ∈ Sp(A), where the right side is an element of R^{n} or C^{n}, as appropriate. This is the same as the composition of ψ̂ with (13.5.6). Equivalently,

ψ̂(α) = h_{ϕ(α)} (13.6.5)

for every α ∈ Sp(A), This means that

(ψ̂(α))(p(T)) = α(ψ(p(T))) = α(p_{A}(a)) = p(ϕ(α)) (13.6.6)

for every α ∈ Sp(A) and p(T) ∈ R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], as appropriate. We may also express (13.6.4) as

ϕ(α) = (â_{1}(α), …, â_{n}(α)), (13.6.7)

where â(α) = α(a) for each a ∈ A, as in Section 12.14. This implies that

(13.6.8) ϕ is continuous as a mapping from Sp(A) into R^{n} or C^{n},

as appropriate, sith respect to the usual topology on Sp(A). This could be obtained from (13.5.9) and (13.6.3) as well.

Clearly (13.6.9) ker ψ ⊆ ker α ∘ ψ = ker ψ̂(α)

for every α ∈ Sp(A). The kernel of ψ consists of the p(T) ∈ R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], as appropriate, such that

p_{A}(a) = 0. (13.6.10)

In this case, we have that p(ϕ(a)) = 0, (13.6.11)

as in (13.6.6).

Suppose now that

(13.6.12) ψ maps R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}] onto A,

as appropriate. This implies that

(13.6.13)  ψ̂ is one-to-one on Sp(A),

as in Section 12.15. Under these conditions, ψ̂(Sp(A)) consists exactly of the h ∈ Sp(R[T_{1}, …, T_{n}]) or Sp(C[T_{1}, …, T_{n}]), as appropriate, such that

ker ψ ⊆ ker h, (13.6.14)

as before. This means that ϕ(Sp(A)) consists exactly of the w ∈ R^{n} or C^{n}, as appropriate, such that ker ψ ⊆ ker h_{w}. (13.6.15)

This is the same as saying that ϕ(Sp(A)) consists exactly of the w ∈ R^{n} or C^{n}, as appropriate, such that p(w) = 0 (13.6.16)

for every p(T) ∈ R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], as appropriate, that satisfies (13.6.10).

13.7 A class of Banach algebras

Let (A, ∥ · ∥_{A}) be a commutative Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Also let a_{1}, …, a_{n} be finitely many elements of A, and let A_{0} be the subalgebra of A generated by e_{A} and a_{1}, …, a_{n}. Suppose that

(13.7.1) A_{0} is dense in A,

with respect to the metric associated to the norm. This corresponds to Example 11.13 (d) on p271 of [286], at least in the complex case.

Let ψ be the algebra homomorphism from R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], as appropriate, into A as in (13.6.1), with a = (a_{1}, …, a_{n}). Observe that A_{0} is the same as the image of R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}], as appropriate, under ψ.

If α ∈ Sp(A), then put

ϕ(α) = (â_{1}(α), …, â_{n}(α)) = (α(a_{1}), …, α(a_{n})), (13.7.2)

as in (13.6.4) and (13.6.7). This defines a mapping from Sp(A) into R^{n} or C^{n}, as appropriate. This mapping is continuous with respect to the topology defined on Sp(A) as in Section 12.10 and the standard metric on R^{n} or C^{n}, as appropriate, because â is continuous on Sp(A) for every a ∈ A, as before. Thus

K = ϕ(Sp(A)) (13.7.3)

is a compact subset of R^{n} or C^{n}, as appropriate, because Sp(A) is compact, as in Section 12.10.

If α ∈ Sp(A), then α is uniquely determined by its restriction to A_{0}, because of (13.7.1), and the fact that α is a bounded linear functional on A, as in Section 6.9. It is easy to see that the restriction of α to A_{0} is uniquely determined by α(a_{1}), …, α(a_{n}). This means that

(13.7.4) ϕ is one-to-one on Sp(A).

Alternatively, ψ̂ is one-to-one on Sp(A) under these conditions, as in Section 12.15. One can use this to get (13.7.4) from (13.6.5).

It follows that

(13.7.5) ϕ is a homeomorphism from Sp(A) onto K,

with respect to the restriction to K of the standard metric on R^{n} or C^{n}, as appropriate. This uses the compactness of Sp(A) and a well-known result in topology. This could also be obtained more directly here, because of the way that the topology on Sp(A) is defined, and because the dual norm of every h ∈ Sp(A) is equal to 1. This corresponds to some remarks in Section 12.15, using (13.5.9) and (13.6.5).

Remember that Sp(A) ≠ ∅ in the complex case. If A is a Banach algebra over the real numbers, then let us suppose that Sp(A) ≠ ∅ for the rest of the section. Of course, this means that

K ≠ ∅. (13.7.6)

Note that ϕ^{−1} is a homeomorphism from K onto Sp(A). If b ∈ A, then b̂ is a continuous function on Sp(A), so that

η(b) = b̂ ∘ ϕ^{−1} (13.7.7)

is a continuous real or complex-valued function on K, as appropriate. This defines an algebra homomorphism from A into C(K, R) or C(K, C), as appropriate.

Remember that â_{j}(α) is the jth coordinate of ϕ(α) for α ∈ Sp(A) and j = 1, …, n, as in (13.7.2). This implies that η(a_{j}) is the same as the restriction to K of the projection onto the jth coordinate for each j = 1, …, n. It follows that η(A_{0}) (13.7.8)

is the subalgebra of C(K, R) or C(K, C), as appropriate, consisting of the restrictions to K of polynomials on R^{n} or C^{n} with real or complex coefficients, as appropriate. In fact, η ∘ ψ (13.7.9) is the homomorphism from R[T_{1}, …, T_{n}] or C[T_{1}, …, T_{n}] into C(K, R) or C(K, C), as appropriate, that sends a formal polynomial to the restriction of the corresponding polynomial function to K.

If b ∈ A, then ∥η(b)∥_{sup,K} = ∥b̂∥_{sup,Sp(A)} ≤ ∥b∥_{A}, (13.7.10)

where ∥η(b)∥_{sup,K} is the supremum norm of η(b) on K, and the second step is as in Section 12.10. It follows that

η(A) (13.7.11)

is contained in the closure of the subalgebra of C(K, R) or C(K, C), as appropriate, consisting of restrictions to K of polynomials with real or complex coefficients, as appropriate, with respect to the supremum metric. In the real case, the restrictions to K of polynomials are dense in C(K, R) with respect to the supremum metric, by the Stone–Weierstrass theorem.

13.8 Polynomial convexity

Let us continue with the same notation and hypotheses as in the previous section, except that now we suppose that A is a complex Banach algebra. Let w ∈ C^{n} be given, and suppose that

|p(w)| ≤ sup_{z∈K} |p(z)| (13.8.1)

for every polynomial p on C^{n} with complex coefficients. We would like to show that w ∈ K (13.8.2)

under these conditions. This means that K is polynomially convex in C^{n}, as on p272 of [286].

If p is a polynomial on C^{n} with complex coefficients, then p corresponds to a formal polynomial in n commuting indeterminates with complex coefficients, as in Section 13.3. Thus we can define p_{A}(a_{1}, …, a_{n}) as an element of A as before. We would like to put

α_{w}(p_{A}(a_{1}, …, a_{n})) = p(w). (13.8.3)

This basically corresponds to an argument mentioned in Section 12.15.

Observe that η(p_{A}(a_{1}, …, a_{n})) (13.8.4)

is the same as the restriction of p to K, as an element of C(K, C). It follows that

sup_{z∈K} |p(z)| ≤ ∥p_{A}(a_{1}, …, a_{n})∥_{A}, (13.8.5)

because of (13.7.10). This means that

|p(w)| ≤ ∥p_{A}(a_{1}, …, a_{n})∥_{A}, (13.8.6) by (13.8.1). This corresponds to the condition that was used in Section 12.15.

Every element of A_{0} is of the form p_{A}(a_{1}, …, a_{n}) for some polynomial p, by definition of A_{0}. Using (13.8.6), we get that α_{w} is well-defined as a linear functional on A_{0}. In fact, α_{w} is a bounded linear functional on A_{0}, with respect to the restriction of ∥ · ∥_{A} to A_{0}. This implies that α_{w} has a unique extension to a bounded linear functional on A, because of (13.7.1), as in Section 2.2.

It is easy to see that α_{w} is an algebra homomorphism from A_{0} into C. It follows that the extension of α_{w} to A mentioned in the preceding paragraph is an algebra homomorphism into C as well. Clearly

α_{w}(e_{A}) = 1, (13.8.7)

by (13.8.3). Let us also use α_{w} to denote this extension to A, which is an element of Sp(A).

Similarly, α_{w}(a_{j}) = w_{j} (13.8.8)

for each j = 1, …, n. This means that

ϕ(α_{w}) = (α_{w}(a_{1}), …, α_{w}(a_{n})) = w. (13.8.9)

Thus (13.8.2) holds, by the definition (13.7.3) of K.

Suppose now that n = 1, so that

ϕ(α) = â_{1}(α) = α(a_{1}) (13.8.10)

for every α ∈ Sp(A), as in (13.7.2). We also have that

K = ϕ(Sp(A)) = σ_{A}(a_{1}), (13.8.11)

where the second step is as in Sections 6.9 and 6.12. We would like to show that

(13.8.12) C \ σ_{A}(a_{1}) is connected,

as mentioned on p272 of [286]. This corresponds to Exercise (2) on p33 of [11] and Exercise 7 on p401 of [284] as well.

Remember that σ_{A}(a_{1}) is a closed and bounded set in C, as in Section 6.8. This implies that C \ σ_{A}(a_{1}) is an open set with exactly one unbounded connected component. If U is a bounded component of C \ σ_{A}(a_{1}), then the boundary of U is contained in σ_{A}(a_{1}). This means every w ∈ U satisfies (13.8.1), by the maximum principle. Thus (13.8.2) implies that U is contained in σ_{A}(a_{1}), which is a contradiction.

13.9 A class of C^{∗} algebras

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra with nonzero multiplicative identity element e_{A} and involution b ↦ b^{∗}. Suppose that a_{1} is a normal element of A, so that a_{1} commutes with a_{1}^{∗}, and let A_{1} be the subalgebra of A generated by e_{A}, a_{1}, and 13.9. A CLASS OF C^{∗} ALGEBRAS 305

a_{1}^{∗}. Note that A_{1} is a commutative algebra, and that if b ∈ A_{1}, then b^{∗} ∈ A_{1} too. Suppose also that (13.9.1) A_{1} is dense in A,

with respect to the metric associated to the norm. Of course, this implies that A is commutative as well.

Let Sp(A) be the space of nonzero algebra homomorphisms from A into C, with its usual topology, as in Section 12.10. If b ∈ A, then b̂(α) = α(b) defines a continuous mapping from Sp(A) onto σ_{A}(a), as before. Remember that α(b) ∈ R for every α ∈ Sp(A) when b ∈ A is self-adjoint, as in Section 12.8. One can use this to get that

α(b^{∗}) = \overline{α(b)} (13.9.2)

for every b ∈ A, which was mentioned in slightly different notation in Section 12.11. In particular, α(a_{1}^{∗}) = \overline{α(a_{1})}. (13.9.3)

If α ∈ Sp(A), then the restriction of α to A_{1} is uniquely determined by α(a_{1}), because of (13.9.3). This implies that α is uniquely determined on A by α(a_{1}), because of (13.9.1), and because α is a bounded linear functional on A, as in Section 6.9. It follows that

(13.9.4)  â_{1} is one-to-one on Sp(A).

In fact,

(13.9.5)  â_{1} is a homeomorphism from Sp(A) onto σ_{A}(a_{1}),

with respect to the restriction of the standard metric on C to σ_{A}(a_{1}). This can be obtained from the compactness of Sp(A) and a well-known result in topology, or using more direct arguments in this case.

Remember that the Gelfand map b ↦ b̂ is an isometric algebra isomorphism from A onto C(Sp(A), C), as in Section 12.11. We also have that the involution on A corresponds to complex-conjugation on C(Sp(A), C), as before. It follows that b ↦ b̂ ∘ â_{1}^{−1} (13.9.6)

is an isometric algebra isomorphism from A onto C(σ_{A}(a_{1}), C), with respect to which the involution on A corresponds to complex-conjugation on C(σ_{A}(a_{1}), C).

If f ∈ C(σ_{A}(a_{1}), C), then let f(a_{1}) = f_{A}(a_{1}) be the unique element of A such that ˆ (f_{A}(a_{1})) ∘ â_{1}^{−1} = f. (13.9.7)

Of course, this is the same as saying that

ˆ (f_{A}(a_{1})) = f ∘ â_{1} (13.9.8)

on Sp(A). Thus f ↦ f_{A}(a_{1}) (13.9.9) defines an isometric algebra isomorphism from C(σ_{A}(a_{1}), C) onto A, which is the inverse of (13.9.6). Note that

(\overline{f})_{A}(a_{1}) = f_{A}(a_{1})^{∗}, (13.9.10)

as before.

If f ≡ 1 on σ_{A}(a_{1}), then f_{A}(a_{1}) = e_{A}. If f(z) = z on σ_{A}(a_{1}), then f_{A}(a_{1}) = a_{1}. If f(z) = \overline{z} on σ_{A}(a_{1}), then it follows that f_{A}(a_{1}) = a_{1}^{∗}. If f is given by a polynomial in z and \overline{z} on σ_{A}(a_{1}), with complex coefficients, then we get that f_{A}(a_{1}) is given by the same polynomial in a_{1} and a_{1}^{∗}. It is well known that every continuous complex-valued function on σ_{A}(a_{1}) can be uniformly approxiimated by polynomials in z and \overline{z} with complex coefficients, by the Stone–Weierstrass theorem.

These remarks basically correspond to Theorem 11.19 on p277 of [286]. This basically corresponds to Theorem 2.3.1 on p51 of [11], which is stated for bounded normal operators on complex Hilbert spaces, and uses an additional fact that will be discussed in the next section. This is also related to Corollary 1 on p263 of [291].

13.10 Spectral permanence and C^{∗} algebras

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra with a nonzero multiplicative identity element e_{A} and involution a ↦ a^{∗}. Also let B be a subalgebra of A that is a closed set with respect to the metric associated to ∥ · ∥_{A}, that contains e_{A}, and for which b^{∗} ∈ B for every b ∈ B. If b ∈ B, then we would like to show that

σ_{A}(b) = σ_{B}(b). (13.10.1)

This corresponds to Corollary 2 on p49 of [11], and Theorem 11.29 on p283 of [286]. A version of this is also mentioned in the proof of Corollary 1 on p263 of [291].

Of course, any invertible element of B is invertible as an element of A. It suffices to show that if x ∈ B and x has an inverse x^{−1} in A, then

x^{−1} ∈ B. (13.10.2)

Note that x^{∗} x is a self-adjoint element of B. This implies that

σ_{B}(x^{∗} x) ⊆ R, (13.10.3)

because B is a C^{∗} algebra, as in Section 12.8. This means that

∂σ_{B}(x^{∗} x) = σ_{B}(x), (13.10.4)

where the left side is the boundary of σ_{B}(x^{∗} x) in the complex plane. Remember that ∂σ_{B}(x^{∗} x) ⊆ σ_{A}(x^{∗} x), (13.10.5) as in Section 7.3. It follows that

σ_{B}(x^{∗} x) ⊆ σ_{A}(x^{∗} x). (13.10.6)

If x is invertible in A, then x^{∗} is invertible in A too. This implies that x^{∗} x is invertible in A, so that 0 ∉ σ_{A}(x^{∗} x). Thus 0 ∉ σ_{B}(x^{∗} x), by (13.10.6), which means that (x^{∗} x)^{−1} ∈ B. (13.10.7)

We also have that (x^{∗} x)^{−1} x^{∗} x = e_{A}, so that

x^{−1} = (x^{∗} x)^{−1} x^{∗}. (13.10.8)

The right side is an element of B, because of (13.10.7), and the fact that x^{∗} ∈ B.

Let a_{1} ∈ A be given, and let A_{1}(a_{1}) be the subalgebra of A generated by e_{A}, a_{1}, and a_{1}^{∗}. It is easy to see that A_{1}(a_{1}) is invariant under the involution on A, and that the closure B(a_{1}) of A_{1}(a_{1}) in A is a subalgebra over A that is invariant under the involution as well. This means that

σ_{A}(a_{1}) = σ_{B(a_{1})}(a_{1}), (13.10.9)

as in (13.10.1). If a_{1} is a normal element of A, then A_{1}(a_{1}) and thus B(a_{1}) are commutative subalgebras of A. In this case, the remarks in the previous section can be used for B(a_{1}), and with the spectrum of a_{1} with respect to A.

13.11 Another continuity property

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra with a nonzero multiplicative identity element e_{A} and involution a ↦ a^{∗}. If a_{1} ∈ A, then let B(a_{1}) be the closure of the subalgebra A_{1}(a_{1}) of A generated by e_{A}, a_{1}, and a_{1}^{∗}, as in the previous section. Thus A(a_{1}) and B(a_{1}) are invariant under the involution, as before, so that B(a_{1}) is a C^{∗} algebra with respect to the restriction of ∥ · ∥_{A} to B.

Suppose that a_{1} is a normal element of A, so that A_{1}(a_{1}) and B(a_{1}) are commutative subalgebras of A, as before. If f is a continuous complex-valued function on σ_{A}(a_{1}), then f(a_{1}) = f_{B(a_{1})}(a_{1}) ∈ B(a_{1}) may be defined as in Section 13.9, because of (13.10.9). We may also use f_{A}(a_{1}) to denote f(a_{1}), because B(a_{1}) is determined by a_{1} and A. Remember that f ↦ f_{A}(a_{1}) defines an isometric algebra isomorphism from C(σ_{A}(a_{1}), C) onto B(a_{1}). Thus

∥f_{A}(a_{1})∥_{A} = ∥f∥_{sup,σ_{A}(a_{1})} (13.11.1)

for every f ∈ C(σ_{A}(a_{1}), C), where the right side is the supremum norm of f on σ_{A}(a_{1}).

Now let f be any continuous complex-valued function on the complex plane. We can define f(a_{1}) = f_{A}(a_{1}) ∈ B(a_{1}) using the restriction of f to σ_{A}(a_{1}). If K is a compact subset of C and

σ_{A}(a_{1}) ⊆ K, (13.11.2) then it follows that ∥f_{A}(a_{1})∥_{A} ≤ ∥f∥_{sup,K} (13.11.3)

where the right side is the supremum norm of f on K. Let us check that

a_{1} ↦ f_{A}(a_{1}) (13.11.4)

is a continuous mapping from

(13.11.5) {a_{1} ∈ A : a_{1} is normal}

into A, with respect to the metric associated to ∥ · ∥_{A} and its restriction to (13.11.5). This corresponds to Exercise (3) on p51 of [11].

More precisely, if r is a positive real number, then (13.11.4) is uniformly continuous on (13.11.6) {a_{1} ∈ A : a_{1} is normal, and ∥a_{1}∥_{A} ≤ r}.

To see this, one can start with the case where f is given by a polynomial in z, \overline{z} with coefficients complex coefficients. In this case, one can check directly that (13.11.4) is Lipschitz on (13.11.6).

Of course, if ∥a_{1}∥_{A} ≤ r, then σ_{A}(a_{1}) is contained in the closed disk

{λ ∈ C : |λ| ≤ r}, (13.11.7)

as in Section 6.8. If f is any continuous complex-valued function on C, then f can be uniformly approximated on (13.11.7) by polynomials in z and \overline{z} with complex coefficients, by the Stone–Weierstrass theorem. This implies that (13.11.4) can be uniformly approximated on (13.11.6) by analogous functions associated to polynomials in z and \overline{z} with complex coefficients, because of (13.11.3). It follows that (13.11.4) is uniformly continuous on (13.11.6), as desired.

13.12 Holomorphic functions in the plane

Let U be a nonempty open subset of the complex plane. It is well known that the space H(U) of holomorphic functions on U is a subalgebra of the space C(U, C) of all continuous complex-valued functions on U, as an algebra over the complex numbers. Consider the space Sp(H(U)) of nonzero algebra homomorphisms from H(U) into C, as in Section 12.14. If w ∈ U, then

h_{w}(f) = f(w) (13.12.1)

defines an element of Sp(H(U)). Thus

w ↦ h_{w} (13.12.2)

defines a mapping from U into Sp(H(U)).

If w ∈ C, then put a_{w}(z) = z − w, (13.12.3)

13.12. HOLOMORPHIC FUNCTIONS IN THE PLANE 309 considered as an element of H(U), as a function of z. Note that h_{w}(a_{0}) = w (13.12.4) for every w ∈ U, so that (13.5.3) is one-to-one. If w ∉ U, then a_{w} is invertible as an element of H(U). This implies that h(a_{w}) ≠ 0 (13.12.5) for every h ∈ Sp(H(U)). This means that h(a_{0}) ≠ w (13.12.6) when w ∉ U, so that h(a_{0}) ∈ U (13.12.7) for every h ∈ H(U).

If h ∈ Sp(H(U)) satisfies h(a_{0}) = w (13.12.8) for some w ∈ U, then we would like to check that h = h_{w}. (13.12.9) If f ∈ H(U), then it is well known that there is a g_{w} ∈ H(U) such that f(z) = f(w) + (z − w) g_{w}(z) (13.12.10) on U. Note that h(a_{w}) = 0, (13.12.11) by (13.12.8). This implies that h(f) = f(w) + h(a_{w} g_{w}) = f(w). (13.12.12)

It follows that (13.12.2) is a one-to-one mapping from U onto H(U), with inverse given by h ↦ h(a_{0}). (13.12.13) Let us take Sp(H(U)) to be equipped with the weak topology associated to the functions f̂(h) = h(f), f ∈ H(U), as usual. Observe that (13.12.2) is continuous as a mapping from U into Sp(H(U)) with respect to this topology, because holomorphic functions are continuous. We also have that (13.12.13) is continuous, because of the way that this topology n Sp(H(U)) is defined. This means that (13.12.2) is a homeomorphism from U onto Sp(H(U)).

13.13 Disk algebras

Let r be a positive real number, and let

U_{r} = {z ∈ C : |z| < r} (13.13.1)

be the open disk in the complex plane centered at 0 with radius r. Of course, the closure of U_{r} with respect to the standard Euclidean metric on C is the corresponding closed disk

\overline{U_{r}} = \{z ∈ C : |z| ≤ r\}. (13.13.2)

Consider the space A(U_{r}) of continuous complex-valued functions on \overline{U_{r}} that are holomorphic on U_{r}, as in Section 1.8. This is a closed linear subspace of the space C(\overline{U_{r}}, C) of continuous complex-valued functions on \overline{U_{r}}, with respect to the supremum metric, as before. Note that A(U_{r}) is a subalgebra of C(\overline{U_{r}}, C), and thus a commutative Banach algebra with respect to the supremum norm.

One can use dilations on C to get that the A(U_{r})’s are all isometrically isomorphic to each other, as Banach algebras. Often A(U_{1}) is mostly considered, which is known as the disk algebra, as in Example 1.3.6 on p8 of [11], and Example 10.3 (c) on p230 of [286].

Suppose that f ∈ A(U_{r}), and remember that f is uniformly continuous on \overline{U_{r}}, because \overline{U_{r}} is compact with respect to the standard Euclidean metric on C. If 0 < t < 1, then f_{t}(z) = f(t z) (13.13.3)

defines an element of A(U_{r/t}), whose restriction to \overline{U_{r}} is an element of A(U_{r}). We also have that

(13.13.4) f_{t} → f uniformly on \overline{U_{r}} as t → 1−,

because f is uniformly continuous on \overline{U_{r}}.

It is well known that any holomorphic function on U_{r} can be expressed as an absolutely convergent power series. The partial sums of this power series converge uniformly on any closed disk in C centered at 0 with radius less than r. If 0 < t < 1, then it follows that f_{t} can be expressed as a power series whose partial sums converge uniformly on \overline{U_{r}}.

In particular, f_{t} can be approximated uniformly by polynomials in z with compelx coefficients on \overline{U_{r}}. This implies that

(13.13.5) f can be approximated uniformly by polynomials on \overline{U_{r}},

because of (13.13.4).

Let us now consider the space Sp(A(\overline{U_{r}})) of nonzero algebra homomorphisms from A(\overline{U_{r}}) into C, as in Sections 12.10 and 12.14. If w ∈ \overline{U_{r}}, then

h_{w}(f) = f(w) (13.13.6) defines an element of Sp(A(U_{r})), so that

w ↦ h_{w} (13.13.7)

defines a mapping from \overline{U_{r}} into Sp(A(U_{r})). Put a_{0}(z) = z, considered as an element of A(U_{r}), so that h_{w}(a_{0}) = w. Thus (13.13.7) is one-to-one on \overline{U_{r}}.

Let h ∈ Sp(A(U_{r})) be given, and remember that

|h(f)| ≤ ∥f∥_{sup,\overline{U_{r}}} (13.13.8)

for every f ∈ A(U_{r}), as in Section 6.9, where the right side is the supremum norm of f on \overline{U_{r}}. In particular,

|h(a_{0})| ≤ r. (13.13.9)

If we put w = h(a_{0}), then would like to check that

h(f) = h_{w}(f) (13.13.10)

for every f ∈ A(U_{r}). This can be verified directly when f is a polynomial. Otherwise, one can reduce to that case, using (13.13.5) and (13.13.8).

This shows that (13.13.7) maps \overline{U_{r}} onto Sp(A(U_{r})), with inverse given by h ↦ h(a_{0}).

As usual, we take Sp(A(U_{r})) to be equipped with the weak topology associated to the functions f̂(h) = h(f), f ∈ A(U_{r}). It is easy to see that (13.13.7) is a homeomorphism from \overline{U_{r}} onto Sp(A(U_{r})) with respect to this topology. This basically corresponds to the n = 1 case of Example 11.13 (c) on p271 of [286]. One can also use the maximum principle to identify A(U_{r}) with a subalgebra of the algebra C(∂U_{r}, C) of continuous complex-valued function on the circle

∂U_{r} = {z ∈ C : |z| = r}, (13.13.11)

which is the boundary of U_{r} in C. This corresponds to Example 1.11.2 on p31 of [11].

13.14 Polydisk algebras

Let n be a positive integer. It is well known that a number of equivalent conditions may be used to characterize holomorphicity of functions of several complex variables. It will be convenient here to use the following, as in Definition 7.20 on p180 of [286]. A continuous complex-valued function f on an open set in C^{n} is holomorphic if it is holomorphic in each variable separately.

Let r = (r_{1}, …, r_{n}) be an n-tuple of positive real numbers, and let

U_{r} = {z ∈ C^{n} : |z_{j}| < r_{j} for j = 1, …, n} (13.14.1)

be the corresponding open polydisk in C^{n} centered at 0. The closure of U_{r} with respect to the standard Euclidean metric on C^{n} is the corresponding closed polydisk \overline{U_{r}} = \{z ∈ C^{n} : |z_{j}| ≤ r_{j} for j = 1, …, n\} (13.14.2)

Consider the space A(U_{r}) of continuous complex-valued functions on \overline{U_{r}} that are holomorphic on U_{r}. This is a closed linear subspace of the space C(\overline{U_{r}}, C) of continuous complex-valued functions on \overline{U_{r}}, with respect to the supremum metric, for essentially the same reasons as when n = 1. We also have that A(\overline{U_{r}}) is a subalgebra of C(\overline{U_{r}}, C), and thus a commutative Banach algebra with respect to the supremum norm.

One can use linear mappings on C^{n} corresponding to dilations in each coordinate to get that the A(U_{r})’s are all isometrically isomorphic to each other, as Banach algebras. If r_{j} = 1 for each j, then A(U_{r}) may be called the polydisk algebra, as in Exercise 4 on p288 of [286], which is often mostly considered.

Let f ∈ A(U_{r}) be given, and note that f is uniformly continuous on \overline{U_{r}}, because \overline{U_{r}} is compact in C^{n}. If 0 < t < 1, then

f_{t}(z) = f(t z) (13.14.3)

defines an element of A(U_{t^{−1} r}), where t^{−1} r = (t^{−1} r_{1}, …, t^{−1} r_{n}), as usual. It is easy to see that

(13.14.4) f_{t} → f uniformly on \overline{U_{r}} as t → 1−,

because f is uniformly continuous on \overline{U_{r}}.

It is well known that any holomorphic function on U_{r} can be expressed as an absolutely convergent multiple power series. Without getting into the details too much, this can be obtained using the Cauchy integral formula in each variable separately. The main point for the moment is that f_{t} can be approximated uniformly by polynomials in z_{1}, …, z_{n} with complex coefficients on \overline{U_{r}} when 0 < t < 1. One can use this and (13.14.4) to get that

(13.14.5) f can be approximated uniformly by polynomials on \overline{U_{r}}.

This corresponds to Exercise 4 on p288 of [286].

Remember that Sp(A(U_{r})) is the space of nonzero algebra homomorphisms from A(U_{r}) into C, as in Sections 12.10 and 12.14. If w ∈ \overline{U_{r}}, then

h_{w}(f) = f(w) (13.14.6)

defines an element of Sp(A(U_{r})), so that

w ↦ h_{w} (13.14.7)

defines a mapping from \overline{U_{r}} into Sp(A(U_{r})), as before. Put

a_{j}(z) = z_{j} (13.14.8)

for each j = 1, …, n, considered as an element of A(U_{r}). Thus

h_{w}(a_{j}) = w_{j} (13.14.9)

for each j = 1, …, n and w ∈ \overline{U_{r}}. This implies that (13.14.7) is one-to-one on \overline{U_{r}}.

If h ∈ Sp(A(U_{r})), then

|h(f)| ≤ ∥f∥_{sup,\overline{U_{r}}} (13.14.10)

for every f ∈ A(U_{r}), as in Section 6.9, where the right side is the supremum norm of f on \overline{U_{r}}. This implies that

|h(a_{j})| ≤ r_{j} (13.14.11)

for each j = 1, …, n. This means that

w = (h(a_{1}), …, h(a_{n})) (13.14.12)

is an element of \overline{U_{r}}.

Using this choice of w, one can check that

h(f) = h_{w}(f) (13.14.13)

for every f ∈ A(\overline{U_{r}}). More precisely, if f is a polynomial in z_{1}, …, z_{n}, then this can be obtained from the fact that h is an algebra homomorphism. Otherwise, one can approximate f by polynomials, as in (13.14.5), and use the continuity of h on A(U_{r}), as in (13.14.10). It follows that (13.14.7) maps \overline{U_{r}} onto Sp(A(U_{r})).

As before, we take Sp(A(U_{r})) to be equipped with the weak topology associated to the functions f̂(h) = h(f), f ∈ A(U_{r}). One can verify that (13.14.7) is a homeomorphism from \overline{U_{r}} onto Sp(A(U_{r})) with respect to this topology, and the restriction of the standard Euclidean metric on C^{n} to \overline{U_{r}}. This corresponds to Example 11.13 (c) on p271 of [286].

13.15 Continuity and semisimplicity

Let (A, ∥·∥_{A}), (B, ∥·∥_{B}) be commutative Banach algebras, both real or both complex, with multiplicative identitye elements e_{A}, e_{B}, respectively, and ∥e_{A}∥_{A} = ∥e_{B}∥_{B} = 1. Remember that Sp(A), Sp(B) are the spaces of nonzero algebra homomorphisms from A, B, respectively, into R or C, as appropriate, as in Section 12.10. These homomorphisms are bounded linear funtionals on A, B, respectively, as in Section 6.9.

Let ϕ be an algebra homomorphism from A into B. If h ∈ Sp(B), then

h ∘ ϕ (13.15.1)

is an algebra homomorphism from A into R or C, as appropriate. Thus (13.15.1) is either an element of Sp(A), or it is identically equal to 0 on A. In both cases, (13.15.1) is a bounded linear functional on A. Of course, if ϕ(e_{A}) = e_{B}, then (13.15.1) is automatically nonzero.

Suppose that

(13.15.2) the elements of Sp(B) separate points in B.

This is the same as saying that B is semisimple in the complex case, as in Section 6.12. Under these conditions, we have that

(13.15.3) ϕ is bounded as a linear mapping from A into B,

as in part (b) of Exercise (5) on p27 of [11], and Theorem 11.10 on p269 of [286]. To see this, one can use the closed graph theorem, as in [11, 286]. This corresponds to a criterion for the boundedness of a linear mapping mentioned in Section 8.7.

Suppose that

(13.15.4) the elements of Sp(A) separate points in A

as well. If ϕ is an algebra isomorphism from A onto B, then we get that

(13.15.5) ϕ^{−1} is bounded as a linear mapping from B into A,

as before. In particular, algebra automorphisms of A are bounded and have bounded inverses in this case. This corresponds to part (c) of Exercise (5) on p27 of [11], and to the corollary and remark after Theorem 11.10 on p270 of

Algebras, norms, and involutions

14.1 Some more remarks about involutions

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}. Suppose that

a ↦ a^{∗} (14.1.1)

is an involution on A, which may be conjugate-linear in the complex case. If the involution is linear, then it is easy to see that

σ_{A}(a^{∗}) = σ_{A}(a) (14.1.2)

for every a ∈ A. Similarly, if A is complex and the involution is conjugate-linear, then σ_{A}(a^{∗}) = \{\overline{λ} : λ ∈ σ_{A}(a)\}. (14.1.3)

This corresponds to part (e) of Theorem 11.15 on p275 of [286], and to part (i) of 2.1 on p262 of [291].

Let Sp(A) be the set of all nonzero algebra homomorphisms from A into R or C, as appropriate, as in Section 12.14. If h ∈ Sp(A) and the involution on A is linear, then h̃(a) = h(a^{∗}) (14.1.4)

defines an element of Sp(A) too. If A is complex and the involution is conjugatelinear, then h̃(a) = \overline{h(a^{∗})} (14.1.5)

is an element of Sp(A), as mentioned in the proof of Theorem 11.16 on p276 of [286]. It is easy to see that

˜h̃ = h (14.1.6) in both cases. In particular, this implies that

h ↦ h̃ (14.1.7)

is a one-to-one mapping from Sp(A) onto itself.

If a ∈ A, then â(h) = h(a) defines a real or complex-valued function on Sp(A), and we take Sp(A) to be equipped with the weakest topology for which these functions are continuous, as before. If h ∈ Sp(A), then

â(h̃) = h̃(a) = h(a^{∗}) = ˆ(a^{∗})(h) (14.1.8)

when the involution on A is linear, and

â(h̃) = h̃(a) = \overline{h(a^{∗})} = \overline{ˆ(a^{∗})(h)} (14.1.9)

when A is complex and the involution is conjugate-linear. One can use these to get that (14.1.7) is a homeomorphism from Sp(A) onto itself in both cases.

Suppose that (A, ∥ · ∥_{A}) is a Banach algebra, so that every h ∈ Sp(A) is a bounded linear functional on A, as in Section 6.9. This implies that

(14.1.10)  h̃ is a bounded linear functional on A

for every h ∈ Sp(A). Suppose also that A is commutative, and that

(14.1.11) the elements of Sp(A) separate points in A,

which is the same as saying that A is semisimple in the complex case, as in Section 6.12. Under these conditions, there is a nonnegative real number C such that ∥a^{∗}∥_{A} ≤ C ∥a∥_{A} (14.1.12)

for every a ∈ A, as in Theorem 11.16 on p276 of [286].

More precisely, if the involution on A is linear, then it may be considered as an algebra automorphism of A, because A is commutative, and (14.1.12) follows from the remarks in Section 13.15. If A is complex and the involution is conjugate-linear, then (14.1.12) is the same as saying that the involution is is bounded as a real-linear mapping from A into itself. This can be obtained from the closed graph theorem, in essentially the same way as before. In particular, one can use the criterion for the boundedness of linear mappings mentioned in Section 8.7.

14.2 Involutions and ideals

Let A be an algebra in the strict sense over the real or complex numbers with an involution a ↦ a^{∗}, which may be conjugate-linear in the complex case. Also let I be a two-sided ideal in A, and let q be the corresponding quotient mapping from A onto A/I. If

(14.2.1) I is invariant under the involution on A, 14.2. INVOLUTIONS AND IDEALS 317 then the involution on A induces a mapping from A/I into itself, defined by q(a)^{∗} = q(a^{∗}) (14.2.2) for every a ∈ A. This defines an algebra involution on A/I, which is conjugatelinear in the complex case when the involution on A is conjugate-linear.

Suppose now that A is commutative and associative, and that A has a nonzero multiplicative identity element e_{A}. Note that the involution on A takes maximal proper ideals in A to maximal proper ideals in A. Remember that the Jacobson radical R(A) of A is the intersection of all of the maximal proper ideals in A. It follows that (14.2.3) R(A) is invariant under the involution on A. This implies that we get an induced involution on A/R(A), as in (14.2.2).

Remember that Sp(A) is the set of all nonzero algebra homomorphisms from A into R or C, as appropriate. Put (14.2.4) rad_{Sp}(A) = ⋂{ker h : h ∈ Sp(A)}, which may be described as the radical of A with respect to Sp(A). This is interpreted as being equal to A when Sp(A) = ∅. Of course, R(A) ⊆ rad_{Sp}(A), (14.2.5) because the kernel of every element of Sp(A) is a maximal proper ideal in A, as in Section 6.12. If A is a complex Banach algebra, then R(A) = rad_{Sp}(A), (14.2.6) as before.

Let h ∈ Sp(A) be given, and let h̃ ∈ Sp(A) be as in (14.1.4) or (14.1.5), as appropriate. In both cases, we have that ker h̃ = {a ∈ A : a^{∗} ∈ ker h}. (14.2.7) This implies that (14.2.8) rad_{Sp}(A) is invariant under the involution on A. Thus we get an induced involution on A/ rad_{Sp}(A) too, as in (14.2.2).

If I is any ideal in A, then any maximal proper ideal in A/I corresponds to a maximal proper ideal in A, by taking the inverse image under the quotient mapping. The maximal proper ideals in A/I correspond exactly to the maximal proper ideals in A that contain I in this way. If I ⊆ R(A), (14.2.9) then every maximal proper ideal in A contains I, and thus corresponds to a maximal proper ideal in A/I, so that

R(A/I) = R(A)/I. (14.2.10)

In particular, R(A/R(A)) = {0}. (14.2.11)

Similarly, every element of Sp(A/I) determines an element of Sp(A), by composition with the quotient mapping. In this case, the kernel of the corresponding element of Sp(A) contains I, by construction. Conversely, if h ∈ Sp(A) and I ⊆ ker h, (14.2.12)

then h corresponds to a unique element of Sp(A/I) in this way. If

I ⊆ rad_{Sp}(A), (14.2.13)

then every element of Sp(A) corresponds to a unique element of Sp(A/I) in this way, and (14.2.14) rad_{sp}(A/I) = rad_{Sp}(A)/I.

Thus (14.2.15) rad_{Sp}(A/rad_{Sp}(A)) = {0}.

Suppose now that (A, ∥ · ∥_{A}) is a Banach algebra, and that Sp(A) ≠ ∅. Remember that every h ∈ Sp(A) is a bounded linear functional on A, as in Section 6.9, so that ker h is a closed set in A, with respect to the metric associated to ∥·∥_{A}. This implies that rad_{Sp}(A) is a closed set in A too, so that A/ rad_{sp}(A) is a Banach algebra with respect to the corresponding quotient norm, as in Section 6.11. Note that A/ rad_{Sp}(A) ≠ {0}, because Sp(A) ≠ ∅.

We also have that Sp(A/ rad_{Sp}(A)) separates points in A/ rad_{Sp}(A), as in (14.2.11). It follows that the induced involution on A/ rad_{sp}(A) is continuous under these conditions, as in the previous section. This corresponds to an argument in the proof of Theorem 11.20 on p278 of [286].

14.3 Involutions and square roots

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A}. Suppose that a ∈ A satisfies ∥a∥_{A} < 1, and put ϕ_{a}(x) = (1/2) a − (1/2) x^{2} (14.3.1)

for every x ∈ A, as in Section 11.15. Under these conditions, we saw before that ϕ_{a} defines a contraction from the closed ball \overline{B}(0, ∥a∥_{A}) in A into itself, so that ϕ_{a} has a unique fixed point x_{a} in \overline{B}(0, ∥a∥_{A}), by the contraction mapping theorem. We also saw that ϕ_{a}(x) = x (14.3.2) if and only if (e_{A} + x)^{2} = e_{A} + a. (14.3.3)

More precisely, in the proof of the contraction theorem, we choose a point u_{1} ∈ \overline{B}(0, ∥a∥_{A}), and we let \{u_{j}\}_{j=1}^{∞} be the sequence of elements of \overline{B}(0, ∥a∥_{A}) defined recursively by u_{j+1} = ϕ_{a}(u_{j}) (14.3.4)

for each j, as in Section 7.12. This sequence converges to the unique fixed point x_{a} of ϕ_{a} in \overline{B}(0, ∥a∥_{A}), as before.

Suppose now that x ↦ x^{∗} is an involution on A, which may be conjugatelinear in the complex case. Suppose also that a^{∗} = a, and observe that

ϕ_{a}(x)^{∗} = ϕ_{a}(x^{∗}) (14.3.5)

for every x ∈ A. We can take u_{1} = 0 in the preceding paragraph, to get that

u_{j}^{∗} = u_{j} (14.3.6)

for each j. If the involution is continuous on A, then it follows that

x_{a}^{∗} = x_{a}. (14.3.7)

Self-adjoint square roots for a larger class of self-adjoint elements are considered in the complex case in Theorems 11.20 and 11.26 on p278, 281 of [286], respectively, without asking that the involution be continuous. The argument for commutative Banach algebras will be discussed in the next section. An extension for Banach algebras that may not be commutative will be discussed in Section 14.9.

Remember that e_{A}^{∗} = e_{A}, as in Section 6.4, so that

(e_{A} + a)^{∗} = e_{A} + a. (14.3.8)

We also have that (14.3.9) e_{A} + a is invertible in A,

because ∥a∥_{A} < 1, as in Section 6.5. These conditions are used in the statement dicsussed in the next section.

14.4 Self-adjoint square roots

Let (A, ∥ · ∥_{A}) be a complex commutative Banach algebra with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1, and let v ↦ v^{∗} be an involution on A, which may be conjugate-linear. Also let {y_{j}}_{j=1}^{∞} be a sequence of elements of A that converges to y ∈ A, and suppose that

y_{j}^{∗} = y_{j} (14.4.1) for each j. If the involution is continuous on A, then we get that

y^{∗} = y, (14.4.2)

and thus (y^{2})^{∗} = y^{2}. (14.4.3)

We would like to show that if

(14.4.4) y^{2} is invertible in A,

and if (14.4.3) holds then (14.4.2) holds, without asking that the involution be continuous on A. This corresponds to part of the proof of Theorem 11.20 on p278 of [286].

Let rad(A) be the radical of A, as in Section 6.12, which is the same as the Jacobson radical R(A) of A, as well as the radical rad_{Sp}(A) of A with respect to Sp(A) in this case, as before. Remember that Sp(A) ≠ ∅ under these conditions, and that rad(A) is a proper closed ideal in A. Thus A/ rad(A) is a Banach algebra with respect to the corresponding quotient norm, as in Section 6.11. Let q be the natural quotient mapping from A onto A/ rad(A), so that q(e_{A}) is the multiplicative identity element in A/ rad(A), as before. We also have that the quotient norm of q(e_{A}) is equal to 1, as before.

The involution on A induces an involution on A/ rad(A), with

q(u)^{∗} = q(u^{∗}) (14.4.5)

for every u ∈ A, as in Section 14.2. Remember that the induced involution on A/ rad(A) is continuous, as before. Of course, {q(y_{j})}_{j=1}^{∞} converges to q(y) in A/ rad(A), because {y_{j}}_{j=1}^{∞} converges to y in A, by hypothesis. This implies that (14.4.6) q(y)^{∗} = lim_{j→∞} q(y_{j})^{∗} = lim_{j→∞} q(y_{j}^{∗}) = lim_{j→∞} q(y_{j}) = q(y),

using (14.4.1) in the third step. It follows that

y^{∗} − y ∈ rad(A), (14.4.7)

because of (14.4.5).

Put w = (1/2) (y + y^{∗}) and z = (1/2) (y − y^{∗}), so that

y = w + z, w^{∗} = w, z^{∗} = −z, (14.4.8)

as in Section 7.5. Note that z ∈ rad(A), (14.4.9)

by (14.4.7). Of course, y^{2} = w^{2} + z^{2} + 2 w z. (14.4.10)

We also have that

(w^{2})^{∗} = w^{2}, (z^{2})^{∗} = z^{2}, (w z)^{∗} = −w z. (14.4.11)

14.5. CENTRALIZERS IN ASSOCIATIVE ALGEBRAS 321

Using our hypothesis (14.4.3), we get that w z = 0. (14.4.12)

We would like to show that z = 0, to get (14.4.2). To do this, it suffices to show that w is invertible in A, because of (14.4.12). Suppose for the sake of a contradiction that w is not invertible in A, so that there is an algebra homomorphism h from A into C such that h(w) = 0 (14.4.13) and h(e_{A}) = 1, as in Section 6.12. Note that h(y^{2}) ≠ 0, (14.4.14) by (14.4.4), and that h(z) = 0, (14.4.15) by (14.4.9). However, (14.4.13) nad (14.4.15) imply that h(y^{2}) = 0, (14.4.16) because of (14.4.10).

14.5 Centralizers in associative algebras

Let A be an associative algebra over the real or complex numbers. If E is a nonempty subset of A, then the centralizer of E in A is defined by (14.5.1) C(E) = C_{A}(E) = {a ∈ A : a x = x a for every x ∈ E}. It is easy to see that (14.5.2) C_{A}(E) is a subalgebra of A. If A has a multiplicative identity element e_{A}, then e_{A} ∈ C_{A}(E). (14.5.3) If ∥ · ∥_{A} is a submultiplicatve norm on A, then (14.5.4) C_{A}(E) is a closed set in A, with respect to the metric associated to ∥ · ∥_{A}. We also have that E ⊆ C_{A}(C_{A}(E)). (14.5.5) This corresponds to (a) and (b) in Section 11.21 on p280 of [286].

Suppose that A has a nonzero multiplicative identity element e_{A}, and that a ∈ C_{A}(E) is invertible in A. Observe that a^{−1} ∈ C_{A}(E). (14.5.6)

322 CHAPTER 14. ALGEBRAS, NORMS, AND INVOLUTIONS This implies that σ_{A}(y) = σ_{C_{A}(E)}(y) (14.5.7) for every y ∈ C_{A}(E). This corresponds to part of Theorem 11.22 on p280 of [286], and its proof.

If E_{1}, E_{2} are nonempty subsets of A with E_{1} ⊆ E_{2}, (14.5.8) then C_{A}(E_{2}) ⊆ C_{A}(E_{1}). (14.5.9) Note that the elements of E commute with each other if and only if E ⊆ C_{A}(E). (14.5.10) This implies that C_{A}(C_{A}(E)) ⊆ C_{A}(E), (14.5.11) as in (14.5.9).

It is easy to see that (14.5.12) C_{A}(E_{1}) is a commutative subalgebra of A when C_{A}(E_{1}) ⊆ E_{1}. (14.5.13) This means that (14.5.14) C_{A}(C_{A}(E)) is a commutative subalgebra of A when the elements of E commute with each other, because of (14.5.11). This corresponds to (c) in Section 11.21 on p280 of [286].

If E is any nonempty subset of A, then B = C_{A}(C_{A}(E)) (14.5.15) is a subalgebra of A that contains E, as before. If A has a nonzero multiplicative identity element e_{A}, then e_{A} ∈ B, and σ_{A}(b) = σ_{B}(b) (14.5.16) for every b ∈ B, as in (14.5.7). In particular, this holds when b ∈ E. If the elements of E commute with each other, and E is maximal with respect to inclusion among subsets of A with this property, then B = E, (14.5.17) by (14.5.5) and (14.5.14).

14.6 Spectra of sums and products

If E_{1}, E_{2} are nonempty subsets of R or C, then put

(14.6.1)  E_{1} + E_{2} = {t_{1} + t_{2} : t_{1} ∈ E_{1}, t_{2} ∈ E_{2}},
(14.6.2)  E_{1} − E_{2} = {t_{1} − t_{2} : t_{1} ∈ E_{1}, t_{2} ∈ E_{2}},

and E_{1} E_{2} = {t_{1} t_{2} : t_{1} ∈ E_{1}, t_{2} ∈ E_{2}}, (14.6.3)

as usual.

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}, and let B be a commutative subalgebra of A that contains e_{A}. Also let h be a nonzero algebra homomorphism from B into R or C, as appropriate. If x, y ∈ B, then

h(x + y) = h(x) + h(y) ∈ σ_{B}(x) + σ_{B}(y) (14.6.4)

and h(x y) = h(x) h(y) ∈ σ_{B}(x) σ_{B}(y). (14.6.5)

More precisely, the second step in each of (14.6.4) and (14.6.5) is as in Section 6.9.

Suppose now that (A, ∥ · ∥_{A}) is a complex Banach algebra with ∥e_{A}∥_{A} = 1, and that B is a closed subalgebra of A, with respect to the metric associated to ∥ · ∥_{A}. Thus B is a commutative complex Banach algebra with respect to the restriction of ∥ · ∥_{A} to B. If b ∈ B, then σ(b) consists of all complex numbers of the form h(b), where h is a nonzero algebra homomorphism from B into C, as in Section 6.12. It follows that

σ_{B}(x + y) ⊆ σ_{B}(x) + σ_{B}(y) (14.6.6)

and σ_{B}(x y) ⊆ σ_{B}(x) σ_{B}(y) (14.6.7)

for every x, y ∈ B, by (14.6.4) and (14.6.5). Of course, if (14.5.16) holds for every b ∈ B, then we get that

σ_{A}(x + y) ⊆ σ_{A}(x) + σ_{A}(y) (14.6.8)

and σ_{A}(x y) ⊆ σ_{A}(x) σ_{A}(y) (14.6.9)

for every x, y ∈ B.

Let x, y be any two commuting elements of A, and put E = {x, y}. If B is as in (14.5.15), then B is a closed commutative subalgebra of A that contains e_{A}, x, and y, and (14.5.16) holds, as before. Thus (14.6.8) and (14.6.9) hold under these conditions, as in Theorem 11.23 on p280 of [286].

14.7 Invertibility of L_{a}, R_{a}

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}. If a ∈ A, then let L_{a} and R_{a} be the corresponding left and right multiplication operators on A, as in Sections 6.3 and 6.4. If a is invertible in A, then L_{a} and R_{a} are invertible as linear mappings from A into itself, with

L_{a}^{−1} = L_{a^{−1}}, R_{a}^{−1} = R_{a^{−1}}. (14.7.1)

If b ∈ A, then it follows that the spectra of L_{b} and R_{b} in the algebra L(A) of linear mappings from A into itself satisfy

σ_{L(A)}(L_{b}), σ_{L(A)}(R_{b}) ⊆ σ_{A}(b). (14.7.2)

Similarly, if ∥ · ∥_{A} is a submultiplicative norm on A, then the spectra of L_{b}, R_{b} in the algebra BL(A) of bounded linear mappings from A into itself satisfy

σ_{BL(A)}(L_{b}), σ_{BL(A)}(R_{b}) ⊆ σ_{A}(b). (14.7.3)

In fact, σ_{BL(A)}(L_{b}), σ_{BL(A)}(R_{b}) = σ_{A}(b), (14.7.4)

as indicated in the proof of the corollary to Theorem 11.23 on p281 of [286]. Similarly, σ_{L(A)}(L_{b}), σ_{L(A)}(R_{b}) = σ_{A}(b), (14.7.5)

without using a submultiplicative norm ∥ · ∥_{A} on A.

To see this, it suffices to show that if L_{a} or R_{a} is invertible in L(A) for some a ∈ A, then a is invertible in A. In particular, if ∥ · ∥_{A} is a submultiplicative norm on A, and L_{a} or R_{a} is invertible in BL(A), then a is invertible in A.

Put L_{A} = {L_{a} : a ∈ A}, R_{A} = {R_{a} : a ∈ A}, (14.7.6)

which are subalgebras of L(A). Note that

L_{a} ∘ R_{b} = R_{b} ∘ L_{a} (14.7.7)

on A for every a, b ∈ A, because A is associative, which was mentioned in Section 10.6 when a = b. In fact,

C_{L(A)}(L_{A}) = R_{A}, C_{L(A)}(R_{A}) = L_{A}, (14.7.8)

using the notation for centralizers in Section 14.5. Indeed, if T is a linear mapping from A into itself that commutes with all left multiplication operators, then one can check that T is the same as right multiplication by T (e_{A}). Similarly, if T commutes with all right multilpication oprators, then T is the same as left moltiplication by T (e_{A}).

If A is equipped with a submultiplicative norm ∥ · ∥_{A}, then L_{A}, R_{A} may be considered as subalgebras of BL(A). In this case, we have that

C_{BL(A)}(L_{A}) = R_{A}, C_{BL(A)}(R_{A}) = L_{A}, (14.7.9)

as in (14.7.8).

If L_{a} is invertible in L(A) for some a ∈ A, then L_{a}^{−1} commutes with all right multiplication operators on A. This implies that L_{a}^{−1} = L_{b} for some b ∈ A, and one can use this to get that b is the inverse of a in A. Similarly, if R_{a} is invertible in L(A) for some a ∈ A, then R_{a}^{−1} = R_{b} for some b ∈ A, which implies that b is the inverse of a in A.

Suppose now that (A, ∥ · ∥_{A}) is a complex Banach algebra, so that BL(A) is a complex Banach algebra with respect to the corresponding operator norm. Let a ∈ A be given, and put δ_{a} = L_{a} − R_{a}, as in Section 10.6. Observe that

σ_{BL(A)}(δ_{a}) ⊆ σ_{BL(A)}(L_{a}) + σ_{BL(A)}(−R_{a}), (14.7.10)

as in (14.6.8), because L_{a} and R_{a} commute with each other. This implies that

σ_{BL(A)}(δ_{a}) ⊆ σ_{BL(A)}(L_{a}) − σ_{BL(A)}(R_{a}). (14.7.11)

It follows that σ_{BL(A)}(δ_{a}) ⊆ σ_{A}(a) − σ_{A}(a), (14.7.12)

because of (14.7.3), as in the corollary to Theorem 11.23 on p281 of [286].

14.8 Involutions and centralizers

Let A be an associative algebra over the real or complex numbers with an involution a ↦ a^{∗}, which may be conjugate-linear in the complex case. Also let E be a nonempty subset of A, so that the centralizer C_{A}(E) of E in A may be defined as in Section 14.5. If

(14.8.1) E is invariant under the involution on A,

then (14.8.2) C_{A}(E) is invariant under the involution on A.

Indeed, if a ∈ C_{A}(E) and (14.8.1) holds, then

a x^{∗} = x^{∗} a (14.8.3)

for every x ∈ E. This implies that

a^{∗} x = x a^{∗} (14.8.4)

for every x ∈ E, so that a^{∗} ∈ C_{A}(E).

It follows that

(14.8.5) C_{A}(C_{A}(E)) is invariant under the involution on A under these conditions. Remember that

B = C_{A}(C_{A}(E)) (14.8.6)

is a subalgebra of A that contains E, as in Section 14.5. Of course, if A has a multiplicative identity element e_{A}, then e_{A} is contained in the centralizer of any nonempty subset of A, and in B in particular. If the elements of E commute with each other, then B is a commutative subalgebra of A, as before. If E is a subset of A whose elements commute with each other, and if E is invariant under the involution on A and maximal with respect to inclusion among subsets of A with these properties, then it follows that

B = E. (14.8.7)

Theorem 11.25 on p281 of [286] discusses some properties of maximal subsets E of Banach algebras with involution such that the elements of E commute with each other and E is invariant under the involution. These properties can be obtained from (14.8.7) and the remarks in Section 14.5, which use similar arguments. In particular, E is a closed set under these conditions, although the involution is not asked to be continuous, as in [286].

14.9 Another criterion for self-adjointness

Let (A, ∥ · ∥_{A}) be a complex Banach algebra with a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1, and an involution a ↦ a^{∗} that may be conjugatelinear. Also let {y_{j}}_{j=1}^{∞} be a sequence of self-adjoint elements of A that converges to an element y of A, and suppose that y^{2} is self-adjoint and invertible in A. Let us ask as well that y_{j} y_{l} = y_{l} y_{j} (14.9.1)

for all j, l, which implies that

y_{j} y = y y_{j} (14.9.2)

for each j. We would like to show that y is self-adjoint under these conditions. Of course, if the involution on A is continuous, then the self-adjointness of y follows from the self-adjointness of the y_{j}’s.

Remember that this was already shown in Section 14.4 when A is commutative. We would like to reduce to the previous result, as in Theorem 11.26 on p281 in [286], and its proof. To do this, we take

E = {y_{j} : j ≥ 1}, (14.9.3)

and B to be as in (14.8.6). The elements of E commute with each other and are self-adjoint, by hypothesis, so that E is invariant under the involution on A. This implies that B is a commutative subalgebra of A that is invariant under the involution on A, as before.

We also have that B contains e_{A} and is a closed set with respect to the metric associated to ∥ · ∥_{A}, as in Section 14.5. Note that y ∈ B, so that y^{2} ∈ B as well. This implies that (y^{2})^{−1} ∈ B, (14.9.4)

because y^{2} is invertible in A, and B is the centralizer of a subset of A, as in Section 14.5. Thus we can use the argument in Section 14.4, with B in place of A, to get that y is self-adjoint.

In practive, we may start with an invertible self-adjoint element b of A with some additional properties, and find a square root of b as a limit of a sequence {y_{j}}_{j=1}^{∞} of self-adjoint elements of A. The y_{j}’s may be elements of the subalgebra of A generated by e_{A} and b, which implies that they commute with each other. In this case, we can take E = {b} (14.9.5)

in the previous argument. This is a bit closed to the formulations in [286].

This includes the analogous question for complex Banach algebras in Section 14.3 too. In the previous notation, b = e_{A} + a, and y_{j} = e_{A} + u_{j} for each j. Note that the inverse of b is automatically contained in any closed subalgebra of A that contains e_{A} and a or equivalently b, because ∥a∥_{A} < 1, by hypothesis.

14.10 A nonnegativity condition

Let A be an associative algebra over the complex numbers with a nonzero multiplicative identity element e_{A} and a conjugate-linear involution a ↦ a^{∗}. Let us say that a ∈ A is nonnegative if a^{∗} = a and

σ_{A}(a) ⊆ {λ ∈ R : λ ≥ 0}. (14.10.1)

This may be expressed by a ≥ 0, (14.10.2)

as in Definition 11.27 on p282 of [286].

Let X be a nonempty metric or topological space, and consider the algebra C(X, C) of all continuous complex-valued functions on X. The complexconjugate of such a function is clearly continuous on X too, which defines a conjugate-linear involution on C(X, C). An element of C(X, C) is self-adjoint with respect to this involution if and only if it is real-valued on X. The spectrum of an element of C(X, C) is its image in the complex plane, as mentioned in Section 8.15. The nonnegative elements of C(X, C) in the sense described in the preceding paragraph are the same as the elements of C(X, C) that are real-valued and nonnegative on X.

Similarly, complex-conjugation defines a conjugate-linear involution on the algebra C_{b}(X, C) of all bounded continuous complex-valued functions on X. This is a subalgebra of C(X, C) that is invariant under complex-conjugation. In particular, an element of C_{b}(X, C) is self-adjoint with respect to this involution if and only if it is real-valued on X, as before. The spectrum of an element of C_{b}(X, C) is the closure of its image in the complex plane, as in Section 12.12. The nonnegative elements of C_{b}(X, C) in the sense described earlier are the same as the elements of C_{b}(X, C) that are real-valued and nonnegative on X.

If a ∈ A is self-adjoint, then a satisfies

a ≥ 0 and − a ≥ 0 (14.10.3)

if and only if σ_{A}(a) ⊆ {0}. (14.10.4)

This implies that a = 0 when A is a semisimple commutative Banach algebra, as in Section 6.12. This also implies that a = 0 when A is a C^{∗} algebra and a is normal, as in (14.12.1).

It is perhaps worth emphasizing that (14.10.1) includes the condition that

σ_{A}(a) ⊆ R. (14.10.5)

If A is a C^{∗} algebra, then this holds when a is self-adjoint, as in Section 12.8. If a is any element of A, then

σ_{A}(a^{2}) = {λ^{2} : λ ∈ σ_{A}(a)}, (14.10.6)

as in Section 8.13. If (14.10.5) holds, then it follows that

σ_{A}(a^{2}) ⊆ {μ ∈ R : μ ≥ 0}. (14.10.7)

If a is also self-adjoint, then a^{2} is self-adjoint as well, and we get that

a^{2} ≥ 0. (14.10.8)

Some additional properties of nonnegative elements of Banach algebras will be discussed in the next section. Afterwards, we shall consider nonnegativity in C^{∗} algebras.

14.11 More on nonnegativity

Let (A, ∥ · ∥_{A}) be a complex Banach algebra, with a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1 and a conjugate-linear involution, as before. Suppose for the moment that A is commutative, and let Sp(A) be the set of nonzero algebra homomorphisms from A into C, as usual. Remember that the Gelfand transform of a is the complex-valued function defined on Sp(A) by â(h) = h(a), as in Section 12.10. We have seen that the spectrum of a ∈ A is the same as the image of â on Sp(A) in C, as in Sections 6.9 and 6.12. Thus (14.10.5) is the same as saying that â is real-valued on Sp(A), and (14.10.1) is the same as saying that â is nonnegative on Sp(A) too.

Suppose for the moment again that a, b are commuting elements of A. If a satisfies (14.10.5) and σ_{A}(b) ⊆ R, (14.11.1) then σ_{A}(a + b) ⊆ R (14.11.2) and σ_{A}(a b) ⊆ R, (14.11.3) as in Section 14.6. Similarly, if a satisfies (14.10.1) and σ_{A}(b) ⊆ {λ ∈ R : λ ≥ 0}, (14.11.4) then σ_{A}(a + b) ⊆ {λ ∈ R : λ ≥ 0} (14.11.5) and σ_{A}(a b) ⊆ {λ ∈ R : λ ≥ 0}. (14.11.6) If a and b are self-adjoint too, then a + b and a b are self-adjoint, so that a + b ≥ 0 (14.11.7) and a b ≥ 0. (14.11.8)

Suppose for the moment that a ∈ A is normal, so that a commutes with a^{∗}. We can express a as a = a_{1} + i a_{2}, (14.11.9) where a_{1}, a_{2} ∈ A are self-adjoint, because the involution on A is conjugatelinear, as in Section 7.5. Note that a_{1} and a_{2} commute, because a is normal. This implies that a^{∗} a = a_{1}^{2} + a_{2}^{2}. (14.11.10) Of course, a^{∗} a is self-adjoint, and in fact a^{∗} a ≥ 0, (14.11.11) because of (14.10.8) and (14.11.7).

If a is any element of A and t ∈ C, then it is easy to see that σ_{A}(t e_{A} − a) = t − σ_{A}(a) = {t − λ : λ ∈ σ_{A}(a)}. (14.11.12) Suppose that (14.10.5) holds, so that σ_{A}(a) ⊆ [−∥a∥_{A}, ∥a∥_{A}], (14.11.13) as in Section 6.8. If t ∈ R, then it follows that σ_{A}(t e_{A} − a) ⊆ [t − ∥a∥_{A}, t + ∥a∥_{A}]. (14.11.14) In particular, this means that σ_{A}(t e_{A} − a) ⊆ {λ ∈ R : λ ≥ 0} (14.11.15) when t ≥ ∥a∥_{A}.

Suppose that b ∈ A, t ∈ R, and ∥t e_{A} − b∥_{A} ≤ t. (14.11.16) If (14.11.1) holds, then (14.11.4) holds as well. This is the same as (14.11.15), with a = t e_{A} − b.

330 CHAPTER 14. ALGEBRAS, NORMS, AND INVOLUTIONS

14.12 Sums in C^{∗} algebras

Let (A, ∥·∥_{A}) be a C^{∗} algebra, with nonzero multiplicative identity element e_{A}. If a is a normal element of A, then ∥a∥_{A} = r_{A}(a) = max{|λ| : λ ∈ σ_{A}(a)}, (14.12.1) where the first step is as in Section 7.7, and the second step is as in Section 6.14. In particular, this holds when a is self-adjoint.

Suppose that u, v are nonnegative elements of A, in the sense defined in Section 14.10. We would like to show that u + v ≥ 0 (14.12.2) too. This corresponds to Lemma 4.8.1 on p126 of [11], part (d) of Theorem 11.28 on p282 of [286], and part of part (i) of Theorem 3.1 on p267 of [291]. Of course, u + v is self-adjoint, because u, v are self-adjoint, by hpothesis, and so we want to show that σ_{A}(u + v) ⊆ {λ ∈ R : λ ≥ 0}. (14.12.3) This was discussed in the previous section when u and v commute.

Note that σ_{A}(u) ⊆ [0, ∥u∥_{A}], (14.12.4) because of (14.11.13) and the hypothesis that u ≥ 0. This implies that σ_{A}(∥u∥_{A} e_{A} − u) ⊆ [0, ∥u∥_{A}], (14.12.5) as in (14.11.12). It follows that ∥∥u∥_{A} e_{A} − u∥_{A} ≤ ∥u∥_{A}, (14.12.6) by (14.12.1). Similarly, ∥∥v∥_{A} e_{A} − v∥_{A} ≤ ∥v∥_{A}. (14.12.7)

Using (14.12.6) and (14.12.7), we get that

∥(∥u∥_{A} + ∥v∥_{A}) e_{A} − (u + v)∥_{A} ≤ ∥∥u∥_{A} e_{A} − u∥_{A} + ∥∥v∥_{A} e_{A} − v∥_{A}
(14.12.8)  ≤ ∥u∥_{A} + ∥v∥_{A}.

Put t = ∥u∥_{A} + ∥v∥_{A}, so that (14.12.8) is the same as saying that ∥t e_{A} − (u + v)∥_{A} ≤ t. (14.12.9) This implies (14.12.3), as mentioned at the end of the previous section, with b = u + v.

14.13. NONNEGATIVITY OF A^{∗} A 331

14.13 Nonnegativity of a^{∗} a

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra again, with a nonzero multiplicative identity element e_{A}. If a ∈ A, then a^{∗} a is self-adjoint, and it is well known that

a^{∗} a ≥ 0, (14.13.1)

in the sense of Section 14.10. This is Theorem 4.8.3 on p127 of [11], part (e) of Theorem 11.28 on p282 of [286], and part of part (iii) of Theorem 3.1 on p267 of [291]. A couple of proofs of this will be discussed in the next two sections. Remember that this was discussed in Section 14.11 when a is normal.

Suppose that σ_{A}(a^{∗} a) ⊆ {λ ∈ R : λ ≤ 0}, (14.13.2)

which means that −a^{∗} a ≥ 0. (14.13.3)

If (14.13.1) holds, then σ_{A}(a^{∗} a) = {0}, (14.13.4)

as in Section 14.10. This implies that

a^{∗} a = 0, (14.13.5)

as in (14.12.1), because a^{∗} a is self-adjoint. This means that

a = 0, (14.13.6)

because A is a C^{∗} algebra.

Lemma 4.8.2 on p127 of [11] states that (14.13.2) implies (14.13.6), without using (14.13.1). In fact, this is used to get (14.13.1) afterwards, as in the next section.

To see this, remember that the nonzero elements of the spectra of a^{∗} a and a a^{∗} are the same, as in Section 11.5. Thus (14.13.2) implies that

σ_{A}(a a^{∗}) ⊆ {λ ∈ R : λ ≤ 0}. (14.13.7)

It follows that σ_{A}(a^{∗} a + a a^{∗}) ⊆ {λ ∈ R : λ ≤ 0}, (14.13.8)

as in the previous section.

Let us express a as a_{1} + i a_{2}, where a_{1}, a_{2} ∈ A are self-adjoint, as in Section 7.5. Observe that a^{∗} a + a a^{∗} = 2 a_{1}^{2} + 2 a_{2}^{2}. (14.13.9)

This means that −2 a_{1}^{2} − 2 a_{2}^{2} ≥ 0, (14.13.10)

by (14.13.8).

Note that a_{1}^{2}, a_{2}^{2} ≥ 0, because a_{1}, a_{2} are self-adjoint, as in Section 14.10. This implies that 2 a_{1}^{2}, 2 a_{2}^{2} ≥ 0 as well. It follows that

−2 a_{1}^{2} = (−2 a_{1}^{2} − 2 a_{2}^{2}) + 2 a_{2}^{2} ≥ 0, (14.13.11)

as in the previous section. This means that 2 a_{1}^{2} = 0, as before. Thus a_{1} = 0, because a_{1} is self-adjoint, and A is a C^{∗} algebra.

Similarly, a_{2} = 0. This shows that (14.13.6) holds, because a = a_{1} + i a_{2}. Observe that (14.13.9) implies that

a^{∗} a + a a^{∗} ≥ 0, (14.13.12)

using the result in the previous section. This could also be obtained from (14.13.1) and the analogous property of

a a^{∗} = (a^{∗})^{∗} a^{∗}. (14.13.13)

14.14 A proof of a^{∗} a ≥ 0

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra with a nonzero multiplicative identity element e_{A}, and let a ∈ A be given. We would like to show that a^{∗} a ≥ 0, as mentioned in the previous section, following the proof of Theorem 4.8.3 on p127 of [11]. Let A_{1}(a^{∗} a) (14.14.1)

be the subalgebra of A generated by e_{A} and a^{∗} a, and let

B(a^{∗} a) (14.14.2)

be the closure of A_{1}(a^{∗} a) in A with respect to the metric associated to ∥ · ∥_{A}. Note that A_{1}(a^{∗} a) is a commutative subalgebra of A that is invariant under the involution, because a^{∗} a is self-adjoint. This implies that B(a^{∗} a) has the same properties, so that it is a commutative C^{∗} algebra with respect to the restriction of ∥ · ∥_{A} to B(a^{∗} a).

Remember that the spectrum of a^{∗} a as an element of B(a^{∗} a) is the same as σ_{A}(a^{∗} a), as in Section 13.10. If f is a continuous complex-valued function on σ_{A}(a^{∗} a), then f(a^{∗} a) ∈ B(a^{∗} a) may be defined as in Section 13.9. The mapping f ↦ f(a^{∗} a) (14.14.3)

defines an isometric algebra isomorphism from C(σ_{A}(a^{∗} a), C) onto B(a^{∗} a), as before. We also have that

(\overline{f})(a^{∗} a) = f(a^{∗} a)^{∗} (14.14.4)

for every f ∈ C(σ_{A}(a^{∗} a), C). This means that

f(a^{∗} a)^{∗} = f(a^{∗} a) (14.14.5)

14.14. A PROOF OF A^{∗} A ≥ 0 333

when f is real-valued on σ_{A}(a).

Note that σ_{A}(a^{∗} a) ⊆ R, (14.14.6) because a^{∗} a is self-adjoint. Consider the continuous real-valued functions f, g defined on σ_{A}(a^{∗} a) by

(14.14.7)  f(t) = \sqrt{t} when t ≥ 0
= 0 when t < 0

and

(14.14.8)  g(t) = 0  when t > 0
= \sqrt{−t} when t ≤ 0.

Thus f(t)^{2} − g(t)^{2} = t (14.14.9) and f(t) g(t) = 0 (14.14.10) for every t ∈ σ_{A}(a^{∗} a). Put

x = f(a^{∗} a), y = g(a^{∗} a), (14.14.11)

which are self-adjoint elements of B(a^{∗} a), as in the previous paragraph. These elements satisfy x^{2} − y^{2} = a^{∗} a (14.14.12) and x y = y x = 0, (14.14.13) because of (14.14.9) and (14.14.10).

Consider (a y)^{∗} (a y) = y a^{∗} a y = y (x^{2} − y^{2}) y = −y^{4}, (14.14.14) as on p128 of [11]. Observe that

y^{4} = (y^{2})^{2} ≥ 0, (14.14.15)

as in Section 14.10, so that

σ_{A}((a y)^{∗} (a y)) ⊆ {λ ∈ R : λ ≤ 0}. (14.14.16)

This implies that a y = 0, (14.14.17) as in the previous section. This means that

y^{4} = 0, (14.14.18)

because of (14.14.14). It follows that y^{2} = 0, because A is a C^{∗} algebra, and y is self-adjoint. Similarly, y = 0. (14.14.19) Thus a^{∗} a = x^{2} ≥ 0, (14.14.20) as in Section 14.10.

14.15 Some remarks about related arguments

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra with a nonzero multiplicative identity element e_{A} again, and let a ∈ A be given. The fact that a^{∗} a ≥ 0 corresponds to part (e) of Theorem 11.28 on p282 of [286], as mentioned in Section 14.13. The proof is basically similar to the one discussed in the previous two sections, aside from one main difference near the beginning.

One starts by choosing a commutative subalgebra B of A that contains e_{A} and a^{∗} a, is invariant under the involution, is closed with respect to the metric associated to ∥ · ∥_{A}, and for which

σ_{A}(b) = σ_{B}(b) (14.15.1)

for every b ∈ B. In fact, this last property holds automatically under these conditions, as in Section 13.10. This was discussed afterwards in Theorem 11.29 on p283 of [286], and B was taken instead to be a maximal subset of A whose elements commute with each other, and which is invariant under the involution. This is related to the remarks in Section 14.8. Using the result in Section 13.10, one can take B = B(a^{∗} a) as in the previous section.

Remember that B is isometrically isomorphic to the algebra of continuous complex-valued functions on the compact Hausdorff space Sp(B), as in Section 12.11. One can use this to deal with functions of a^{∗} a as elements of B, instead of the version in Section 13.9, as in the previous section.

Note that nonnegativity of a self-adjoint element b of B is the same with respect to A and B, because of (14.15.1). If b is any element of B, then

σ_{B}(b) = b̂(Sp(B)), (14.15.2)

where b̂ is the Gelfand transform of b. This is the same as the spectrum of b̂ in C(Sp(B), C). The nonnegativity of b in B is equivalent to the nonnegativity of b̂ in C(Sp(B), C), which means that b̂ is real-valued and nonnegative on Sp(B).

The nonnegativity of a^{∗} a is also part of part (iii) of Theorem 3.1 on p267 of [291], as mentioned in Section 14.13. The proof is basically similar to the one in the previous two sections as well, and some additional related facts are discussed too.

Norms, weights, and power series

15.1 Norms, isometries, and G(A)

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}, and remember that G(A) is the group of invertible elements of A. Of course, if x ∈ G(A), then 0 is not an element of either σ_{A}(x) or σ_{A}(x^{−1}). One can check that

σ_{A}(x^{−1}) = {1/λ : λ ∈ σ_{A}(x)} (15.1.1)

in this case. Suppose that ∥·∥_{A} is a submultiplicative norm on A with ∥e_{A}∥_{A} = 1. Observe that 1 = ∥e_{A}∥_{A} ≤ ∥x∥_{A} ∥x^{−1}∥_{A} (15.1.2)

for every x ∈ G(A).

One can check that

{u ∈ G(A) : ∥u∥_{A}, ∥u^{−1}∥_{A} ≤ 1} (15.1.3)

is a subgroup of G(A). This is the same as

{u ∈ G(A) : ∥u∥_{A} = ∥u^{−1}∥_{A} = 1}, (15.1.4)

because of (15.1.2). It is easy to see that this is a relatively closed set in G(A), because x ↦ x^{−1} is continuous on G(A). If A is a Banach algebra, then one can verify that (15.1.5) (15.1.3) is a closed set in A,

using a remark in Section 6.7.

If u is an element of (15.1.3) and λ ∈ σ_{A}(u), then λ ≠ 0, as before, and |λ| ≤ 1, as in Section 6.8. Similarly,

|1/λ| ≤ 1, (15.1.6) because 1/λ ∈ σ_{A}(u^{−1}), as in (15.1.1). This means that

|λ| = 1. (15.1.7)

If u is an element of (15.1.3) and a ∈ A, then

∥u a∥_{A} ≤ ∥u∥_{A} ∥a∥_{A} = ∥a∥_{A} (15.1.8)

and ∥a∥_{A} = ∥u^{−1} u a∥_{A} ≤ ∥u^{−1}∥_{A} ∥u a∥_{A} = ∥u a∥_{A}. (15.1.9)

This implies that ∥u a∥_{A} = ∥a∥_{A}. (15.1.10)

Similarly, ∥a u∥_{A} = ∥a∥_{A}. (15.1.11)

Conversely, if u ∈ A and (15.1.10) or (15.1.11) holds with a = e_{A}, then ∥u∥_{A} = 1. If u ∈ G(A) and (15.1.10) or (15.1.11) holds with a = u^{−1}, then ∥u^{−1}∥_{A} = 1.

Let V, W ≠ {0} be vector spaces, both real or both complex, with norms ∥·∥_{V}, ∥·∥_{W}, respectively, and let T be a one-tone bounded linear mapping from V onto W, with bounded inverse. Remember that ∥T^{−1}∥_{op,W V} is the smallest nonnegative real number such that

∥T^{−1}(w)∥_{V} ≤ ∥T^{−1}∥_{op,W V} ∥w∥_{W} (15.1.12)

for every w ∈ W. Equivalently, ∥T^{−1}∥_{op,W V} is the smallest nonnegative real number such that ∥v∥_{V} ≤ ∥T^{−1}∥_{op,W V} ∥T (v)∥_{W} (15.1.13)

for every v ∈ V. It follows that T is an isometric linear mapping from V onto W if and only if ∥T ∥_{op,V W} , ∥T^{−1}∥_{op,W V} ≤ 1, (15.1.14)

in which case ∥T ∥_{op,V W} = ∥T^{−1}∥_{op,W V} = 1. In particular, if A is the algebra BL(V) of bounded linear mappings from V into itself, then (15.1.3) is the group of isometric linear mappings from V onto itself.

15.2 Some remarks about unitary elements

Let (A, ∥·∥_{A}) be a C^{∗} algebra, with nonzero multiplicative identity element e_{A}. An element u of A is said to be unitary if

u^{∗} u = u u^{∗} = e_{A}, (15.2.1)

as on p261 of [291]. Of course, this is the same as saying that u is invertible in A, with u^{−1} = u^{∗}. (15.2.2)

The collection U(A) of unitary elements of A is a subgroup of the group G(A) of invertible elements of A. This was mentioned in Section 7.5 for arbitrary associative algebras with involutions.

Using (15.2.1), we get that

∥u∥_{A} = 1. (15.2.3)

Similarly, ∥u^{−1}∥_{A} = 1, (15.2.4)

because (u^{−1})^{∗} u^{−1} = u u^{−1} = e_{A}. This shows that U(A) is contained in (15.1.3).

Suppose now that u ∈ G(A) satisfies ∥u∥_{A}, ∥u^{−1}∥_{A} ≤ 1, so that

∥u^{∗} u∥_{A} = ∥u∥_{A}^{2} ≤ 1 (15.2.5)

and ∥(u^{−1})^{∗}∥_{A} = ∥u^{−1}∥_{A} ≤ 1. (15.2.6)

This uses the fact that the involution on a C^{∗} algebra preserves the norm, as in Section 7.7. It follows that

∥u^{−1} (u^{−1})^{∗}∥_{A} = ∥((u^{−1})^{∗})^{∗} (u^{−1})^{∗}∥_{A} = ∥(u^{−1})^{∗}∥_{A}^{2} ≤ 1. (15.2.7)

Observe that (u^{∗} u)^{−1} = u^{−1} (u^{∗})^{−1} = u^{−1} (u^{−1})^{∗}, (15.2.8)

so that ∥(u^{∗} u)^{−1}∥_{A} ≤ 1. (15.2.9)

This shows that u^{∗} u is an element of (15.1.3), so that

σ_{A}(u^{∗} u) ⊆ {λ ∈ C : |λ| = 1}, (15.2.10)

as in (15.1.7). Of course, u^{∗} u is self-adjoint, so that

σ_{A}(u^{∗} u) ⊆ R, (15.2.11)

as in Section 12.8. Combining this with (15.2.10), we get that

σ_{A}(u^{∗} u) ⊆ {1, −1}. (15.2.12)

This implies that σ_{A}(u^{∗} u) = {1}, (15.2.13)

because u^{∗} u ≥ 0, as in Section 14.13.

This means that σ_{A}(u^{∗} u − e_{A}) = {0}, (15.2.14)

as in Section 14.11. It follows that u^{∗} u = e_{A}, because u^{∗} u − e_{A} is self-adjoint, as in Section 14.12. Thus u ∈ U(A), because u ∈ G(A), by hypothesis. This shows that U(A) is equal to (15.1.3) in this case.

15.3 Weighted ℓ^{p} spaces

Let X be a nonempty set, let p be a positive real number, and let Z be a vector space over the real or complex numbers with a norm ∥ · ∥_{Z}. Also let w be a positive real-valued function on X, and consider the space

ℓ_{w}^{p}(X, Z) (15.3.1)

of Z-valued functions f on X such that

∥f(x)∥_{Z} w(x) (15.3.2)

is p-summable as a nonnegative real-valued function on X. Equivalently, this means that ∥f(x)∥_{Z}^{p} w(x)^{p} (15.3.3)

is summable on X. Sometimes one may consider w^{p} as the relevant weight here, and use somewhat different notation to reflect that. This version can be more convenient in some ways, in connection with the depedence on p.

Similarly, let ℓ_{w}^{∞}(X, Z) be the space of Z-valued functions f on X such that (15.3.2) is bounded on X. If f ∈ ℓ_{w}^{p}(X, Z), 0 < p ≤ ∞, then put

∥f∥_{p,w} = ∥f∥_{ℓ_{w}^{p}(X,Z)} = ( ∑_{x∈X} ∥f(x)∥_{Z}^{p} w(x)^{p})^{1/p} (15.3.4)

when p < ∞, and

∥f∥_{∞,w} = ∥f∥_{ℓ_{w}^{∞}(X,Z)} = sup_{x∈X}(∥f(x)∥_{Z} w(x)) (15.3.5)

when p = ∞. Of course, ℓ_{w}^{p}(X, Z) is the same as ℓ^{p}(X, Z), as in Section 2.6, for every p > 0 when w ≡ 1 on X, in which case ∥f∥_{p,w} is the same as ∥f∥_{p} = ∥f∥_{ℓ^{p}(X,Z)}. Otherwise,

(15.3.6) f ∈ ℓ_{w}^{p}(X, Z) if and only if w f ∈ ℓ^{p}(X, Z),

so that f ↦ w f (15.3.7)

defines a one-to-one mapping from ℓ_{w}^{p}(X, Z) onto ℓ^{p}(X, Z). In particular, ℓ_{w}^{p}(X, Z) is a linear subspace of the space of all Z-valued functions on X, because of the analogous statement for ℓ^{p}(X, Z).

Equivalently, ∥f∥_{p,w} = ∥w f∥_{p} (15.3.8)

for every f ∈ ℓ_{w}^{p}(X, Z) and p > 0. If p ≥ 1, then ∥ · ∥_{p,w} defines a norm on ℓ_{w}^{p}(X, Z), because of the analogous statement for ℓ^{p}(X, Z). If 0 < p ≤ 1, then (15.3.4) satisfies the usual homogeneity property of a norm, and

∥f + g∥_{p,w}^{p} ≤ ∥f∥_{p,w}^{p} + ∥g∥_{p,w}^{p} (15.3.9)

15.3. WEIGHTED ℓ^{P} SPACES 339 for all f, g ∈ ℓ_{w}^{p}(X, Z), because of the analogous statement for ℓ^{p}(X, Z). It follows that ∥f − g∥_{p,w}^{p} (15.3.10) defines a metric on ℓ_{w}^{p}(X, Z) when p ≤ 1, as before. If Z is complete with respect to the metric associated to ∥ · ∥_{Z}, then (15.3.11) ℓ_{w}^{p}(X, Z) is complete with respect to the appropriate metric for each p > 0, because of the analogous statement for ℓ^{p}(X, Z).

Remember that c_{00}(X, Z) is the space of Z-valued functions on X with finite support, as in Section 2.3. Let c_{0,w}(X, Z) (15.3.12) be the space of all Z-valued functions on X such that (15.3.2) vanishes at infinity on X. This is the same as c_{0}(X, Z) when w ≡ 1 on X, and otherwise (15.3.13) f ∈ c_{0,w}(X, Z) if and only if w f ∈ c_{0}(X, Z). Thus (15.3.7) is a one-to-one mapping from c_{0,w}(X, Z) onto c_{0}(X, Z), and (15.3.14) c_{0,w}(X, Z) is a linear subspace of ℓ_{w}^{∞}(X, Z), because of the analogous statement for c_{0}(X, Z). In fact, (15.3.15) c_{0,w}(X, Z) is the same as the closure of c_{00}(X, Z) in ℓ_{w}^{∞}(X, Z) with respect to the metric associated to (15.3.5), because of the analogous statement for c_{0}(X, Z).

If 0 < p_{1} ≤ p_{2} ≤ +∞, then ℓ_{w}^{p_{1}}(X, Z) ⊆ ℓ_{w}^{p_{2}}(X, Z), (15.3.16) and ∥f∥_{p_{2},w} ≤ ∥f∥_{p_{1},w} (15.3.17) for all f ∈ ℓ_{w}^{p_{1}}(X, Z), because of the analogous staements for ℓ^{p}(X, Z). If 0 < p < ∞, then c_{00}(X, Z) ⊆ ℓ_{w}^{p}(X, Z) ⊆ c_{0,w}(X, Z), (15.3.18) where the second inclusion follows from the analogous statement for ℓ^{p}(X, Z), as usual. We also have that (15.3.19) c_{00}(X, Z) is dense in ℓ_{w}^{p}(X, Z) with respect to the appropriate metric when p < ∞, because of the analogous statement for ℓ^{p}(X, Z).

15.4 Polynomials with vector coefficients

Let V be a vector space over the real or complex numbers, and let T be an indeterminate. A formal polynomial in T with coefficients in V may be expressed as

f(T) = ∑_{j=0}^{n} f_{j} T^{j}, (15.4.1)

with f_{j} ∈ V for each j. The coefficients f_{j} should be considered as being defined for all j ≥ 0, with f_{j} = 0 when j > n. The space of all of these formal polynomials may be denoted V [T]. This may be defined more precisely as the space c_{00}(Z_{+} ∪ {0}, V) (15.4.2)

of all V -valued functions on the set Z_{+} ∪ {0} of nonnegative integers with finite support.

Note that V [T] is a vector space over the real or complex numbers, as appropriate, with respect to termwise addition and scalar multiplication of formal polynomials. Of course, this corresponds to pointwise addition and scalar multiplication on (15.4.2). If f(T) ∈ V [T] is as in (15.4.1) and t ∈ R or C, as appropriate, then

f(t) = ∑_{j=0}^{n} f_{j} t^{j} (15.4.3)

defines an element of V. Clearly

f(T) ↦ f(t) (15.4.4)

is a linear mapping from V [T] into V.

It is sometimes convenient to identify v ∈ V with the formal polynomial in T for which the coefficient of T^{j} is equal to v when j = 0, and to 0 otherwise. Thus V corresponds to a linear subspace of V [T] in this way.

If A is an algebra in the strict sense over the real or complex numbers, then A[T] may be defined initially as a vector space over the real or complex numbers, as appropriate, as before. Multiplication on A can be used to define a bilinear operation of multiplication on A[T], with

(a T^{j}) (b T^{l}) = (a b) T^{j+l} (15.4.5)

for all a, b ∈ A and j, l ≥ 0. This makes A[T] an algebra in the strict sense as well. If A is commutative or associative, then one can check that A[T] has the same property. If t ∈ R or C, as appropriate, then (15.4.4) defines an algebra homomorphism from A[T] into A.

Let us identify a ∈ A with the formal polynomial in T whose constant term is equal to a, and for which the coefficient of T^{j} is 0 when j ≥ 1, as before. It is easy to see that A corresponds to a subalgebra of A[T] in this way. If A has a multiplicative identity element e_{A}, then the corresponding formal polynomial is the multiplicative identity element in A[T].

Suppose for the moment that A = R or C, considered as an algebra over itself. In this case, A[T] is the same as R[T] or C[T], as in Section 13.3, with n = 1. We may identify R and C with subalgebras of R[T] and C[T], respectively, as in the preceding paragraph.

One can also consider formal polynomials in n commuting indeterminates T_{1}, …, T_{n} with coefficients in a vector space V or an algebra A for any positive integer n, as in Section 13.3 for polynomials with real or complex coefficients. If n ≥ 2, then a formal polynomial in T_{1}, …, T_{n} corresponds to a formal polynomial in T_{n} whose coefficients are formal polynomials in T_{1}, …, T_{n−1}.

Let V be a vector space over the real or complex numbers again. If a(T) is an element of R[T] or C[T], as appropriate, and f(T) ∈ V [T], then one can define a(T) f(T) ∈ V [T] (15.4.6)

in an obvious way.

Let f(T) ∈ V [T] be as in (15.4.1). We may say that f(T) has degree less than or equal to n in this case, or equal to n if f_{n} ≠ 0. If n ≥ 1 and t_{0} ∈ R or C, as appropriate, then

f(T) = f(t_{0}) + (T − t_{0}) g(T) (15.4.7)

for some g(T) ∈ V [T] of degree less than or equal to n − 1. In particular, if f(t_{0}) = 0, then f(T) = (T − t_{0}) g(T). (15.4.8)

It follows that f(t) has at most n zeros in R or C, as appropriate, unless f(T) = 0.

15.5 Formal power series

Let V be a vector space over the real or complex numbers, and let T be an indeterminate again. A formal power series in T with coefficients in V may be expressed as

f(T) = ∑_{j=0}^{∞} f_{j} T^{j}, (15.5.1)

with f_{j} ∈ V for each j. The space of all of these formal power series may be denoted V [[T]], which may be defined more precisely as the space of all V - valued functions on Z_{+} ∪ {0}. This is a vector space over the real or complex numbers, as appropriate, with respect to termwise addition and scalar multiplication, which corresponds to pointwise addition and scalar multiplication of V -valued functions on Z_{+} ∪ {0}. The space V [T] of formal polynomials in T with coefficients in V may be considered as a linear subspace of V [[T]].

If A is an algebra in the strict sense over the real or complex numbers, then A[[T]] may be defined initially as a vector space over the real or complex numbers, as appropriate, as in the preceding paragraph. Multiplication in A[[T]] may be defined using Cauchy products, as in Section 9.10, as follows. Let f(T), g(T) ∈ A[[T]] be given, with f(T) as in (15.5.1), and

g(T) = ∑_{l=1}^{∞} g_{l} T^{l}. (15.5.2)

Put

h_{n} = ∑_{j=0}^{n} f_{j} g_{n−j} (15.5.3)

for each n ≥ 0, and

h(T) = ∑_{n=0}^{∞} h_{n} T^{n}. (15.5.4)

The product of f(T) and g(T) in A[[T]] is defined by

f(T) g(T) = h(T). (15.5.5)

It is easy to see that this definition of multiplication is bilinear, so that A[[T]] is an algebra in the strict sense. One can check that this definition of multiplication agrees with multiplication on A[T], so that A[T] is a subalgebra of A[[T]]. In particular, we may consider A as a subalgebra of A[[T]]. If A has a multiplicative identity element e_{A}, then the corresponding element of A[T] is the multiplicative identity element in A[[T]]. Note that

f(T) ↦ f_{0} (15.5.6)

defines an algebra homomorphism from A[[T]] onto A.

Suppose that f(T), g(T) ∈ A[[T]] are as in (15.5.1) and (15.5.2) again, with

f_{j} = 0 when j ≤ j_{0} (15.5.7)

and g_{l} = 0 when l ≤ l_{0} (15.5.8)

for some j_{0}, l_{0} ≥ 0. If h_{n} is as in (15.5.3), then

h_{n} = 0 when n ≤ j_{0} + l_{0}. (15.5.9)

If A is commutative or associative, then one can check that A[[T]] has the same property. One way to do this is to reduce to the case of formal polynomials using the previous remark. In particular, R[[T]] and C[[T]] are commutative associative algebras over the real and complex numbers, respectively.

Let V be a vector space over the real or complex numbers again, and let f(T) ∈ R[[T]] or C[[T]], as appropriate, be given, as in (15.5.1). If g(T) ∈ V [[T]] is as in (15.5.2), then (15.5.3) defines an element of V for each n ≥ 0, so that (15.5.4) defines an element of V [[T]]. Thus one can define f(T) g(T) ∈ V [[T]] as in (15.5.5). If f(T) ∈ R[T] or C[T], as appropriate, and g(T) ∈ V [T], then f(T) g(T) ∈ V [T], as in the previous section. We also have that (15.5.7) and (15.5.8) imply (15.5.9), as before.

15.6. INVERTIBILITY IN A[[T]] 343

15.6 Invertibility in A[[T]]

Let A be an associative algebra over the real or complex numbers with a nonzero multiplicative identity element e_{A}, and let T be an indeterminate. Also let a(T) ∈ A[[T]] be given, so that a(T) T is a formal power series in T with constant term equal to 0. If n is a nonnegative integer, then consider

∑_{j=0}^{n}(a(T) T)^{j} = ∑_{j=0}^{n} a(T)^{j} T^{j} (15.6.1)

where (a(T) T)^{j} = a(T)^{j} T^{j} (15.6.2)

is interpreted as being e_{A} when j = 0. If l is a nonnegative integer, then the coefficient of T^{l} in (15.6.2) is equal to 0 when j > l. It follows that the coefficient of T^{l} in (15.6.1) does not depend on n when n ≥ l.

We would like to define

∑_{j=0}^{∞}(a(T) T)^{j} = ∑_{j=0}^{∞} a(T)^{j} T^{j} (15.6.3)

as an element of A[[T]]. Namely, for each l ≥ 0, the coefficient of T^{l} in (15.6.3) is defined to be the same as in (15.6.1) when n ≥ l.

Note that

(e_{A} − a(T) T) ∑_{j=0}^{n}(a(T) T)^{j} = ( ∑_{j=0}^{n}(a(T) T)^{j}) (e_{A} − a(T) T)
(15.6.4)  = e_{A} − (a(T) T)^{n+1}

for every n ≥ 0, as in Section 6.5. This implies that

(e_{A} − a(T) T) ∑_{j=0}^{∞}(a(T) T)^{j} = ( ∑_{j=0}^{∞}(a(T) T)^{j}) (e_{A} − a(T) T) = e_{A}. (15.6.5)

More precisely, if n ≥ r ≥ 0, then the coefficient of T^{r} in each of these three expressions is the same as for the corresponding expression in (15.6.4). In fact, the coefficient of T^{r} in the first two expressions in (15.6.5) only involves the coefficients of T^{l} in (15.6.3) for 0 ≤ l ≤ r, which are the same as for (15.6.1), because r ≤ n. It follows that

(15.6.6) e_{A} − a(T) T is invertible in A[[T]],

with inverse equal to (15.6.3).

Suppose that f(T) ∈ A[[T]] is as in (15.5.1), with f_{0} an invertible element of A. In this case, f(T) can be expressed as

f(T) = f_{0} (e_{A} − a(T) T) (15.6.7)

344 CHAPTER 15. NORMS, WEIGHTS, AND POWER SERIES

for some a(T) ∈ A[[T]]. This implies that f(T) is invertible in A[[T]], with f(T)^{−1} = (e_{A} − a(T) T)^{−1} f_{0}^{−1}. (15.6.8) Conversely, if f(T) is invertible in A[[T]], then one can check that f_{0} is invertible in A, using the homomorphism (15.5.6).

15.7 Some spaces of power series

Let V be a vector space over the real or complex numbers with a norm ∥ · ∥_{V}, and let r be a positive real number. Put w_{r}(j) = r^{j} (15.7.1) for each nonnegative integer j, which is interpreted as being equal to 1 when j = 0, as usual. This is a positive real-valued function on the set Z_{+} ∪ {0} of nonnegative integers, which can be used to define the spaces ℓ_{w_{r}}^{p}(Z_{+} ∪ {0}, V) for 0 < p ≤ ∞ as in Section 15.3, as well as the spaces c_{0,w_{r}}(Z_{+} ∪ {0}, V).

Suppose that 0 < r < t < +∞, (15.7.2) so that w_{r} ≤ w_{t} (15.7.3) on Z_{+} ∪ {0}. This implies that ℓ_{w_{t}}^{p}(Z_{+} ∪ {0}, V) ⊆ ℓ_{w_{r}}^{p}(Z_{+} ∪ {0}, V), (15.7.4) with ∥a∥_{p,w_{r}} ≤ ∥a∥_{p,w_{t}} (15.7.5) for every a ∈ ℓ_{w_{t}}^{p}(Z_{+} ∪ {0}, V). Similarly, c_{0,w_{t}}(Z_{+} ∪ {0}, V) ⊆ c_{0,w_{r}}(Z_{+} ∪ {0}, V). (15.7.6)

If p < ∞, then ℓ_{w_{t}}^{∞}(Z_{+} ∪ {0}, V) ⊆ ℓ_{w_{r}}^{p}(Z_{+} ∪ {0}, V). (15.7.7) More precisely, if a ∈ ℓ_{w_{t}}^{∞}(Z_{+} ∪ {0}, V), then

∥a∥_{p,w_{r}} = ( ∑_{j=0}^{∞} ∥a(j)∥_{V}^{p} w_{r}(j)^{p})^{1/p} ≤ ∥a∥_{∞,w_{t}} ( ∑_{j=0}^{∞} w_{r}(j)^{p} w_{t}(j)^{−p})^{1/p}
(15.7.8) = ∥a∥_{∞,w_{t}} ( ∑_{j=0}^{∞}(r/t)^{j p})^{1/p} = (1 − (r/t)^{p})^{−1/p} ∥a∥_{∞,w_{t}}.

Let T be an indeterminate, so that V [[T]] is the space of formal power series f(T) = ∑_{j=0}^{∞} f_{j} T^{j} in T with coefficients in V, as in Section 15.5. In this case, we may also use f to refer to the V -valued function on Z_{+} ∪ {0} defined by the coefficients f_{j} of f(T). If 0 < p ≤ ∞, then let V_{r}^{p}[[T]] be the space of f(T) ∈ V [[T]] such that

f ∈ ℓ_{w_{r}}^{p}(Z_{+} ∪ {0}, V), (15.7.9)

and put

∥f(T)∥_{p,r} = ∥f(T)∥_{V_{r}^{p}[[T]]} = ∥f∥_{p,w_{r}}. (15.7.10)

Similarly, let V_{0,r}[[T]] be the space of f(T) ∈ V [[T]] such that

f ∈ c_{0,w_{r}}(Z_{+} ∪ {0}, V). (15.7.11)

If p < ∞, then

V [T] ⊆ V_{r}^{p}[[T]] ⊆ V_{0,r}[[T]], (15.7.12)

as in Section 15.3. In fact, V [T] is dense in V_{r}^{p}[[T]] when p < ∞, and V [T] is dense in V_{0,r}[[T]], with respect to the appropriate metrics, as before. If 0 < p_{1} ≤ p_{2} ≤ ∞, then

V_{r}^{p_{1}}[[T]] ⊆ V_{r}^{p_{2}}[[T]], (15.7.13)

as before. If 0 < r_{1} < r_{2} < ∞, then

V_{r_{2}}^{p}[[T]] ⊆ V_{r_{1}}^{p}[[T]] (15.7.14)

and

V_{0,r_{2}}[[T]] ⊆ V_{0,r_{1}}[[T]], (15.7.15)

as in (15.7.4) and (15.7.6), respectively. In this case, if p < ∞, then

V_{r_{2}}^{∞}[[T]] ⊆ V_{r_{1}}^{p}[[T]], (15.7.16)

as in (15.7.7).

If 0 < ρ ≤ ∞, then put

V_{ρ}[[T]] = ⋂_{0<r<ρ} V_{r}^{p}[[T]]. (15.7.17)

It is easy to see that the right side does not depend on p > 0, because of (15.7.13) and (15.7.16).

Of course, R and C may be considered as one-dimensional vector spaces over themselves, with their standard absolute value functions as norms. If V = R or C, then we may use (R)_{r}^{p}[[T]] or (C)_{r}^{p}[[T]], as appropriate, for V_{r}^{1}[[T]]. Similarly, we may use (R)_{0,r}[[T]] or (C)_{0,r}[[T]], as appropriate, for V_{0,r}[[T]]. We may also use (R)_{ρ}[[T]] or (C)_{ρ}[[T]], as appropriate, for V_{ρ}[[T]] in this case.

15.8 Absolute convergence of power series

Let (V, ∥ · ∥_{V}) be a Banach space over the real or complex numbers, let T be an indeterminate, and let r be a positive real number. Also let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ V_{r}^{1}[[T]] be given, and let t be a real or complex number, as appropriate, with |t| ≤ r. Under these conditions,

f(t) = ∑_{j=0}^{∞} f_{j} t^{j} (15.8.1)

defines an element of V, because the sum on the right converges absolutely, as in Section 1.7. More precisely,

∥f(t)∥_{V} ≤ ∑_{j=0}^{∞} ∥f_{j}∥_{V} |t|^{j} ≤ ∑_{j=0}^{∞} ∥f_{j}∥_{V} r^{j} = ∥f(T)∥_{1,r}. (15.8.2)

Thus (15.8.1) defines a V -valued function on

{t ∈ R : |t| ≤ r} = [−r, r] (15.8.3)

or {t ∈ C : |t| ≤ r}, (15.8.4) as appropriate.

The partial sums

∑_{j=0}^{n} f_{j} t^{j} (15.8.5)

converge to (15.8.1) uniformly on (15.8.3) or (15.8.4), as appropriate, with respect to the metric on V associated to ∥ · ∥_{V}, as in Weierstrass’ well-known criterion for uniform convergece. One can check that the partial sums (15.8.5) are continuous as V -valued functions on (15.8.3) or (15.8.4), as appropriate. It follows that (15.8.1) is continuous as a V -valued function on (15.8.3) or (15.8.4), as appropriate. This defines a bounded linear mapping from V_{r}^{1}[[T]] into the space of continuous V -valued functions on (15.8.3) or (15.8.4), as appropriate, with respect to the corresponding supremum norm, because of (15.8.2). One may also consider the right side of (15.8.1) as an absolutely convergent sum in this space of continuous V -valued functions with respect to the supremum norm.

Suppose now that 0 < ρ ≤ ∞, and that f(T) ∈ V_{ρ}[[T]]. If 0 < r < ρ, then f(T) ∈ V_{r}^{1}[[T]], as in (15.7.17). If t ∈ R or C, as appropriate, and |t| < ρ, then (15.8.1) defines an element of V, as before. This defines a V -valued function on

{t ∈ R : |t| < ρ} = (−ρ, ρ) (15.8.6)

or {t ∈ C : |t| < ρ}, (15.8.7) as appropriate. This function is continuous with respect to the metric on V associated to ∥ · ∥_{V}. This can be obtained from the continuity of the restriction of the function to (15.8.3) or (15.8.4), as appropriate, when 0 < r < ρ.

15.9 Products and absolute convergence

Let A be an algebra in the strict sense over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}, let T be an indeterminate, and let r be a positive real number. Suppose that f(T) = ∑_{j=0}^{∞} f_{j} T^{j} and g(T) = ∑_{l=0}^{∞} g_{l} T^{l} are elements of A_{r}^{1}[[T]]. Let h(T) = f(T) g(T) be their product in A[[T]], as in Section 15.5, with h(T) = ∑_{n=0}^{∞} h_{n} T^{n}. Observe that

∥h_{n}∥_{A} ≤ ∑_{j=0}^{n} ∥f_{j}∥_{A} ∥g_{n−j}∥_{A} (15.9.1)

for each n ≥ 0, so that

∥h_{n}∥_{A} r^{n} ≤ ∑_{j=0}^{n}(∥f_{j}∥_{A} r^{j}) (∥g_{n−j}∥_{A} r^{n−j}). (15.9.2)

The right side of (15.9.2) is the same as the nth term of the Cauchy product of the sums corresponding to ∥f(T)∥_{1,r} and ∥g(T)∥_{1,r}, as in Section 9.10.

It follows that h(T) ∈ A_{r}^{1}[[T]] (15.9.3)

too, with ∥h(T)∥_{1,r} ≤ ∥f(T)∥_{1,r} ∥g(T)∥_{1,r}. (15.9.4)

Suppose that A is complete with respect to the metric associated to ∥ · ∥_{A}, and let t be a real or complex number, as appropriate, with |t| ≤ r. The series corresponding to h(t) as in the previous section is the same as the Cauchy product of the series corresponding to f(t) and g(t), as in Section 9.11. This implies that h(t) = f(t) g(t), (15.9.5)

as before. If 0 < ρ ≤ ∞ and f(T), g(T) ∈ A_{ρ}[[T]], then we get that

h(T) ∈ A_{ρ}[[T]], (15.9.6)

and that (15.9.5) holds for every t ∈ R or C, as appropriate, with |t| < ρ. In particular,

(15.9.7) A_{r}^{1}[[T]] and A_{ρ}[[T]] are subalgebras of A[[T]].

If A is a Banach algebra, then A_{r}^{1}[[T]] is a Banach algebra with respect to ∥·∥_{1,r}.

Now let V be a vector space over the real or complex numbers with a norm ∥ · ∥_{V}, and let g(T) be an element of V_{r}^{1}[[T]]. Also let f(T) be an element of (R)_{r}^{1}[[T]] or (C)_{r}^{1}[[T]], as appropriate. Thus h(T) = f(T) g(T) may be defined as an element of V [[T]], as in Section 15.5. As before, we have that

∥h_{n}∥_{V} ≤ ∑_{j=0}^{n} |f_{j}| ∥g_{n−j}∥_{V} (15.9.8)

for each n ≥ 0, so that

∥h_{n}∥_{V} r^{n} ≤ ∑_{j=0}^{n}(|f_{j}| r^{j}) (∥g_{n−j}∥_{V} r^{n−j}). (15.9.9)

The right side of (15.9.9) is the same as the nth term of the Cauchy product of the sums corresponding to ∥f(T)∥_{1,r} and ∥g(T)∥_{1,r}.

This implies that h(T) ∈ V_{r}^{1}[[T]], (15.9.10)

and that (15.9.4) holds, as before. Suppose that V is a Banach space, and that t ∈ R or C, as appropriate, satisfies |t| ≤ r. The series corresponding to h(t) as in the previous section may be considered as the Cauchy product of the series corresponding to f(t) and g(t), as before. It follows that (15.9.5) holds in this case as well. Similarly, if 0 < ρ ≤ ∞, f(T) ∈ (R)_{ρ}[[T]] or (C)_{ρ}[[T]], as appropriate, and g(T) ∈ V_{ρ}[[T]], then

h(T) ∈ V_{ρ}[[T]], (15.9.11)

and (15.9.5) holds for every t ∈ R or C, as appropriate, with |t| < ρ.

15.10 Differentiating power series

Let V be a vector space over the real or complex numbers, and let T be an indeterminate. If f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ V [[T]], then the derivative f′(T) of f(T) in V [[T]] is defined by

f′(T) = ∑_{j=1}^{∞} j f_{j} T^{j−1} = ∑_{j=0}^{∞}(j + 1) f_{j+1} T^{j}. (15.10.1)

Note that f(T) ↦ f′(T) (15.10.2)

is a linear mapping from V [[T]] into itself that sends V [T] into itself.

Let A be an algebra in the strict sense over the real or complex numbers. One can check that

(f(T) g(T))′ = f′(T) g(T) + f(T) g′(T) (15.10.3)

for all f(T), g(T) ∈ A[[T]]. Similarly, this holds when g(T) ∈ V [[T]] and f(T) is an element of R[[T]] or C[[T]], as appropriate.

Suppose now that (V, ∥ · ∥_{V}) is a Banach space, and let r be a positive real number. Suppose also that f(T) ∈ V [[T]] satisfies

f′(T) ∈ V_{r}^{1}[[T]], (15.10.4)

and note that

∥f′(T)∥_{1,r} = ∑_{j=1}^{∞} j ∥f_{j}∥_{V} r^{j−1}. (15.10.5)

It is easy to see that f(T) ∈ V_{r}^{1}[[T]] (15.10.6)

in this case. Let t_{1}, t_{2} be real or complex numbers, as appropriate, such that

|t_{1}|, |t_{2}| ≤ r. (15.10.7)

Thus f(t_{1}), f(t_{2}) ∈ V may be defined as in Section 15.8, and we have that

∥f(t_{1}) − f(t_{2})∥_{V} = ∥∑_{j=0}^{∞} f_{j} t_{1}^{j} − ∑_{j=0}^{∞} f_{j} t_{2}^{j}∥_{V} = ∥∑_{j=1}^{∞} f_{j} (t_{1}^{j} − t_{2}^{j})∥_{V}

(15.10.8)  ≤ ∑_{j=1}^{∞} ∥f_{j}∥_{V} |t_{1}^{j} − t_{2}^{j}| ≤ ∑_{j=1}^{∞} j ∥f_{j}∥_{V} r^{j−1} |t_{1} − t_{2}|,

where the last step is as in Section 10.1.

This implies that

∥f(t_{1}) − f(t_{2})∥_{V} ≤ ∥f′(T)∥_{1,r} |t_{1} − t_{2}| (15.10.9)

when t_{1}, t_{2} satisfy (15.10.7). This is a Lipschitz condition for f(t) on the set of t ∈ R or C, as appropriate, with |t| ≤ r. If f(T) ∈ V [T], then completeness of V is not needed here.

If t ∈ R or C, as appropriate, and |t| ≤ r, then

f′(t) = ∑_{j=1}^{∞} j f_{j} t^{j−1} = ∑_{j=0}^{∞}(j + 1) f_{j+1} t^{j} (15.10.10)

defines an element of V, as in Sections 1.7 and 15.8. Of course, this corresponds to differentiating the power series for f(t) termwise. One can use the same type of Lipschitz conditions as before to show that the derivative of the sum is equal to the sum of the derivatives, as mentioned in Section 10.1 for power series with real or complex coefficients.

In the complex case, (15.10.10) should be interpreted as a suitable complex derivative, as before. In particular, f(t) is holomorphic as a V -valued function in a suitable sense.

If 0 < ρ ≤ ∞ and f(T) ∈ V_{ρ}[[T]], then

f′(T) ∈ V_{ρ}[[T]] (15.10.11)

too, by standard arguments. If t ∈ R or C, as appropriate, and |t| < ρ, then f(t) and f′(t) are defined as elements of V, and f′(t) corresponds to the derivative of f(t), as before. In the complex case, f(t) is holomorphic as a V -valued function on {t ∈ C : |t| < ρ}. (15.10.12)

15.11 Coefficients in R, C

Let T be an indeterminate, and let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}. If f(T) is a formal polynomial in T with real or complex coefficients, as appropriate, and x ∈ A, then we may define f(x) as an element of A, as in Section 8.13. We also have that f(T) ↦ f(x) (15.11.1)

defines an algebra homomorphism from R[T] or C[T], as appropriate, into A, as before. This corresponds to a remark in Section 13.4 with n = 1 as well.

Suppose now that (A, ∥ · ∥_{A}) is a Banach algebra, ∥e_{A}∥_{A} = 1, and that f(T) is a formal power series in T with real or complex coefficients, as appropriate. Let r be a positive real number, and suppose that

f(T) ∈ (R)_{r}^{1}[[T]] or (C)_{r}^{1}[[T]], (15.11.2)

as appropriate. If x ∈ A and ∥x∥_{A} ≤ r, then f(x) may be defined as an element of A as in Section 9.14, and we have that

∥f(x)∥_{A} ≤ ∥f(T)∥_{1,r}, (15.11.3)

as before. We also have that (15.11.1) defines an algebra homomorphism from (R)_{r}^{1}[[T]] or (C)_{r}^{1}[[T]], as appropriate, into A, as in Section 9.13.

If f′(T) ∈ (R)_{r}^{1}[[T]] or (C)_{r}^{1}[[T]], (15.11.4)

as appropriate, then (15.11.2) holds, as in the previous section. If x, y ∈ A and ∥x∥_{A}, ∥y∥_{A} ≤ r, then we get that

∥f(x) − f(y)∥_{A} ≤ ∥f′(T)∥_{1,r} ∥x − y∥_{A}, (15.11.5)

as in Section 10.1.

Suppose that 0 < ρ ≤ ∞, and that

f(T) ∈ (R)_{ρ}[[T]] or (C)_{ρ}[[T]], (15.11.6)

as appropriate. If x ∈ A and ∥x∥_{A} < ρ, then f(x) may be defined as an element of A as in Section 9.14 again. Of course, we can take r > 0 such that

∥x∥_{A} ≤ r < ρ, (15.11.7)

so that (15.11.2) holds. Under these conditions, (15.11.1) defines an algebra homomorphism from (R)_{ρ}[[T]] or (C)_{ρ}[[T]], as appropriate, into A, as before.

Let A be any associative algebra over R or C with a multiplicative identity element e_{A} again, and suppose that f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ R[[T]] or C[[T]], as appropriate. If x ∈ A is nilpotent, then

f(x) = ∑_{j=0}^{∞} f_{j} x^{j} (15.11.8)

reduces to a finite sum in A, and thus defines an element of A. One can check that (15.11.1) defines an algebra homomorphism from R[[T]] or C[[T]], as appropriate, into A, as usual.

Remember that A[[T]] is an associative algebra over the real or complex numbers, as appropriate, with a multiplicative identity element, as in Section 15.5. If f(T) ∈ R[T] or C[T], as appropriate, and a(T) ∈ A[[T]], then

f(a(T)) (15.11.9)

may be defined as an element of A[[T]], as in Section 8.13. The mapping

f(T) ↦ f(a(T)) (15.11.10)

is an algebra homomorphism from R[T] or C[T], as appropriate, into A[[T]], as before. If a(T) ∈ A[T], then (15.11.9) is an element of A[T] too.

Suppose that the constant term in a(T) ∈ A[[T]] is equal to 0, and that f(T) ∈ R[[T]] or C[[T]], as appropriate. Note that the coefficient of T^{l} in a(T)^{j} is equal to 0 when j > l. This implies that the coefficient of T^{l} in

∑_{j=0}^{n} f_{j} a(T)^{j} (15.11.11)

does not depend on n when n ≥ l. Let us define

f(a(T)) = ∑_{j=0}^{∞} f_{j} a(T)^{j} (15.11.12)

to be the formal power series in T with coefficients in A such that the coefficient of T^{l} is equal to the coefficient of T^{l} in (15.11.11) when n ≥ l. One can check that (15.11.10) is an algebra homomorphism from R[[T]] or C[[T]], as appropriate, into A[[T]] under these conditions.

15.12 Some formal compositions

Let T be an indeterminate, and let f(T) and g(T) be formal polynomials in T with real or complex coefficients. Thus f(g(T)) may be defined as a formal polynomial with coefficients in R or C, as appropriate, as in the previous section. Let us put (f ∘ g)(T) = f(g(T)), (15.12.1)

which corresponds formally to composing g(T) with f(T).

Let A be an associative algebra over the real or complex numbers, as appropriate, with a multiplicative identity element e_{A}. If x ∈ A, then g(x) may be defined as an element of A as in Section 8.13. Similarly, f(g(x)) and (f ∘ g)(x) may be defined as elements of A. One can check that

(f ∘ g)(x) = f(g(x)) (15.12.2) under these conditions. In particular, one can take A = R[T] or C[T], as appropriate, to get that formal compositions of formal polynomials in T with coefficients in R or C is associative.

Let a(T) be a formal power series in T with coefficients in A, so that g(a(T)) may be defined as an element of A[[T]], as in the previous section. As before, we may put (g ∘ a)(T) = g(a(T)), (15.12.3)

because it also corresponds to formally composing a(T) with g(T). Similarly,

((f ∘ g) ∘ a)(T) = (f ∘ g)(a(T)) (15.12.4)

and (f ∘ (g ∘ a))(T) = f((g ∘ a)(T)) (15.12.5)

are defined as elements of A[[T]]. Note that

(f ∘ g)(a(T)) = f(g(a(T))), (15.12.6)

as in (15.12.2). This implies that

((f ∘ g) ∘ a)(T) = (f ∘ (g ∘ a))(T), (15.12.7)

because of (15.12.3).

Suppose for the moment that a(T) ∈ A[T], so that (15.12.3) is an element of A[T] too, as before. If t ∈ R or C, as appropriate, then a(t) and (g ∘ a)(t) may be defined as elements of A as in Section 15.4. Similarly, g(a(t)) is defined as an element of A, because a(t) ∈ A, as in Section 8.13. One can verify that

(g ∘ a)(t) = g(a(t)). (15.12.8)

If a(T) ∈ A[[T]], then a(0) and (g ∘ a)(0) may be interpreted as the constant terms of these power series, and one can check that (15.12.8) holds, with t = 0.

Suppose now that g(T) is a formal power series in T with real or complex coefficients, so that (15.12.1) is a formal power series in T with real or complex coefficients too, as appropriate. If x ∈ A is nilpotent, then g(x) and (f ∘ g)(x) may be defined as elements of A as in the previous section, and f(g(x)) may be defined as an element of A as well. One can check that (15.12.2) also holds in this case.

15.13 Some more formal compositions

Let T be an indeterminate, and let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{E}. If g is a formal power series in T with real or complex coefficients, and x ∈ A is nilpotent, then g(x) may be defined as an element of A, as in Section 15.11. Suppose that

(15.13.1) the constant term in g(T) is equal to 0.

In this case, one can check that

(15.13.2) g(x) is nilpotent.

Let f(T) be another formal power series in T with coefficients in R or C, as appropriate, and remember that f(g(T)) may be defined as an element of R[[T]] or C[[T]] under these conditions, as appropriate, as in Section 15.11. This corresponds to formally composing g(T) with f(T), as before, and thus may be expressed as in (15.12.1) again. Note that (f ∘ g)(x) and f(g(x)) are defined as elements of A, because x and g(x) are nilpotent. One can verify that (15.12.2) holds in this case too.

Let a(T) be a formal power series in T with coefficients in A, and suppose that (15.13.3) the constant term in a(T) is equal to 0.

If g(T) is a formal power series in R or R, as appropriate, then g(a(T)) may be defined as a formal power series in T with coefficients in A as in Section 15.11. This corresponds to formally composing a(T) and g(T), as usual, and may be expressed as in (15.12.3).

If f(T) is a formal polynomial in T with coefficients in R or C, as appropriate, then f(g(T)) is a formal power series with coefficients in R or C, as appropriate, which may be expressed as in (15.12.1), as before. Similarly, (15.12.4) and (15.12.5) may be defined as formal power series in T with coefficients in A under these conditions. One can check that (15.12.6) holds in this case too, using some of the remarks in Section 15.11. It follows that (15.12.7) holds, as before.

Suppose now that the constant term in g(T) is equal to 0 too. Using this, it is easy to see that

(15.13.4) the constant term in g(a(T)) is equal to 0,

because of (15.13.3). If f(T) is a formal power series in T with coefficients in R or C, as appropriate, then f(g(T)) may be defined as a formal power series in T with coefficients in R or C, as appropriate, as in Section 15.11, and which may be expressed as in (15.12.1). Similarly, (15.12.4) and (15.12.5) may be defined as formal power series in T with coefficients in A, as in Section 15.11. In order to verify (15.12.6), one can approximate f(T) by formal polynomials in T, for which the analogous statement was mentioned in the preceding paragraph. This implies that (15.12.7) holds under these conditions as well.

15.14 Polynomials of power series

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}, and let T be an indeterminate. If f(T) ∈ R[[T]] or C[[T]], as appropriate, and a(T) ∈ A[[T]], then we may be able to define f(a(T)) as an element of A[[T]] in a reasonable way under suitable conditions. In this case, we may put

(f ∘ a)(T) = f(a(T)), (15.14.1)

because it corresponds formally to composing a(T) and f(T), as before. In particular, we would like to have

(f ∘ a)(t) = f(a(t)) (15.14.2)

for suitable t ∈ R or C, as appropriate. Similarly, if the coefficients of a(T) are in R or C, as appropriate, then we would like to have

(f ∘ a)(x) = f(a(x)) (15.14.3)

for suitable x ∈ A.

Of course, we have seen some versions of this in the previous two sections. In this section and the next one, we would like to consider some more versions of this, involving convergence of power series.

Suppose for the rest of this section that f(T) is a formal polynomial in T with real or complex coefficients, as appropriate. Thus f(a(T)) is defined as an element of A[[T]] for each a(T) ∈ A[[T]], as in Section 15.11. Let ∥ · ∥_{A} be a submultiplicative norm on A, let r_{a} be a positive real number, and suppose that

a(T) ∈ A_{r_{a}}^{1}[[T]]. (15.14.4)

This implies that f(a(T)) ∈ A_{r_{a}}^{1}[[T]], (15.14.5)

because A_{r_{a}}^{1}[[T]] is a subalgebra of A[[T]], as in Section 15.9.

Suppose that A is a Banach algebra with respect to ∥ · ∥_{A}, and that t is a real or complex number, as appropriate, that satisfies

|t| ≤ r_{a}. (15.14.6)

This means that a(t) and (f ∘ a)(t) may be defined as elements of A, as in Section 15.8, so that f(a(t)) may be defined as an element of A as well. One can check that (15.14.2) holds, by reducing to the case where f(T) = T^{j} for some j, as usual.

Similarly, let 0 < ρ_{a} ≤ ∞ be given, and suppose that

a(T) ∈ A_{ρ_{a}}[[T]]. (15.14.7)

This implies that f(a(T)) ∈ A_{ρ_{a}}[[T]], (15.14.8)

because A_{ρ_{a}}[[T]] is a subalgebra of A[[T]], as before. If A is a Banach algebra and t ∈ R or C, as appropriate, satisfies

|t| < ρ_{a}, (15.14.9) then a(t), (f ∘ a)(t), and f(a(t)) may be defined as elements of A. We also have that (15.14.2) holds in this case, by the same type of argument as before, or by reducing to the previous case.

Suppose now that a(T) is a formal power series in T with real or complex coefficients, as appropriate, so that f(a(T)) is a formal power series in T with real or complex coefficients, as appropriate, too. Let r_{a} be a positive real number again, and suppose for the moment that

a(T) ∈ (R)_{r_{a}}^{1}[[T]] or (C)_{r_{a}}^{1}[[T]], (15.14.10)

as appropriate, so that

(f ∘ a)(T) = f(a(T)) ∈ (R)_{r_{a}}^{1}[[T]] or (C)_{r_{a}}^{1}[[T]], (15.14.11)

as appropriate. If A is a Banach algebra, and x ∈ A satisfies

∥x∥_{A} ≤ r_{a}, (15.14.12)

then a(x) and (f ∘ a)(x) may be defined as elements of A as in Section 9.14, so that f(a(x)) is defined as an element of A too. One can verify that (15.14.3) holds under these conditions, by reducing to the case where f(T) = T^{j} for some j, as before.

Similarly, let 0 < ρ_{a} ≤ ∞ be given again, and suppose that

a(T) ∈ (R)_{ρ}[[T]] or (C)_{ρ}[[T]], (15.14.13)

as appropriate, so that

(f ∘ a)(T) = f(a(T)) ∈ (R)_{ρ}[[T]] or (C)_{ρ}[[T]], (15.14.14)

as appropriate. If A is a Banach algebra and x ∈ A satisfies

∥x∥_{A} < ρ_{a}, (15.14.15)

then a(x), (f ∘ a)(x), and f(a(x)) may be defined as elements of A. We also get that (15.14.3) holds, by reducing to the previous case.

15.15 Some compositions of power series

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1, and let r_{a}, r_{f} be positive real numbers. Also let a(T) ∈ A_{r_{a}}^{1}[[T]] (15.15.1)

and f(T) ∈ (R)_{r_{f}}^{1} [[T]] or (C)_{r_{f}}^{1} [[T]], (15.15.2)

as appropriate, be given. Remember that A_{r_{a}}^{1}[[T]] is a Banach algebra with respect to the appropriate norm, as in Section 15.9.

If ∥a(T)∥_{1,r_{a}} ≤ r_{f}, (15.15.3)

then we may define f(a(T)) ∈ A_{r_{a}}^{1}[[T]], (15.15.4)

as in Section 9.14. The mapping

f(T) ↦ f(a(T)) (15.15.5)

is an algebra homomorphism from (R)_{r_{f}}^{1} [[T]] or (C)_{r_{f}}^{1} [[T]], as appropriate, into A_{r_{a}}^{1}[[T]], as in Sections 9.13 and 15.11. Of course, this definition of f(a(T)) is the same as the one in the previous section when f(T) is a formal polynomial in T. We may express f(a(T)) as (f ∘ a)(T), as in (15.14.1).

We also have that

∥f(a(T))∥_{1,r_{a}} ≤ ∥f(T)∥_{1,r_{f}} , (15.15.6)

as in Sections 9.14 and 15.11. More precisely, the left side is the norm of f(a(T)) in A_{r_{a}}^{1}[[T]], and the right side refers to the norm of f(T) in (R)_{r_{f}}^{1} [[T]] or (C)_{r_{f}}^{1} [[T]], as appropriate. If f(T) = ∑_{j=0}^{∞} f_{j} T^{j}, as usual, then

lim_{n→∞} ∑_{j=0}^{n} f_{j} (a(T))^{j} = f(a(T)), (15.15.7)

with respect to the metric on A_{r_{a}}^{1}[[T]] associated to its norm, by construction.

If t is a real or complex number, as appropriate, with |t| ≤ r_{a}, then a(t) is defined as an element of A as in Section 15.8, with

∥a(t)∥_{A} ≤ ∥a(T)∥_{1,r_{a}}. (15.15.8)

This implies that ∥a(t)∥_{A} ≤ r_{f}, (15.15.9)

because of (15.15.3), so that f(a(t)) may be defined as an element of A as in Section 9.14. Similarly, (f ∘ a)(t) is defined as an element of A as in Section 15.8. One can check that (15.14.2) holds, by approximating f(T) by formal polynomials, for which the analogous statement was mentioned in the previous section.

Suppose for the moment that constant term in a(T) is equal to 0, so that f(a(T)) may be defined as a formal power series in T with coefficients in A as in Section 15.11. One can check that this is equivalent to the definition in (15.15.4), as a formal power series in T, using (15.15.7).

Let r_{a} be a positive real number again, and suppose now that

a(T) ∈ (R)_{r_{a}}^{1}[[T]] or (C)_{r_{a}}^{1}[[T]], (15.15.10)

as appropriate. If a(T) satisfies (15.15.3), then we may define

f(a(T)) ∈ (R)_{r_{a}}^{1}[[T]] or (C)_{r_{a}}^{1}[[T]], (15.15.11) as appropriate, as in Section 9.14 again. As before, (15.15.5) is an algebra homomorphism from (R)_{r_{f}}^{1} [[T]] or (C)_{r_{f}}^{1} [[T]] into (R)_{r_{a}}^{1}[[T]] or (C)_{r_{a}}^{1}[[T]], as appropriate. If f(T) is a formal polynomial in T, then this definition of f(a(T)) is the same as the one in the previous section, as before.

We also get that (15.15.6) holds, where now the left side is the norm of f(a(T)) in (R)_{r_{a}}^{1}[[T]] or (C)_{r_{a}}^{1}[[T]], as appropriate. This implies f(a(T)) may be given as in (15.15.7), as before.

If x ∈ A and ∥x∥_{A} ≤ r_{a}, then a(x) may be defined as an element of A as in Section 9.14, with ∥a(x)∥_{A} ≤ ∥a(T)∥_{1,r_{a}}. (15.15.12)

This means that ∥a(x)∥_{A} ≤ r_{f}, (15.15.13)

by (15.15.3), so that f(a(x)) may be defined as an element of A in the same way. We may define (f ∘ a)(x) as an element of A too, where (f ∘ a)(T) = f(a(T)), as before. One can use (15.15.7) to get that (15.14.3) holds, because of the analogous statement when f(T) is a formal polynomial, as in the previous section. In particular, if t ∈ R or C, as appropriate, and |t| ≤ r_{a}, then (15.14.2) holds, as before.

Part IV

Algebras, norms, and operators, 3 Algebras, norms, and power series

16.1 More on differentiation

Let A be a commutative associative algebra over the real or complex numbers, and let T be an indeterminate. Remember that the space A[[T]] of formal power series in T with coefficients in A is a commutative associative algebra over R or C, as appropriate, as well, as in Section 15.5. If a(T) ∈ A[[T]] and j is a positive integer, then we can define a(T)^{j} as formal power series in T with coefficients in A in the usual way. The derivative of this power series is given by

(a(T)^{j})′ = j a(T)^{j−1} a′(T), (16.1.1)

because of the product rule, as in Section 15.10. Of course, the factor of a(T)^{j−1} on the right is not needed when j = 1.

Suppose now that A also has a multiplicative identity element e_{A}, which may be considered as the multiplicative identity element in A[[T]] too, as in Section 15.5. If f is a formal polynomial in T with real or complex coefficients, as appropriate, then f(a(T)) may be defined as an element of A[[T]], as in Sections 8.13 and 15.11. Note that the derivative f′(T) of f(T) is a formal polynomial in T with coefficients in R or C, as appropriate, so that

f′(a(T)) (16.1.2)

may be defined as an element of A[[T]], as before. It is easy to see that

f(a(T))′ = f′(a(T)) a′(T), (16.1.3)

as in the chain rule, using (16.1.1).

Suppose that the constant term in a(T) is equal to 0, and that f(T) is a formal power series in T with coefficients in R or C, as appropriate. Remember that f(a(T)) may be defined as a formal power series in T with coefficients in A, as in Section 15.11. Similarly, (16.1.2) may be defined as an element of A[[T]]. One can check that (16.1.3) also holds in this case, by approximating f(T) by formal polynomials in T.

16.2 Some norms of derivatives

Let A be an algebra in the strict sense over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}, let T be an indeterminate, and let r be a positive real number. Suppose that a(T), b(T) are elements of the space A_{r}^{1}[[T]] defined in Section 15.7. This implies that

a(T) b(T) ∈ A_{r}^{1}[[T]], (16.2.1)

with ∥a(T) b(T)∥_{1,r} ≤ ∥a(T)∥_{1,r} ∥b(T)∥_{1,r}, (16.2.2)

as in Section 15.9. Remember that ∥ · ∥_{1,r} = ∥ · ∥_{A_{r}^{1}[[T]]} is as in Section 15.7. We also have that

(a(T) b(T))′ = a′(T) b(T) + a(T) b′(T), (16.2.3)

as in Section 15.10. If a′(T), b′(T) ∈ A_{r}^{1}[[T]], (16.2.4)

then we get that (a(T) b(T))′ ∈ A_{r}^{1}[[T]], (16.2.5)

with

∥(a(T) b(T))′∥_{1,r} ≤ ∥a′(T)∥_{1,r} ∥b(T)∥_{1,r} + ∥a(T)∥_{1,r} ∥b′(T)∥_{1,r}. (16.2.6)

Remember that a(T) ∈ A_{r}^{1}[[T]] when a′(T) ∈ A_{r}^{1}[[T]], as mentioned in Section 15.10. Thus {a(T) ∈ A[[T]] : a′(T) ∈ A_{r}^{1}[[T]]} (16.2.7)

is a subalgebra of A_{r}^{1}[[T]].

Suppose now that A is associative too, and let j be a positive integer. If a′(T) ∈ A_{r}^{1}[[T]], then (a(T)^{j})′ ∈ A_{r}^{1}[[T]], (16.2.8)

as in the preceding paragraph. More precisely, one can check that

∥(a(T)^{j})′∥_{1,r} ≤ j ∥a(T)∥_{1,r}^{j−1} ∥a′(T)∥_{1,r}, (16.2.9)

using induction. This can be obtained from (16.1.1) when A is commutative as well.

Suppose that A has a multiplicative identity element e_{A}, and let f(T) be a formal polynomial with real or complex coefficients. Observe that

f(a(T))′ ∈ A_{r}^{1}[[T]]. (16.2.10)

16.3. MORE ON COMPOSITIONS 361 If ∥a(T)∥_{1.r} ≤ r_{0} (16.2.11) for some positive real number r_{0}, then one can check that ∥f(a(T))′∥_{1,r} ≤ ∥f′(T)∥_{1,r_{0}} ∥a′(T)∥_{1,r}, (16.2.12) where ∥f′(T)∥_{1,r_{0}} is as in Section 15.7. If A is commutative, then this can be obtained from (16.1.3) and the fact that ∥f′(a(T))∥_{1,r} ≤ ∥f′(T)∥_{1,r_{0}}. (16.2.13)

16.3 More on compositions

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1, let T be an indeterminate, and let r_{a}, r_{f} be positive real numbers. Also let a(T) and f(T) be formal power series in T with coefficients in A and R or C, as appropriate, respectively. Suppose that a′(T) ∈ A_{r_{a}}^{1}[[T]], (16.3.1) where a′(T) is as in Section 15.10 and A_{r_{a}}^{1}[[T]] is as in Section 15.7. This implies that a(T) ∈ A_{r_{a}}^{1}[[T]], as mentioned in Section 15.10, and we ask that ∥a(T)∥_{1,r_{a}} ≤ r_{f}, (16.3.2) where ∥a(T)∥_{1,r_{a}} = ∥a(T)∥_{A_{ra}^{1} [[T]]} is as in Section 15.7.

Similarly, suppose that f′(T) ∈ (R)_{r_{f}}^{1} [[T]] or (C)_{r_{f}}^{1} [[T]], (16.3.3) as appropriate, which implies that f(T) is an element of (R)_{r_{f}}^{1} [[T]] or (C)_{r_{f}}^{1} [[T]], as appropriate, as before. This permits us to define f(a(T)) as an element of A_{r_{a}}^{1}[[T]], as in Section 15.15. We can define f′(a(T)) ∈ A_{r_{a}}^{1}[[T]] (16.3.4) in the same way. Note that ∥f′(a(T))∥_{1,r_{a}} ≤ ∥f′(T)∥_{1,r_{f}} , (16.3.5) as before.

Under these conditions, we would like to check that f(a(T))′ ∈ A_{r_{a}}^{1}[[T]], (16.3.6) with ∥f(a(T))′∥_{1,r_{a}} ≤ ∥f′(T)∥_{1,r_{f}} ∥a′(T)∥_{1,r_{a}}. (16.3.7)

This follows from the remarks in the previous section when f(T) is a formal polynomial in T. Otherwise, we can approximate f(T) by its partial sums, which are formal polynomials in T, as follows.

Remember that if f(T) = ∑_{j=0}^{∞} f_{j} T^{j}, as usual, then

∑_{j=0}^{n} f_{j} a(T)^{j} (16.3.8)

converges to f(a(T)) as n → ∞, with respect to the metric on A_{r_{a}}^{1}[[T]] associated to ∥ · ∥_{1,r_{a}}, as in Section 15.15. In particular, this implies that for each nonnegative integer l, the coefficient of T^{l} in (16.3.8) converges to the coefficient of T^{l} in f(a(T)) as n → ∞, as a sequence of real or complex numbers, as appropriate. This means that the coefficients of f(a(T))′ are the same as the limits of the corresponding coefficients of the formal derivative of (16.3.8) as n → ∞.

The analogue of (16.3.7) for formal polynomials implies that

∥( ∑_{j=0}^{n} f_{j} a(T)^{j})′∥_{1,r_{a}} ≤ ∥( ∑_{j=0}^{n} f_{j} T^{j})′∥_{1,r_{f}} ∥a′(T)∥_{1,r_{a}} (16.3.9)

for each n ≥ 0. It is easy to see that

∥( ∑_{j=0}^{n} f_{j} T^{j})′∥_{1,r_{f}} ≤ ∥f′(T)∥_{1,r_{j}} (16.3.10)

for each n, because of the way that the norm is defined. It follows that

∥( ∑_{j=0}^{n} f_{j} a(T)^{j})′∥_{1,r_{a}} ≤ ∥f′(T)∥_{1,r_{j}} ∥a′(T)∥_{1,r_{a}} (16.3.11)

for each n. One can use this to get the desired properties of f(a(T))′, because of the remarks in the preceding paragraph.

In fact, the formal derivative of (16.3.8) converges to f(a(T))′ with respect to the metric on A_{r_{a}}^{1}[[T]] associated to ∥ · ∥_{1,r_{a}}. This can be obtained from (16.3.7), with f(T) replaced with

f(T) − ∑_{j=0}^{n} f_{j} T^{j} = ∑_{j=n+1}^{∞} f_{j} T^{j}. (16.3.12)

If A is commutative, then

f(a(T))′ = f′(a(T)) a′(T). (16.3.13)

This was mentioned in Section 16.1 when f(T) is a formal polynomial in T, and when the constant term in a(T) is equal to 0. Here one can reduce to the case of formal polynomials by approximating f(T) by its partial sums again.

16.4 Some differential equations

Let T be an indeterminate, and let a(T), b(T) be formal power series in T with coefficients in the real or complex numbers. Consider the ordinary differential equation f′(T) = a(T) f(T) + b(T), (16.4.1)

where f(T) is another formal power series in T with coefficients in R or C. It is easy to see that there is a unique solution to this equation with a prescribed constant term. Indeed, (16.4.1) determines the coefficients of f(T) after the constant term in terms of the previous coefficients of f(T) and the coefficients of a(T) and b(T).

The exponential function can be defined as a formal power series in T by

exp(T) = ∑_{j=0}^{∞}(1/j!) T^{j}, (16.4.2)

as in Section 10.4. Of course,

exp′(T) = exp(T), (16.4.3)

as usual. This and the condition that the constant term in (16.4.2) be equal to 1 determines exp(T) uniquely, as in the preceding paragraph.

Similarly, consider

log(1 + T) = ∑_{j=1}^{∞} \frac{(−1)^{j+1}}{j} T^{j}, (16.4.4)

which is the formal power series corresponding to the natural logarithm of 1+T. Observe that

log′(1 + T) = ∑_{j=1}^{∞}(−1)^{j+1} T^{j−1} = ∑_{j=0}^{∞}(−1)^{j} T^{j} = (1 + T)^{−1}, (16.4.5)

where the third step is as in Section 15.6. As before, (16.4.4) is uniquely determined by this and the condition that the constant term be equal to 0.

Because the constant term in (16.4.4) is equal to 0,

f(T) = exp(log(1 + T)) (16.4.6)

may be defined as a formal power series in T as in Section 15.11. Of course, we should have that f(T) = 1 + T. (16.4.7)

This could be obtained from the usual properties of the exponential and logarithm functions on the real line and the set R_{+} of positive real numbers, respectively.

Alternatively,

(16.4.8) f′(T) = exp′(log(1 + T)) log′(1 + T)
= exp(log(1 + T)) (1 + T)^{−1} = f(T) (1 + T)^{−1},

where the first step is as in Section 16.1. The right side of (16.4.7) satisfies the same differential equation. This implies (16.4.7), because both sides have the same constant term.

16.5 More on log(1 + T)

Let T be an indeterminate, and let r be a positive real number strictly less than 1. It is easy to see that log(1 + T) ∈ (R)_{r}^{1}[[T]], (16.5.1)

in the notation of Section 15.7. More precisely,

(16.5.2) ∥ log(1 + T)∥_{1,r} = ∥ log(1 + T)∥_{(R)_{r}^{1}[[T]]} = ∑_{j=1}^{∞} \frac{r^{j}}{j} = − log(1 − r),

using the standard power series representation for log(1−t) when t ∈ R satisfies |t| < 1 in the last step.

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. If x ∈ A and ∥x∥_{A} ≤ r, then we put

log(e_{A} + x) = log_{A}(e_{A} + x) = ∑_{j=1}^{∞} \frac{(−1)^{j+1}}{j} x^{j}, (16.5.3)

as in Sections 9.14 and 15.11. This is an element of A with

(16.5.4) ∥ log(e_{A} + x)∥_{A} ≤ ∥ log(1 + T)∥_{1,r} = − log(1 − r),

as before.

Note that exp(T) ∈ (R)_{r_{1}}^{1}[[T]] (16.5.5)

for every positive real number r_{1}, with

∥ exp(T)∥_{1,r_{1}} = ∑_{j=0}^{∞} \frac{(r_{1})^{j}}{j!} = exp(r_{1}). (16.5.6)

Remember that the exponential function on A was discussed in Section 10.4. In fact, we have that

exp(log(e_{A} + x)) = e_{A} + x (16.5.7) under these conditions. This uses (16.4.7) and the remarks in Section 15.15. This is related to part (b) of Theorem 10.30 on p246 of [286].

Similarly, let A be any associative algebra over the real or complex numbers with a multiplicative identity element e_{A}, and suppose that x ∈ A is nilpotent. This implies that log(e_{A} + x) may be defined as an element of A as in (16.5.3), as in Section 15.11. In fact,

(16.5.8) log(e_{A} + x) is nilpotent in A,

as in Section 15.13. It follows that

exp(log(e_{A} + x)) (16.5.9)

may be defined as an element of A too, as in Sections 11.1 and 15.11. We also have that (16.5.7) holds, as in Section 15.13 again.

16.6 Some more logarithms

Let T be an indeterminate, and consider the formal power series

exp(−T) = ∑_{j=0}^{∞} \frac{(−1)^{j}}{j!} T^{j}. (16.6.1)

This is the same as formally composing −T with exp(T). One can check that

(16.6.2) exp(−T) exp(T) = exp(T) exp(−T) = 1,

using the binomial theorem, as in Section 10.4. This means that exp(−T) is the multiplicative inverse of exp(T) in the algebra of formal power series in T with coefficients in R, as in Section 15.6.

Let a(T) be a formal power series in T with real or complex coefficients, and suppose that the constant term in a(T) is equal to 0. This implies that

f(T) = log(1 + a(T)) = ∑_{j=1}^{∞} \frac{(−1)^{j+1}}{j} a(T)^{j} (16.6.3)

may be defined as a formal power series in T with coefficients in R or C, as appropriate, as in Section 15.11. Note that the constant term in (16.6.3) is equal to 0 too. We also have that

f′(T) = a′(T) (1 + a(T))^{−1}, (16.6.4)

as in Section 16.1. In fact, (16.6.3) is uniquely determined by these two properties.

In particular, we can take

a(T) = exp(T) − 1 = ∑_{l=1}^{∞}(1/l!) T^{l}. (16.6.5)

Note that a′(T) = exp′(T) = exp(T). (16.6.6)

Thus f′(T) = exp(T) exp(T)^{−1} = 1, (16.6.7)

as in (16.6.4). It follows that f(T) = T, (16.6.8)

because the constant term in f(T) is equal to 0 as before. This means that

(16.6.9) log(exp(T)) = log(1 + (exp(T) − 1)) = T,

where the first step may be considered as the definition of the left side.

If r_{1} is a positive real number, then exp(T) − 1 ∈ (R)_{r_{1}}^{1}[[T]], as in (16.5.5). More precisely,

∥ exp(T) − 1∥_{1,r_{1}} = ∑_{l=1}^{∞}(1/l!) r_{1}^{l} = exp(r_{1}) − 1. (16.6.10)

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. If x ∈ A, then

∥exp(x) − e_{A}∥_{A} ≤ exp(∥x∥_{A}) − 1, (16.6.11)

as in Sections 9.14 and 15.11. If

exp(∥x∥_{A}) − 1 < 1, (16.6.12)

then we would like to check that

log(exp(x)) = x. (16.6.13)

More precisely, the left side is interpreted as being

log(e_{A} + (exp(x) − e_{A})), (16.6.14)

as defined as in the previous section.

Put r = exp(∥x∥_{A}) − 1, (16.6.15)

so that r < 1, by hypothesis. Remember that log(1 + T) ∈ R)_{r}^{1}[[T]], as in (16.5.1). One can get (16.6.14) using (16.6.9) and the remarks in Section 15.15. This also uses (16.6.10), with r_{1} = ∥x∥_{A}.

Now let A be any associative algebra over the real or complex numbers witha multiplicative identity element e_{A}, and suppose that x ∈ A is nilpotent. This means that exp(x) can be defined as an element of A as in Sections 11.1 and 15.11, and that (16.6.16) exp(x) − e_{A} is nilpotent in A,

as in Section 15.13. If we interpret log(exp(x)) as in (16.6.14), then this can be defined as an element of A as in Section 15.11. Under these conditions, (16.6.13) follows from (16.6.9), as in Section 15.13.

16.7 A basic type of composition

Let T be an indeterminate, and let a(T) = ∑_{l=0}^{∞} a_{l} T^{l} be a formal power series in T with coefficients in the real or complex numbers. Also let A be an associative algebra over the real or complex numbers, as appropriate, with a multiplicative identity element e_{A}. If x ∈ A, then

a_{x}(T) = ∑_{l=0}^{∞} a_{l} x^{l} T^{l} (16.7.1)

is a formal power series in T with coefficients in A. This may also be expressed as a(x T), which may be considered as the formal power series in T with coefficients in A obtained by composing x T with a(T) as in Section 15.11. Note that

a(T) ↦ a_{x}(T) = a(x T) (16.7.2)

is an algebra homomorphism from R[[T]] or C[[T]], as appropriate, into A[[T]], as before.

Suppose for the moment that a(T) is a formal polynomial in T, so that a_{x}(T) is a formal polynomial in T as well. In this case, a_{x}(t) and a(t x) are defined as elements of A for every t ∈ R or C, as appropriate, and it is easy to see that

a_{x}(t) = a(t x). (16.7.3)

Similarly, if a(T) is a formal power series in T, and x ∈ A is nilpotent, then

a_{x}(T) = a(x T) ∈ R[T] or C[T], (16.7.4)

as appropriate. If t ∈ R or C, then t x is nilpotent in A too, so that a(t x) may be defined as an element of A, as in Section 15.11. Under these conditions, a_{x}(t) defines an element of A as well, and (16.7.3) holds.

Now let (A, ∥·∥_{A}) be a Banach algebra over the real or complex numbers, as appropriate, with a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1. Suppose that x ∈ A has the property that

∑_{l=0}^{∞} |a_{l}| ∥x^{l}∥_{A} (16.7.5)

converges as an infinite series of nonnegative real numbers. In this case, we can put

a(x) = ∑_{l=0}^{∞} a_{l} x^{l}, (16.7.6)

where the series on the right converges in A, because it converges absolutely.

Observe that a_{x}(T) ∈ A_{1}^{1}[[T]], (16.7.7) in the notation of Section 15.7, with

∥a_{x}(T)∥_{1,1} = ∥a_{x}(T)∥_{A_{1}^{1}[[T]]} = ∑_{l=0}^{∞} |a_{l}| ∥x^{l}∥_{A}. (16.7.8)

If t ∈ R or C, as appropriate, with |t| ≤ 1, then

a_{x}(t) = ∑_{l=0}^{∞} a_{l} t^{l} x^{l} (16.7.9)

defines an element of A, as in Section 15.8. Of course,

a_{x}(1) = a(x), (16.7.10)

by construction.

16.8 Some more compositions

Let us return to the same notation and hypotheses as at the beginning of the previous section. Suppose for the moment that f(T) is a formal polynomial in T with coefficients in the real or complex numbers, as appropriate. Remember that (f ∘ a)(T) = f(a(T)) may be defined as a formal power series in T with coefficients in R or C, as appropriate, as in Sections 15.11 and 15.12. Thus

(f ∘ a)_{x}(T) = (f ∘ a)(x T) (16.8.1)

may be defined as a formal power series in T with coefficients in A, as in the previous section. We can also define

(f ∘ a_{x})(T) = f(a_{x}(T)) (16.8.2)

as a formal power series in T with coefficients in A, as in Sections 15.11 and 15.12. It is easy to see that (16.8.1) and (16.8.2) are the same. We may express this formal power series as f(a(x T)). (16.8.3)

The type of formal composition mentioned in the previous section may be considered as a version of one discussed in Section 15.13. This means that the equality of (16.8.1) and (16.8.2) is basically an associativity property of formal compositions like these, which were discussed previously.

Suppose for the moment again that the constant term in a(T) is equal to 0, and let f(T) be a formal power series in T with coefficients in R or C, as appropriate. In this case, (f ∘a)(T) = f(a(T)) may be defined as a formal power series in T with coefficients in R or C, as appropriate, as in Sections 15.11 and 15.13. This means that (16.8.1) may be defined as a formal power series in T with coefficients in A, as in the previous section again. Note that

(16.8.4) the constant term in a_{x}(T) = a(x T) is equal to 0, so that (16.8.2) may be defined as a formal power series in T with coefficients in A as in Sections 15.11 and 15.13. One can check that (16.8.1) and (16.8.2) are the same under these conditions too. This can be seen by approximating f(T) by formal polynomials in T, as in Section 15.11. This may also be considered as an instance of an associativity property of formal compositions like these, as in Section 15.13. The resulting formal power series may be expressed as in (16.8.3), as before.

Suppose for the moment that a(T) and f(T) are formal polynomials in T, so that (f ∘ a)(T), a_{x}(T), and (f ∘ a_{x})(T) (16.8.5)

are formal polynomials in T too. If t ∈ R or C, as appropriate, then

(f ∘ a_{x})(t) = (f ∘ a)_{x}(t) = (f ∘ a)(t x) = f(a(t x)), (16.8.6)

where the second step is as in (16.7.3), and the third step is as in Section 15.12. Alternatively, (f ∘ a_{x})(t) = f(a_{x}(t)) = f(a(t x)), (16.8.7)

where the first step is as in Section 15.12, and the second step is as in (16.7.3).

Let a(T) be a formal power series in T again, and suppose that x ∈ A is nilpotent, so that a_{x}(T) is a formal polynomial in T, as in (16.7.4). If f(T) is a formal polynomial in T, then (f ∘ a)(T) is a formal power series in T, and

(16.8.8) (f ∘ a_{x})(T), (f ∘ a)_{x}(T) are formal polynomials in T.

If t ∈ R or C, as appropriate, then t x is nilpotent in A, a(t x) is defined as an element of A, and (16.8.6) holds, where the second step is as in the previous section, and the third step is as in Section 15.12. Alternatively, (16.8.7) holds, where the first step is as in Section 15.12, and the second step is as in the previous section.

Suppose that the constant term in a(T) is equal to 0. If x ∈ A is nilpotent, then it is easy to see that

(16.8.9) a_{x}(T) = a(x T) is nilpotent in A[T].

If f(T) is a formal power series in T again, then it follows that

(f ∘ a_{x})(T) = f(a_{x}(T)) ∈ A[T]. (16.8.10)

More precisely, f(a_{x}(T)) may be defined initially as a formal power series in T, because of (16.8.4), as in Section 15.11. It is easy to see that (16.8.10) holds in this case, because of (16.8.9). However, one can also define f(a_{x}(T)) as an element of A[T] more directly, because of (16.8.9), as in Section 15.11. One can check that these two ways of defining f(a(T)) are equivalent.

Note that (f ∘ a)(T) is defined as a formal power series in T, and that (f ∘ a)_{x}(T) is a formal polynomial in T, as before. If t ∈ R or C, then t x is nilpotent in A, and a(t x) is nilpotent as well, because the constant term in a(T) is equal to 0, as in Section 15.13. Thus f(a(t x)) is defined as an element of A, as in Section 15.11. Under these conditions, (16.8.6) holds, where the second step is as in the previous section again, and the third step is as in Section 15.13.

To get the first step in (16.8.7), one can approximate f(T) by a formal polynomial, because of (16.8.9). The analogous statement for formal polynomials was mentioned in Section 15.12, as before. The second step in (16.8.7) is as in the previous section, as before.

16.9 Some compositions with convergence

Let (A, ∥·∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1, and let T be an indeterminate. Also let a(T) = ∑_{l=0}^{∞} a_{l} T^{l} be a formal power series in T with coefficients in R or C, as appropriate, let x be an element of A, and suppose that ∑_{l=0}^{∞} |a_{l}| ∥x^{l}∥_{A} converges, as an infinite series of nonnegative real numbers. Thus

a_{x}(T) = ∑_{l=0}^{∞} a_{l} x^{l} T^{l} ∈ A_{1}^{1}[[T]], (16.9.1)

in the notation of Section 15.7, as in Section 16.7.

Let f(T) be a formal polynomial in T with coefficients in R or C, as appropriate. Observe that f(a_{x}(T)) ∈ A_{1}^{1}[[T]], (16.9.2)

as in Section 15.14. This may be expressed as (f ∘ a_{x})(T), as in (16.8.2). If t ∈ R or C, as appropriate, and

|t| ≤ 1, (16.9.3)

then a_{x}(t) and (f ∘ a_{x})(t) may be defined as elements of A, as in Section 15.8. This implies that f(a_{x}(t)) may be defined as an element of A, and

(f ∘ a_{x})(t) = f(a_{x}(t)), (16.9.4)

as in Section 15.14.

Remember that (f ∘ a)(T) = f(a(T)) is a formal power series in T with coefficients in R or C, as appropriate, which may be expressed as

(f ∘ a)(T) = ∑_{m=0}^{∞}(f ∘ a)_{m} T^{m}. (16.9.5)

We also have that (f ∘ a_{x})(T) = f(a_{x}(T)) is a formal power series in T with coefficients in A, which may be expressed as

(f ∘ a_{x})(T) = (f ∘ a)_{x}(T) = ∑_{m=0}^{∞}(f ∘ a)_{m} x^{m} T^{m}, (16.9.6)

where the first step is as in the previous section. Note that

∑_{m=0}^{∞} |(f ∘ a)_{m}| ∥x^{m}∥_{A} (16.9.7)

converges as an infinite series of nonnegative real numbers, as in (16.9.2). If t ∈ R or C, as appropriate, satisfies (16.9.3), then we get that

f(a_{x}(t)) = (f ∘ a_{x})(t) = ∑_{m=0}^{∞}(f ∘ a)_{m} t^{m} x^{m}, (16.9.8)

using (16.9.4) in the first step. Of course, the right side converges absolutely in this case, because (16.9.7) converges.

16.10 Some more convergence conditions

Let us return to the same notation and hypotheses as at the beginning of the previous section. Now let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} be a formal power series in T with coefficients in R or C, as appropriate, with

f(T) ∈ (R)_{r_{f}}^{1} [[T]] or (C)_{r_{f}}^{1} [[T]], (16.10.1)

as appropriate, for some positive real number r_{f}, in the notation of Section 15.7. Suppose that

∥a_{x}(T)∥_{1,1} = ∥a_{x}(T)∥_{A_{1}^{1}[[T]]} = ∑_{l=0}^{∞} |a_{l}| ∥x^{l}∥_{A} ≤ r_{f}. (16.10.2)

Under these conditions, we may define

f(a_{x}(T)) ∈ A_{1}^{1}[[T]], (16.10.3)

as in Section 15.15. We also have that

∥f(a_{x}(T))∥_{1,1} ≤ ∥f(T)∥_{1,r_{f}} , (16.10.4)

as before. Remember that the left side is the norm of f(a_{x}(T)) in A_{1}^{1}[[T]], and that the right side is the norm of f(T) in (R)_{r_{f}}^{1} [[T]] or (C)_{r_{f}}^{1} [[T]], as appropriate. We may express f(a_{x}(T)) as (f∘a_{x})(T), as before. If f(T) is a formal polynomial in T, then this is equivalent to the definition of f(a(T)) as in the previous section, as mentioned in Section 15.15.

If t ∈ R or C, as appropriate, and |t| ≤ 1, then a_{x}(t) may be defined as an element of A as in Section 15.8, with

∥a_{x}(t)∥_{A} ≤ ∥a_{x}(T)∥_{1,1}. (16.10.5)

It follows that ∥a_{x}(t)∥_{A} ≤ r_{f}, (16.10.6) because of (16.10.2), so that f(a_{x}(t)) may be defined as an element of A as in Section 9.14. Similarly, (f ∘ a_{x})(t) may be defined as an element of A as in Section 15.8. In fact, we have that

(f ∘ a_{x})(t) = f(a_{x}(t)), (16.10.7)

as in Section 15.15.

Suppose that the constant term in a(T) is equal to 0, so that (f ∘ a)(T) = f(a(T)) may be defined as a formal power series in T with coefficients in R or C, as appropriate, as in Sections 15.11 and 15.13. This implies that the constant term in a_{x}(T) is equal to 0, so that (f ∘ a_{x})(T) = f(a_{x}(T)) may be defined as a formal power series in T with coefficients in A, as in Sections 15.11 and 15.13 again. This is the same as in (16.10.3), as a formal power series in T, as mentioned in Section 15.15. Remember that (f ∘ a)_{x}(T) may be defined as a formal power series in T with coefficients in A as in Section 16.7. We also have that (f ∘ a)_{x}(T) = (f ∘ a_{x})(T), (16.10.8)

as in Section 16.8.

If (f ∘a)(T) is as in (16.9.5), then (f ∘a_{x})(T) is as in (16.9.6), as before. This implies that (16.9.7) converges as an infinite series of nonnegative real numbers, because of (16.10.3). In fact,

∑_{m=0}^{∞} |(f ∘ a)_{m}| ∥x^{m}∥_{A} = ∥(f ∘ a_{x})(T)∥_{1,1} ≤ ∥f(T)∥_{1,r_{j}}, (16.10.9)

as in (16.10.4). If t ∈ R or C, as appropriate, and |t| ≤ 1, then (16.9.8) holds, where the first step is as in (16.10.7).

16.11 Some more exponentials of logarithms

Let T be an indeterminate, and let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}. Put

a(T) = log(1 + T) = ∑_{l=1}^{∞} \frac{(−1)^{l+1}}{l} T^{l}, (16.11.1)

as in Section 16.4. If x ∈ A, then

a_{x}(T) = a(x T) = log(1 + x T) = ∑_{l=1}^{∞} \frac{(−1)^{l+1}}{l} x^{l} T^{l} (16.11.2)

is a formal power series in T with coefficients in A, as in Section 16.7. Note that the constant term in (16.11.2) is equal to 0, so that

exp(a_{x}(T)) (16.11.3)

16.11. SOME MORE EXPONENTIALS OF LOGARITHMS 373 may be defined as a formal power series in T with coefficients in A as in Section 15.11. Remember that exp(log(1 + T)) = 1 + T, (16.11.4) as in Section 16.4. Using this, we get that exp(a_{x}(T)) = 1 + x T. (16.11.5) as in Section 16.8.

Suppose now that (A, ∥ · ∥_{A}) is a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1, and that x ∈ A has the property that

∑_{l=1}^{∞}(1/l) ∥x^{l}∥_{A} (16.11.6)

converges as an infinite series of nonnegative real numbers. This means that a_{x}(T) ∈ A_{1}^{1}[[T]], (16.11.7) in the notation of Section 15.7, with

∥a_{x}(T)∥_{1,1} = ∥a_{x}(T)∥_{A_{1}^{1}[[T]]} = ∑_{l=1}^{∞}(1/l) ∥x^{l}∥_{A}. (16.11.8)

Remember that f(T) = exp(T) ∈ (R)_{r_{1}}^{1}[[T]] (16.11.9) for every r_{1} > 0, as in Section 16.5. Thus (f ∘ a_{x})(T) = f(a_{x}(T)) = exp(a_{x}(T)) ∈ A_{1}^{1}[[T]] (16.11.10) may be defined as in Section 15.15. This is the same as (16.11.5), as a formal power series in T with coefficients in A, as before.

If t ∈ R or C, as appropriate, and |t| ≤ 1, then a_{x}(t) may be defined as an element of A as in Section 15.8. This means that f(a_{x}(t)) = exp(a_{x}(t)) (16.11.11) may be defined as an element of A too, as in Sections 9.14 and 10.4. Under these conditions, we have that exp(a_{x}(t)) = f(a_{x}(t)) = (f ∘ a_{x})(t) = e_{A} + t x, (16.11.12) as in the previous section. We may consider a_{x}(t) as a definition of the logarithm of e_{A} + t x in A in this case, as in Section 16.5. This is related to part (b) of Theorem 10.30 on p246 of [286].

16.12 The binomial theorem

Let n be a positive integer, and let

\binom{n}{j} = \frac{n!}{j! (n − j)!} (16.12.1)

be the usual binomial coefficient for j = 0, 1, …, n. Here “l factorial” l! is defined in the usual way for each positive integer l, and is interpreted as being equal to 1 when l = 0, so that \binom{0}{0} is interpreted as being equal to 1 as well. It is well known that

(x + y)^{n} = ∑_{j=0}^{n} \binom{n}{j}x^{j} y^{n−j} (16.12.2)

when x and y are commuting elements of an associative algebra A over the real or complex numbers, or a ring, for that matter, by the binomial theorem. If A has a multiplicative identity element e_{A}, then both sides of the equation may be interpreted as being equal to e_{A} when n = 0.

It is easy to see that (x + y)^{n} can be expressed as a sum of 2^{n} terms, each of which is of the form x^{j} y^{n−j} (16.12.3)

for some j. Thus (x + y)^{n} may be expressed as a sum of terms of this form with positive integer coefficients. The coefficients do not depend on x, y, or A, and are the same as the number of subsets of {1, …, n} with j elements for each j.

The expression (16.12.1) for these coefficients can be verified by induction. Alternatively, one can take x, y ∈ R, and consider the derivatives of (x + y)^{n} in x or y. More precisely, one can take y = 1, and consider the derivatives of (x + 1)^{n} at x = 0.

In particular, the binomial coefficients are uniquely determined by the condition that

(x + 1)^{n} = ∑_{j=0}^{n} \binom{n}{j} x^{j} (16.12.4)

on R. In fact, \binom{n}{j} is uniquely determined for j = 0, 1, …, n by the condition that (16.12.4) holds for n + 1 elements x of R, because (x + 1)^{n} is a polynomial of degree n.

Let T be an indeterminate, and for each positive integer l, consider the formal polynomial

\binom{T}{l} = \frac{T (T − 1) ⋯ (T − l + 1)}{l!} (16.12.5)

of degree l in T with rational coefficients, as in Exercise 8 on p74 of [62]. If l = 0, then this is interpreted as the constant polynomial equal to 1. This corresponds to the polynomial function

\binom{t}{l} = \frac{t (t − 1) ⋯ (t − l + 1)}{l!} (16.12.6)

of t in R, C, or any commutative associative algebra over the real or rational numbers with a multiplicative identity element. If t ≥ l is an integer, then this is the same as the usual binomial coefficient, as before. If t < l is a nonnegative integer, then this is equal to 0.

If t_{1} and t_{2} are nonnegative integers, then

(x + 1)^{t_{1}+t_{2}} = (x + 1)^{t_{1}} (1 + x)^{t_{2}} (16.12.7)

for every x ∈ R. One can check that

\binom{t_{1} + t_{2}}{l} = ∑_{j=0}^{l} \binom{t_{1}}{j} \binom{t_{2}}{l − j} (16.12.8)

for every nonnegative integer l, using the appropriate binomial expansions for both sides of (16.12.7).

If t_{2} is fixed, then both sides of (16.12.8) are polynomials in t_{1}. It follows that (16.12.8) holds for all t_{1} ∈ R, because it holds for all nonnegative integers t_{1}, as before.

If t_{1} ∈ R is fixed, then both sides of (16.12.8) are polynomials in t_{2} with real coefficients. This implies that (16.12.8) holds for all t_{2} ∈ R, because it holds for all nonnegative integers t_{2}, as in the preceding paragraph.

Thus (16.12.8) holds for all t_{1}, t_{2} ∈ R. The same argument could be used to get that (16.12.8) holds for all t_{1}, t_{2} ∈ C.

Let T_{1}, T_{2} be commuting indeterminates, and observe that

\binom{T_{1} + T_{2}}{l} (16.12.9)

defines a formal polynomial in T_{1}, T_{2} with rational coefficients, as in Section 13.3. In fact,

\binom{T_{1} + T_{2}}{l} = ∑_{j=0}^{l} \binom{T_{1}}{j} \binom{T_{2}}{l − j}, (16.12.10)

as formal polynomials in T_{1}, T_{2} for every nonnegative integer l. This means that the coefficients of the monomials in T_{1}, T_{2} on both sides of the equation are the same.

Indeed, the corresponding polynomial functions on R^{2} are the same, as in (16.12.8). To get that the coefficients are the same, one can consider the derivatives of these polynomials at (0, 0). This corresponds to part (i) of Exercise 8 on p74 of [62]. It follows that (16.12.8) holds for all t_{1}, t_{2} in any commutative associative algebra over the real or rational numbers with a multiplicative identity element.

16.13 Binomial series

Let X be an indeterminate. If α is a real or complex number, then

b_{α}(X) = ∑_{j=0}^{∞} \binom{α}{j} X^{j} (16.13.1)

is a formal power series in X with coefficients in R or C, as appropriate. This is the binomial series in X associated to α. If α is a nonnegative integer, then

b_{α}(X) = (1 + X)^{α}, (16.13.2)

as before. Otherwise, this may be considered as a definition of the right side. If β is another real or complex number, then

b_{α+β}(X) = b_{α}(X) b_{β}(X). (16.13.3)

This follows from (16.12.8), with t_{1} = α and t_{2} = β. This means that

(1 + X)^{α+β} = (1 + X)^{α} (1 + X)^{β}, (16.13.4)

using the notation in (16.13.2).

Observe that

\binom{−1}{l} = (−1)^{l} (16.13.5)

for each nonnegative integer l. This implies that (16.13.2) is the same as the usual definition of (1 + X)^{−1}, as in Section 15.6, when α = −1. Similarly, if α is a negative integer, then (1 + X)^{−α} is a formal polynomial in X, and (1 + X)^{α} may be defined as its multiplicative inverse in R[[X]]. One can check that this is the same as (16.13.2), using (16.13.4).

Of course,

b′_{α}(X) = ∑_{j=1}^{∞} j \binom{α}{j} X^{j−1} = ∑_{j=0}^{∞}(j + 1) \binom{α}{j + 1} X^{j}, (16.13.6)

as in Section 15.10. If l is a positive integer, then it is easy to see that

l \binom{α}{l} = α \binom{α − 1}{l − 1}. (16.13.7)

This implies that

b′_{α}(X) = ∑_{j=0}^{∞} α \binom{α − 1}{j} X^{j} = α b_{α−1}(X). (16.13.8)

If l is a nonnegative integer, then

\binom{α}{l} = ∑_{j=0}^{l} \binom{1}{j} \binom{α − 1}{l − j}, (16.13.9)

by (16.12.8), with t_{1} = 1, and t_{2} = α − 1. This means that

\binom{α}{l} = \binom{α − 1}{l} + \binom{α − 1}{l − 1} (16.13.10)

when l ≥ 1.

It is easy to see that

(1 + X) b′_{α}(X) = ∑_{j=0}^{∞}(j + 1) \binom{α}{j + 1} X^{j} + ∑_{j=1}^{∞} j \binom{α}{j} X^{j}. (16.13.11)

It follows that

(1 + X) b′_{α}(X) = ∑_{j=0}^{∞} α \binom{α − 1}{j} X^{j} + ∑_{j=1}^{∞} α \binom{α − 1}{j − 1} X^{j}, (16.13.12)

because of (16.13.7). One can use this and (16.13.10) to obtain that

(1 + X) b′_{α}(X) = α b_{α}(X), (16.13.13)

which is another way to look at (16.13.8). In fact, b_{α}(X) is uniquely determined by (16.13.13) and the condition that its constant term be equal to 1, as in Section 16.4.

Observe that f_{α}(X) = exp(α log(1 + X)) (16.13.14)

defines a formal power series in X with real or complex coefficients, as appropriate, where log(1 + X) is as in Section 16.4. More precisely, the constant term in log(1 + X) is equal to 0, so that the formal composition may be defined as in Section 15.11. We also have that

f′_{α}(X) = exp′(α log(1 + X)) α log′(1 + X)
(16.13.15)  = exp(α log(1 + X)) α (1 + X)^{−1} = α (1 + X)^{−1} f_{α}(X),

where the first step is as in Section 16.1, and the second step is as in Section 16.4. It is easy to see that the constant term in f_{α}(X) is equal to 1, because the constant term in log(1 + X) is equal to 0. It follows that

b_{α}(X) = f_{α}(X), (16.13.16)

as in the preceding paragraph.

16.14 Using the ratio test

If α is a real or complex number that it not a nonnegative integer, then it is easy to see that

\binom{α}{l} ≠ 0 (16.14.1)

for each nonnegative integer l. In this case,

\binom{α}{l} \binom{α}{l + 1}^{−1} = \frac{α − l}{l + 1} (16.14.2)

for every l ≥ 0. This implies that

lim_{l→∞} (\binom{α}{l} \binom{α}{l + 1}^{−1}) = 1, (16.14.3)

by a standard argument. If r is a nonnegative real number with r < 1, then it follows that

∑_{j=0}^{∞}|\binom{α}{j}| r^{j} (16.14.4)

converges as an infinite series of nonnegative real numbers, by the ratio test. If X is an indeterminate, then we get that

b_{α}(X) ∈ (R)_{1}[[X]] or (C)_{1}[[X]] (16.14.5)

when α ∈ R or C, respectively, in the notation of Section 15.7.

If x is a real or complex number with |x| < 1, then put

b_{α}(x) = ∑_{j=0}^{∞} \binom{α}{j}x^{j}, (16.14.6)

where the series on the right converges absolutely, by the comparison test. Of course, if α is a nonnegative integer, then the right side reduces to a finite sum. If β is another real or complex number, then

b_{α+β}(x) = b_{α}(x) b_{β}(x), (16.14.7)

because of (16.13.3), as in Section 9.10.

It is well known that b_{α}(x) = (1 + x)^{α} (16.14.8)

under these conditions, where the right side can be defined in other ways. One can use the principal branch of the complex logarithm to define the right side by exp(α log(1 + x)), (16.14.9)

which is a holomorphic function on a larger open set in the complex plane. The power series expansion for b_{α}(x) corresponds exactly to the Taylor series for this function at x = 0.

Of course, b_{α}(x) is a holomorphic function on the open unit disk, because it is defined by an absolutely convergent power series there. We have that

b′_{α}(x) = α b_{α−1}(x) (16.14.10) on the open unit disk, because of (16.13.8). Alternatively, we have that

(1 + x) b′_{α}(x) = α b_{α}(x) (16.14.11)

on the open unit disk, as in (16.13.13).

Similarly, we may consider b_{α}(x) as a smooth function on the open interval (−1, 1) in the real line. One can obtain (16.14.8) from (16.14.11), as in Exercise 22 on p201 of [282].

16.15 Some related estimates

If α is a real number and α ≤ 0, then it is easy to see that

(−1)^{l} \binom{α}{l} ≥ 0 (16.15.1)

for every nonnegative integer l. If α is any real or complex number, then

|α − l| ≤ |α| + l = |−|α| − l| (16.15.2)

for every nonnegative integer l. This implies that

|\binom{α}{l}| ≤ |\binom{−|α|}{l}| = (−1)^{l} \binom{−|α|}{l} (16.15.3)

for every nonnegative integer l.

If r is a nonnegative real number with r < 1, then we get that

∑_{l=0}^{∞}|\binom{α}{l}| r^{l} ≤ ∑_{l=0}^{∞}|\binom{−|α|}{l}| r^{l} = ∑_{l=0}^{∞}(−1)^{l} \binom{−|α|}{l} r^{l}. (16.15.4)

Remember that these series converge, as in the previous section.

Of course,

∑_{l=0}^{∞}(−1)^{l} \binom{−|α|}{l} r^{l} = b_{−|α|}(−r) = (1 − r)^{−|α|}, (16.15.5)

using (16.14.8) in the second step. Thus

∑_{l=0}^{∞}|\binom{α}{l}| r^{l} ≤ (1 − r)^{−|α|}, (16.15.6)

with equality when α is a real number with α ≤ 0.

Let X be an indeterminate, so that b_{α}(X) may be defined as a formal power series in X with real or complex coefficients as in (16.13.1). In addition to (16.14.5), we get that ∥b_{α}(X)∥_{1,r} ≤ (1 − r)^{−|α|}, (16.15.7)

in the notation of Section 15.7, because of (16.15.6). We also have equality when α is a real number with α ≤ 0, as before.

Norms, power series, and involutions

17.1 Another convergence property

Let α be a real number, so that \binom{α}{l} ∈ R for every nonnegative integer l. Suppose that 0 < α < 1. (17.1.1)

It is easy to see that

(−1)^{l+1} \binom{α}{l} > 0 (17.1.2)

for every l ≥ 1. This corresponds to part (a) of Exercise (1) on p125 of [11]. If x ∈ R and |x| < 1, then

b_{α}(−x) = ∑_{j=0}^{∞} \binom{α}{j} (−x)^{j} = 1 − ∑_{j=1}^{∞}(−1)^{j+1} \binom{α}{j} x^{j}. (17.1.3)

This means that

1 − ∑_{j=1}^{∞}(−1)^{j+1} \binom{α}{j} x^{j} = (1 − x)^{α}, (17.1.4)

because of (16.14.8). This is closer to the formulation in [11].

We can use this to get that

∑_{j=1}^{∞}(−1)^{j+1} \binom{α}{j} = 1, (17.1.5)

as in part (b) of Exercise (1) on p125 of [11]. Indeed,

∑_{j=1}^{n}(−1)^{j+1} \binom{α}{j} x^{j} ≤ 1 − (1 − x)^{α} ≤ 1 (17.1.6)

for each positive integer n when 0 ≤ x < 1, by (17.1.4). This implies that

∑_{j=1}^{n}(−1)^{j+1} \binom{α}{j} ≤ 1 (17.1.7)

for every n ≥ 1, by taking the limit as x → 1−. This means that

∑_{j=1}^{∞}(−1)^{j+1} \binom{α}{j} ≤ 1, (17.1.8)

which includes the convergence of the series on the left.

If 0 ≤ x < 1, then

1 − (1 − x)^{α} = ∑_{j=1}^{∞}(−1)^{j+1} \binom{α}{j} x^{j} ≤ ∑_{j=1}^{∞}(−1)^{j+1} \binom{α}{j}, (17.1.9)

using (17.1.4) in the first step. It follows that (17.1.5) holds, by taking the limit as x → 1−. Of course, (17.1.5) implies that

∑_{j=0}^{∞}|\binom{α}{j}| (17.1.10)

converges. If X is an indeterminate, then we get that

b_{α}(X) ∈ (R)_{1}^{1}[[X]], (17.1.11)

in the notation of Section 15.7.

If α is a nonnegative integer, then (17.1.10) reduces to a finite sum, and b_{α}(X) ∈ R[T], as before. If α is any nonnegative real number, then one can use (16.13.3) to get that (17.1.10) converges, so that (17.1.11) holds, using the remarks in Section 15.9. In this case, if x is a real or complex number, then one can define b_{α}(x) as a real or complex number as in (16.14.6). This defines a continuous real or complex-valued function on the set of x ∈ R or C, as appropriate, such that |x| ≤ 1, as in Section 9.12. It follows that (16.14.8) holds for all x ∈ R or C with |x| ≤ 1, because of the analogous statement when |x| < 1, and because both sides of the equation are continuous on the set where |x| ≤ 1.

17.2 Binomial series and Banach algebras

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥ = 1, and let α be a real or complex number, appropriate. If x ∈ A and

∥x∥_{A} < 1, (17.2.1) then

b_{α}(x) = ∑_{j=0}^{∞} \binom{α}{j} x^{j} (17.2.2)

defines an element of A, as in Section 9.14, because of (16.14.5). If α is a nonnegative real number, then this works when

∥x∥_{A} ≤ 1, (17.2.3)

because of (17.1.11).

Let β be another real or complex number, as appropriate. If (17.2.1) holds, then b_{α+β}(x) = b_{α}(x) b_{β}(x), (17.2.4)

because of (16.13.3), as in Sections 9.13 and 15.11. Similarly, this works when α and β are nonnegative real numbers, and (17.2.3) holds.

Let us put (e_{A} + x)^{α} = b_{α}(x) (17.2.5)

for every real or complex number α when (17.2.1) holds, and for every nonnegative real number α when (17.2.3) holds. If n is a positive integer, then

((e_{A} + x)^{α})^{n} = (e_{A} + x)^{n α} (17.2.6)

in both cases, because of (17.2.4). In particular,

((e_{A} + x)^{1/n})^{n} = e_{A} + x (17.2.7)

when (17.2.3) holds. The n = 2 case corresponds to part (a) of Exercise (2) on p125 of [11]. This is also related to part (a) of Theorem 10.30 on p246 of [286].

Similarly, let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}, and suppose that x ∈ A is nilpotent. If α ∈ R or C, as appropriate, then b_{α}(x) may be defined as an element of A as in (17.2.2), as in Section 15.11. If β is another real or complex number, then (17.2.4) holds, because of (16.13.3), as before. We may also express b_{α}(x) as in (17.2.5) in this case. Of course, (17.2.6) and (17.2.7) hold for all positive integers n, as before.

17.3 Some related formal power series

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}, and let T be an indeterminate. If α is a real or complex number and x ∈ A, then

b_{α,x}(T) = b_{α}(x T) = ∑_{j=0}^{∞} \binom{α}{j} x^{j} T^{j} (17.3.1)

17.3. SOME RELATED FORMAL POWER SERIES 383

is a formal power series in T with coefficients in A, as in Section 16.7. If β is another real or complex number, as appropriate, then b_{α+β,x}(T) = b_{α,x}(T) b_{β,x}(T), (17.3.2) because of (16.13.3) and a remark in Section 16.7. If n is a positive integer, then it follows that b_{α,x}(T)^{n} = b_{n α,x}(T). (17.3.3) In particular, we can take α = 1/n, to get that b_{1/n,x}(T)^{n} = 1 + x T. (17.3.4)

Suppose that (A, ∥·∥_{A}) is a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1, and that x ∈ A has the property that

∑_{j=0}^{∞}|\binom{α}{j}| ∥x^{j}∥_{A} (17.3.5)

converges as an infinite series of nonnegative real numbers. This implies that b_{α,x}(T) ∈ A_{1}^{1}[[T]], (17.3.6) in the notation of Section 15.7, with

∥b_{α,x}(T)∥_{1,1} = ∥b_{α,x}(T)∥_{A_{1}^{1}[[T]]} = ∑_{j=0}^{∞}|\binom{α}{j}| ∥x^{j}∥_{A}. (17.3.7)

If t ∈ R or C, as appropriate, and |t| ≤ 1, then

b_{α,x}(t) = ∑_{j=0}^{∞} \binom{α}{j} t^{j} x^{j} (17.3.8)

may be defined as an element of A as in Sections 15.8 and 16.7.

Suppose for the moment that x also has the property that

∑_{j=0}^{∞}|\binom{β}{j}| ∥x^{j}∥_{A} (17.3.9)

converges as an infinite series of nonnegative real numbers, so that b_{β,x}(T) ∈ A_{1}^{1}[[T]], (17.3.10) with

∥b_{β,x}(T)∥_{1,1} = ∑_{j=0}^{∞}|\binom{β}{j}| ∥x^{j}∥_{A}. (17.3.11)

This implies that b_{α+β,x}(T) ∈ A_{1}^{1}[[T]], (17.3.12) because of (17.3.2), as in Section 15.9. This means that

∑_{j=0}^{∞}|\binom{α + β}{j}| ∥x^{j}∥_{A} (17.3.13)

converges as an infinite series of nonnegative real numbers too, with

∑_{j=0}^{∞}|\binom{α + β}{j}| ∥x^{j}∥_{A} = ∥b_{α+β,x}(T)∥_{1,1} ≤ ∥b_{α,x}(T)∥_{1,1} ∥b_{β,x}(T)∥_{1,1}, (17.3.14)

as before. If t ∈ R or C, as appropriate, and |t| ≤ 1, then b_{β,x}(t) and b_{α+β,x}(t) may be defined as elements of A as well, and

b_{α+β,x}(t) = b_{α,x}(t) b_{β,x}(t), (17.3.15)

as in Section 15.9 again.

If (17.3.5) converges, then it follows that

∑_{j=0}^{∞}|\binom{n α}{j}| ∥x^{j}∥_{A} (17.3.16)

converges as an infinite series of nonnegative real numbers for every positive integer n, so that b_{n α,x}(T) ∈ A_{1}^{1}[[T]]. (17.3.17)

If t ∈ R or C, as appropriate, and |t| ≤ 1, then b_{n α,x}(t) may be defined as an element of A, and b_{α,x}(T)^{n} = b_{α,x}(t)^{n}. (17.3.18)

If α = 1/n, then we get that

b_{1/n,x}(t)^{n} = e_{A} + t x, (17.3.19)

because of (17.3.4). This is related to part (a) of Theorem 10.30 on p246 of [286].

17.4 Logarithms and binomial series

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}, and suppose that x ∈ A is nilpotent. Remember that log(e_{A} + x) may be defined as an element of A as in Section 16.5, and is nilpotent. If α is a real or complex number, as appropriate, then

(17.4.1) α log(e_{A} + x) is nilpotent in A

too, so that exp(α log(e_{A} + x)) (17.4.2)

17.4. LOGARITHMS AND BINOMIAL SERIES 385 may be defined as an element of A, as in Sections 11.1 and 15.11.

Note that b_{α}(x) may be defined as an element of A as well, as in Section 15.11. Under these conditions, b_{α}(x) = exp(α log(e_{A} + x)), (17.4.3) because of the analogous statement for formal power series in Section 16.13, as in Section 15.13.

Let T be an indeterminate, and let r be a positive real number strictly less than 1. Remember that a(T) = log(1 + T) ∈ (R)_{r}^{1}[[T]], (17.4.4) in the notation of Section 15.7, with (17.4.5) ∥a(T)∥_{1,r} = ∥ log(1 + T)∥_{1,r} = ∥ log(1 + t)∥_{(R)_{r}^{1}[[T]]} = − log(1 − r), as in Section 16.5. We also have that f(T) = exp(T) ∈ (R)_{r_{1}}^{1}[[T]] (17.4.6) for every r_{1} > 0, with ∥f(T)∥_{1,r_{1}} = ∥ exp(T)∥_{1,r_{1}} = exp(r_{1}) (17.4.7) as in Section 16.5.

If α is a real or complex number, then α a(T) ∈ (R)_{r}^{1}[[T]] or (C)_{r}^{1}[[T]], (17.4.8) as appropriate, with ∥α a(T)∥_{1,r} = |α| ∥a(T)∥_{1,r} = −|α| log(1 − r). (17.4.9) It follows that

(f ∘ (α a))(T) = f(α a(T))
(17.4.10)  = exp(α log(1 + T)) ∈ (R)_{r}^{1}[[T]] or (C)_{r}^{1}[[T]],

as appropriate, may be defined as in Section 15.15. More precisely, if α ≠ 0, then we can take r_{1} = −|α| log(1 − r), (17.4.11) to get that ∥f(α a(T))∥_{1,r} ≤ ∥f(T)∥_{1,r_{1}} = (1 − r)^{−|α|}. (17.4.12)

Note that the constant term in α a(T) is equal to 0, because of the analogous statement for a(T), as in Section 16.4. This implies that f(α a(T)) = exp(α log(1 + T)) (17.4.13) may be defined as a formal power series in T with coefficients in the real or complex numbers, as appropriate, as in Section 15.11. In fact,

exp(α log(1 + T)) = b_{α}(T) (17.4.14)

as a formal power series in T, as in Section 16.13. This is the same as in (17.4.10), as a formal power series in T, as in Section 15.15.

It follows that b_{α}(T) ∈ (R)_{r}^{1}[[T]] or (C)_{r}^{1}[[T]], (17.4.15)

as appropriate. Remember that this was obtained in Section 16.14 as well, because r < 1. If α ≠ 0, then (17.4.12) implies that

∥b_{α}(T)∥_{1,r} ≤ (1 − r)^{−|α|}. (17.4.16)

Of course, this is clear when α = 0. Note that this is the same as in Section 16.15.

Now let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers, as appropriate, with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. If x ∈ A and ∥x∥_{A} ≤ r, then log(e_{A} + x) may be defined as an element of A as in Section 16.5. This means that (17.4.2) may be defined as an element of A as in Section 10.4. Similarly, b_{α}(x) may be defined as an element of A, as in Sections 9.14 and 17.2. We also have that (17.4.3) holds in this case, because of the remarks in Section 15.15.

17.5 More on logarithms, binomial series

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1, and suppose that x ∈ A has the property that

∑_{l=1}^{∞}(1/l) ∥x^{l}∥_{A} (17.5.1)

converges as an infinite series of nonnegative real numbers. Let T be an indeterminate, and let us take a(T) = log(1 + T) and

a_{x}(T) = log(1 + x T) = ∑_{l=1}^{∞} \frac{(−1)^{l+1}}{l} x^{l} T^{l}, (17.5.2)

as in Section 16.11. Remember that

a_{x}(T) ∈ A_{1}^{1}[[T]], (17.5.3)

as in Section 15.7. We also have that f(T) = exp(T) is an element of (R)_{r_{1}}^{1}[[T]] for every r_{1} > 0, as in Section 16.5.

If α ∈ R or C, as appropriate, then α a_{x}(T) is an element of A_{1}^{1}[[T]], because of (17.5.3). This implies that

(f ∘ (α a_{x}))(T) = f(α a_{x}(T)) = exp(α a_{x}(T)) ∈ A_{1}^{1}[[T]] (17.5.4) may be defined as in Section 15.15.

Remember that the constant term in a(T) is equal to 0, so that α a(T) has the same property. This means that

(17.5.5) exp(α a(T)) = exp(α log(1 + T))

may be defined as a formal power series in T with coefficients in R or C, as appropriate, as in Section 15.11. This is equal to b_{α}(T), as in Section 16.13.

Similarly, the constant term in α a_{x}(T) is equal to 0, because of the analogous property of a_{x}(T), so that exp(α a_{x}(T)) may be defined as a formal power series in T with coefficients in A as in Section 15.11. One can use this to get that

exp(α a_{x}(T)) = b_{α,x}(T), (17.5.6)

as in Section 16.8.

This definition of exp(α a_{x}(T)) as a formal power series in T with coefficients in A is equivalent to the one in Section 15.15 under these conditions, as before. It follows that b_{α,x}(T) ∈ A_{1}^{1}[[T]], (17.5.7)

because of (17.5.4). Another way to look at this will be mentioned in a moment.

If t ∈ R or C, as appropriate, and |t| ≤ 1, then a_{x}(t) may be defined as an element of A, as in Sections 15.8 and 16.11. This implies that

f(α a_{x}(t)) = exp(α a_{x}(t)) (17.5.8)

may be defined as an element of A as well, as in Sections 9.14 and 10.4. Similarly, b_{α,x}(t) may be defined as an element of A as in Section 15.8 and the previous section.

It is easy to see that

exp(α a_{x}(t)) = f(α a_{x}(t)) = (f ∘ (α a_{x}))(t) = b_{α,x}(t), (17.5.9)

using some remarks in Section 15.15 in the second step, and (17.5.6) and the remark at the beginning of the paragraph that followed in the third step. This reduces to a statement in Section 16.11 when α = 1.

Observe that the convergence of the series (17.5.1) implies that

∥x^{l_{0}}∥_{A} < 1 (17.5.10)

for some positive integer l_{0}. This means that

∥x^{l_{0}}∥_{A}^{1/l_{0}} < 1, (17.5.11)

so that r_{A}(x) < 1, (17.5.12)

in the notation of Section 6.13. Remember that ∥x^{l}∥_{A}^{1/l} → r_{A}(x) as l → ∞, as in Section 6.14. It is easy to see that (17.5.12) implies (17.5.7), using this and the remarks in Section 16.14. Similarly, (17.5.12) implies that (17.5.1) converges.

17.6 Power series and involutions

Let T be an indeterminate, and let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} be a formal power series in T with complex coefficients. Put

\overline{f}(T) = ∑_{j=0}^{∞} \overline{f_{j}} T^{j}, (17.6.1)

which is another element of C[[T]]. It is easy to see that

f(T) ↦ \overline{f}(T) (17.6.2)

defines a conjugate-linear involution on C[[T]], as an algebra over the complex numbers, as in Section 6.4. Of course, f(T) is a formal polynomial in T if and only if \overline{f}(T) ∈ C[T].

Similarly, if r is a positive real number, then

f(T) ∈ (C)_{r}^{1}[[T]] (17.6.3)

if and only if \overline{f}(T) ∈ (C)_{r}^{1}[[T]], (17.6.4)

in the notation of Section 15.7. In this case,

∥\overline{f}(T)∥_{1,r} = ∥f(T)∥_{1,r}. (17.6.5)

If 0 < ρ ≤ ∞, then f(T) ∈ (C)_{ρ}[[T]] (17.6.6)

if and only if \overline{f}(T) ∈ (C)_{ρ}[[T]], (17.6.7)

using the notation in Section 15.7 again.

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A} and an algebra involution x ↦ x^{∗}, which may be conjugate-linear in the complex case. If x ∈ A and f(T) is a formal polynomial in T with real or complex coefficients, as appropriate, then

f(x)^{∗} = f(x^{∗}) (17.6.8)

in the real case, and in the complex case when the involution is complex-linear, and f(x)^{∗} = \overline{f}(x^{∗}) (17.6.9)

in the complex case when the involution is conjugate-linear.

If x ∈ A is nilpotent, then it is easy to see that

(17.6.10) x^{∗} is nilpotent in A

too. If f(T) is a formal power series in T with real or complex coefficients, as appropriate, then f(x) may be defined as an element of A as in Section 15.11, 17.7. COEFFICIENTS IN A 389

and similarly for f(x^{∗}), and for \overline{f}(x^{∗}) in the complex case. We also have (17.6.8) or (17.6.9), as appropriate, as before.

Now let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A}, ∥e_{A}∥_{A} = 1, and an algebra involution x ↦ x^{∗}, which may be conjugate-linear in the complex case. Suppose that there is a real number C ≥ 1 such that

∥x^{∗}∥_{A} ≤ C ∥x∥_{A} (17.6.11)

for every x ∈ A. Of course, this implies that

∥x∥_{A} = ∥(x^{∗})^{∗}∥_{A} ≤ C ∥x^{∗}∥_{A} (17.6.12)

for every x ∈ A. If C = 1, then we get that ∥x^{∗}∥_{A} = ∥x∥_{A} for every x ∈ A.

Suppose that f(T) is an element of (R)_{r}^{1}[[T]] or (C)_{r}^{1}[[T]], as appropriate, for some r > 0. If x ∈ A and ∥x∥_{A} ≤ r, then f(x) may be defined as an element of A, as in Sections 9.14 and 15.11. Similarly, if

∥x^{∗}∥_{A} ≤ r, (17.6.13)

then f(x^{∗}) may be defined as an element of A, and \overline{f}(x^{∗}) may be defined as an element of A in the complex case. It is easy to see that (17.6.8) or (17.6.9) holds under these conditions, as appropriate.

Of course, (17.6.13) follows from the hypothesis that ∥x∥_{A} ≤ r when C = 1. We shall consider another version of this with a simpler role for C in Section 17.8.

17.7 Coefficients in A

Let A be an algebra in the strict sense over the real or complex numbers with an involution x ↦ x^{∗} that may be conjugate-linear in the complex case, and let T be an indeterminate. If f(T) = ∑_{j=0}^{∞} f_{j} T^{j} is a formal power series in T with coefficients in A, then put

f^{∗}(T) = ∑_{j=0}^{∞} f_{j}^{∗} T^{j}, (17.7.1)

which is another element of A[[T]]. One can check that

f(T) ↦ f^{∗}(T) (17.7.2)

defines an algebra involution on A[[T]], which is conjugate-linear in the complex case when x ↦ x^{∗} is conjugate-linear on A. Note that (17.7.2) maps A[T] onto itself.

If f(T) is a formal polynomial in T and t ∈ R or C, as appropriate, then f(t) may be defined as an element of A, as in Section 15.4. It is easy to see that

f(t)^{∗} = f^{∗}(t) (17.7.3) in the real case, and in the complex case when the involution is complex-linear, and that f(t)^{∗} = f^{∗}(\overline{t}) (17.7.4)

in the complex case when the involution is conjugate-linear.

Let ∥ · ∥_{A} be a submultiplicative norm on A, and suppose that (17.6.11) holds on A for some C ≥ 1. Also let r be a positive real number, and suppose that f(T) ∈ A_{r}^{1}[[T]], in the notation of Section 15.7. This implies that

f^{∗}(T) ∈ A_{r}^{1}[[T]] (17.7.5)

too, with ∥f^{∗}(T)∥_{1,r} ≤ C ∥f(T)∥_{1,r}. (17.7.6)

Similarly, if 0 < ρ ≤ ∞ and f(T) ∈ A_{ρ}[[T]], then

f^{∗}(T) ∈ A_{ρ}[[T]] (17.7.7)

as well.

Suppose that A is complete with respect to the metric associated to ∥ · ∥_{A}, and that f(T) ∈ A_{r}^{1}[[T]] for some r > 0, as before. If t ∈ R or C, as appropriate, and |t| ≤ r, then f(t) and f^{∗}(t) may be defined as elements of A as in Section 15.8, and f^{∗}(\overline{t}) may be defined as an element of A in the complec case. We also have that (17.7.3) or (17.7.4) holds, as appropriate, under these conditions.

17.8 Convergence conditions and involutions

Let A be an associative algebra over the real or complex numbers with a multiplicative identity element e_{A}, and let T be an indeterminate. If a(T) = ∑_{l=0}^{∞} a_{l} T^{l} is a formal power series in T with real or complex coefficients, as appropriate, and x ∈ A, then put

a_{x}(T) = ∑_{l=0}^{∞} a_{l} x^{l} T^{l}, (17.8.1)

as in Section 16.7. In the complex case, \overline{a}(T) ∈ C[[T]] may be defined as in Section 17.6. Let y ↦ y^{∗} be an involution on A, which may be conjugate-linear in the complex case, so that (a_{x})^{∗}(T) ∈ A[[T]] may be defined as in the previous section. Observe that (a_{x})^{∗}(T) = a_{x^{∗}}(T) (17.8.2)

in the real case, and in the complex case when the involution is complex-linear, and that (a_{x})^{∗}(T) = \overline{a}_{x^{∗}}(T) (17.8.3)

in the complex case when the involution is conjugate-linear.

Now let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A}, ∥e_{A}∥_{A} = 1, and an algebra involution y ↦ y^{∗}, which may be conjugate-linear in the complex case. Suppose that

∥y^{∗}∥_{A} ≤ C ∥y∥_{A} (17.8.4)

for some C ≥ 1 and every y ∈ A, as before. Suppose also that x ∈ A has the property that

∑_{l=0}^{∞} |a_{l}| ∥x^{l}∥_{A} (17.8.5)

converges as an infinite series of nonnegative real numbers. Note that

∥(x^{∗})^{l}∥_{A} = ∥(x^{l})^{∗}∥_{A} ≤ C ∥x^{l}∥_{A} (17.8.6)

for each l. It follows that

∑_{l=0}^{∞} |a_{l}| ∥(x^{∗})^{l}∥_{A} (17.8.7)

converges too, by the comparison test, with

∑_{l=0}^{∞} |a_{l}| ∥(x^{∗})^{l}∥_{A} ≤ C ∑_{l=0}^{∞} |a_{l}| ∥x^{l}∥_{A}. (17.8.8)

This means that a_{x}(T), a_{x^{∗}}(T) ∈ A_{1}^{1}[[T]], (17.8.9)

in the notation of Section 15.7, with

∥a_{x^{∗}}(T)∥_{1,1} ≤ C ∥a_{x}(T)∥_{1,1}. (17.8.10)

In the complex case, we also have that

\overline{a}_{x}(T), \overline{a}_{x^{∗}}(T) ∈ A_{1}^{1}[[T]], (17.8.11)

with ∥\overline{a}_{x}(T)∥_{1,1} = ∥a_{x}(T)∥_{1,1}, ∥\overline{a}_{x^{∗}}(T)∥_{1,1} = ∥a_{x^{∗}}(T)∥_{1,1}. (17.8.12)

If t ∈ R or C, as appropriate, and |t| ≤ 1, then a_{x}(t) and a_{x^{∗}}(t) may be defined as elements of A, as in Sections 15.8 and 16.7. In the complex case, \overline{a}_{x}(t) and \overline{a}_{x^{∗}}(t) may be defined as elements of A too, and of course these statements hold with t replaced by \overline{t} as well. Under these conditions, one can check that

(a_{x}(t))^{∗} = (a_{x})^{∗}(t) = a_{x^{∗}}(t) (17.8.13)

in the real case, and in the complex case when the involution is complex-linear, and that (a_{x}(t))^{∗} = (a_{x})^{∗}(\overline{t}) = \overline{a}_{x^{∗}}(\overline{t}) (17.8.14)

in the complex case when the involution is conjugate-linear.

17.9 Square roots from binomial series

Let (A, ∥·∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Also let T be an indeterminate, and for each x ∈ A, put

b_{1/2,x}(T) = ∑_{j=0} \binom{1/2}{j} x^{j} T^{j}, (17.9.1)

as in Section 17.3. This is a formal power series in T with coefficients in A such that b_{1/2,x}(T)^{2} = 1 + x T, (17.9.2)

as before. Remember that

∑_{j=0}^{∞}|\binom{1/2}{j}| (17.9.3)

converges as an infinite series of nonnegative real numbers, as in Section 17.1. Suppose that

∑_{j=0}^{∞}|\binom{1/2}{j}| ∥x^{j}∥_{A} (17.9.4)

converges as an infinite series of nonnegative real numbers, so that

b_{1/2,x}(T) ∈ A_{1}^{1}[[T]], (17.9.5)

in the notation of Section 15.7. If t ∈ R or C, as appropriate, and |t| ≤ 1, then

b_{1/2,x}(t) = ∑_{j=0}^{∞} \binom{1/2}{j} t^{j} x^{j} (17.9.6)

may be defined as an element of A, as in Sections 15.8, 16.7, and 17.3. We also have that b_{1/2,x}(t)^{2} = e_{A} + t x, (17.9.7)

as in Section 17.3.

Let y ↦ y^{∗} be an algebra involution on A that may be conjugate-linear in the complex case, and that satisfies (17.8.4) on A for some C ≥ 1. This implies that

∑_{j=0}^{∞}|\binom{1/2}{j}| ∥(x^{∗})^{j}∥_{A} (17.9.8)

converges as an infinite series of nonnegative real numbers, as in the previous section. This means that

b_{1/2,x^{∗}}(T) ∈ A_{1}^{1}[[T]], (17.9.9)

so that b_{1/2,x^{∗}}(t) may be defined as an element of A when |t| ≤ 1, as before. In fact, b_{1/2,x}(t)^{∗} = b_{1/2,x^{∗}}(t) (17.9.10) in the real case, and in the complex case when the involution is complex-linear, and b_{1/2,x}(t)^{∗} = b_{1/2,x^{∗}}(\overline{t}) (17.9.11)

in the complex case when the involution is conjugate-linear, as before. In particular, if x is self-adjoint, so that x^{∗} = x, then

b_{1/2,x}(t)^{∗} = b_{1/2,x}(t) (17.9.12)

in the real case, and in the complex case when the involution is complex-linear, and b_{1/2,x}(t)^{∗} = b_{1/2,x}(\overline{t}) (17.9.13)

in the complex case when the involution is conjugate-linear. This implies that b_{1/2,x}(t) is self-adjoint in A in the real case, in the complex case when the involution is complex-linear, and in the complex case when the involution is conjugate-linear and t ∈ R. This is related to part (b) of Exercise (2) on p125 of [11].

17.10 Some more self-adjointness arguments

Let (A, ∥·∥_{A}) be a real or complex Banach algebra with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Suppose that x ∈ A satisfies

r_{A}(x) < 1, (17.10.1)

where r_{A}(x) is as in Section 6.13. This implies that (17.9.4) converges, as in Section 17.5. Thus

b_{1/2,x}(1) = ∑_{j=1}^{∞} \binom{1/2}{j} x^{j} (17.10.2)

may be defined as an element of A as before, and satisfies

b_{1/2,x}(1)^{2} = e_{A} + x. (17.10.3)

We also have that e_{A} + x is invertible in A, as in Section 6.13.

Let y ↦ y^{∗} be an involution on A that may be conjugate-linear in the complex case. Suppose that x is also self-adjoint, so that the partial sums of the series on the right side of (17.10.2) are self-adjoint elements of A too. If the involution on A is continuous, then

b_{1/2,x}(1)^{∗} = b_{1/2,x}(1), (17.10.4)

as before. If A is complex and commutative, then this holds without asking that the involution be continuous, as in Section 14.4. This is basically a version of Theorem 11.20 on p278 of [286], with simpler hypotheses on x.

The same conclusion holds when A is complex and not necessarily commutative, as in Section 14.9. This uses the fact that the partial sums of the series on the right side of (17.10.2) commute with each other. This corresponds to Theorem 11.26 on p281 of [286].

17.11 Involutions and r_{A}(x)

Let A be an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. If x ∈ A, then r_{A}(x) may be defined as in Section 6.13, as mentioned at the beginning of Section 7.2. Let y ↦ y^{∗} be an involution on A, which may be conjugate-linear in the complex case.

It would sometimes be nice to have

r_{A}(x^{∗}) = r_{A}(x) (17.11.1)

for every x ∈ A. It is easy to see that this holds when

r_{A}(x^{∗}) ≤ r_{A}(x) (17.11.2)

for every x ∈ A. If the involution on A is continuous, so that (17.8.4) holds for some C ≥ 1, then (17.11.2) follows from the fact that

∥(x^{∗})^{l}∥_{A}^{1/l} = ∥(x^{l})^{∗}∥_{A}^{1/l} ≤ C^{1/l} ∥x^{l}∥_{A}^{1/l} (17.11.3)

for every positive integer l, by taking the limit as l → ∞.

If A is a complex Banach algebra with a multiplicative identity element, then (17.11.1) follows from the characterization of r_{A}(·) in terms of the spectrum of an element of A, as in Section 6.14. This also uses the characterization of the spectrum of x^{∗} in terms of the spectrum of x, as in Section 14.1.

Suppose that x ∈ A is normal in the sense that it commutes with x^{∗}, as in Section 7.5. This implies that

r_{A}(x x^{∗}) ≤ r_{A}(x) r_{A}(x^{∗}), (17.11.4)

as in Section 7.2. If (17.11.2) holds, then we get that

r_{A}(x x^{∗}) ≤ r_{A}(x)^{2}. (17.11.5)

17.12 Identity elements and r_{A}(·)

Let A be an algebra in the strict sense over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. If α is a positive real number, then

∥x∥_{A,α} = α ∥x∥_{A} (17.12.1)

defines a norm on A. If α ≥ 1, then it is easy to see that

(17.12.2) ∥ · ∥_{A,α} is submultiplicative on A.

Suppose for the moment that A is an associative algebra. If x ∈ A, then r_{A}(x) may be defined as in Sections 6.13 and 7.2. If α ≥ 1, then we can define r_{A,α}(x) in the same way, with respect to the norm ∥ · ∥_{A,α} on A. Observe that

r_{A,α}(x) = lim_{l→∞} ∥x^{l}∥_{A,α}^{1/l} = lim_{l→∞}(α^{1/l} ∥x^{l}∥_{A}^{1/l}) = r_{A}(x), (17.12.3)

17.12. IDENTITY ELEMENTS AND R_{A}(·) 395

because α^{1/l} → 1 as l → ∞.

Let A_{1} = A × R or A × C, as appropriate, considered as an algebra in the strict sense over the real or complex numbers, as appropriate, as in Section 6.15. Remember that x ↦ (x, 0) (17.12.4)

is an injective algebra homomorphism from A into A_{1}, and that e_{A_{1}} = (0, 1) is the multiplicative identity element in A_{1}, by construction. If A has a multiplicative identity element e_{A}, then (e_{A}, 0) is an idempotent element of A_{1}, as in Section 7.6. If α ≥ 1, then

∥(x, t)∥_{A_{1},α} = ∥x∥_{A,α} + |t| = α ∥x∥_{A} + |t| (17.12.5)

defines a submultiplicative norm on A_{1}, as before. Of course, this is the same as ∥ · ∥_{A_{1}} in Section 6.15 when α = 1.

Suppose that A is an associative algebra again, so that A_{1} is associative as well, as in Section 6.15. If (x, t) ∈ A_{1} and α ≥ 1, then r_{A_{1},α}((x, t)) may be defined in the usual way, with respect to the norm ∥ · ∥_{A_{1},α} on A_{1}. Note that

r_{A_{1},α}((x, 0)) = r_{A,α}(x) = r_{A}(x) (17.12.6)

for every x ∈ A. Similarly,

r_{A_{1},α}((0, t)) = |t| (17.12.7)

for every t ∈ R or C, as appropriate.

If (x, t) ∈ A_{1} and l is a positive integer, then one can check that

∥(x, t)^{l}∥_{A_{1},α} ≥ |t|^{l}. (17.12.8)

This implies that r_{A_{1},α}((x, t)) ≥ |t|. (17.12.9)

It is easy to see that (x, 0) and (0, t) commute in A_{1}. It follows that

r_{A_{1},α}((x, t)) ≤ r_{A_{1},α}((x, 0)) + r_{A_{1},α}((0, t)) = r_{A}(x) + |t|, (17.12.10)

where the first step is as in Section 9.7. We also have that

r_{A_{1},α}((x, t)) = r_{A_{1},1}((x, t)), (17.12.11)

because ∥(x, t)^{l}∥_{A_{1},1}^{1/l} ≤ ∥(x, t)^{l}∥_{A_{1},α}^{1/l} ≤ α^{1/l} ∥(x, t)^{l}∥_{A_{1},1}^{1/l}. (17.12.12)

Let A be an algebra in the strict sense again, and let x ↦ x^{∗} be an involution on A, which may be conjugate-linear in the complex case. If (x, t) ∈ A_{1}, then put (x, t)^{∗} = (x^{∗}, t) (17.12.13)

in the real case, and in the complex case when the involution on A is complexlinear, and (x, t)^{∗} = (x^{∗}, \overline{t}) (17.12.14) in the complex case when the involution on A is conjugate-linear. One can verify that this defines an involution on A_{1}, which is conjugate-linear in the complex case when the involution on A is conjugate linear.

More precisely, this is the unique extension of the involution on A to an involution on A_{1}, that is complex-linear or conjugate-linear in the complex case depending on whether the involution on A is complex-linear or conjugate-linear. This uses the fact that the multiplicative identity element in A_{1} is self-adjoint with respect to any involution on A_{1}, as in Section 7.5.

Let ∥ · ∥_{A} be a submultiplicative norm on A again, and suppose that

∥x^{∗}∥_{A} ≤ C ∥x∥_{A} (17.12.15)

for some C ≥ 1 and all x ∈ A. This implies that

∥x^{∗}∥_{A,α} ≤ C ∥x∥_{A,α} (17.12.16)

for all x ∈ A, so that

∥(x, t)^{∗}∥_{A_{1},α} ≤ C ∥(x, t)∥_{A_{1},α} (17.12.17)

for all (x, t) ∈ A_{1}. In particular, if the involution on A preserves ∥ · ∥_{A}, then it preserves ∥ · ∥_{A,α}, and the corresponding involution on A_{1} preserves ∥ · ∥_{A_{1},α}. This corresponds to part of Proposition 2.5.4 on p58 of [11].

17.13 Limits at infinity

Let X be an infinite set, and remember that c^{lai}(X, R), c^{lai}(X, C) are the spaces of real or complex-valued functions f on X, as appropriate, with a limit at infinity, as in Section 3.3. This means that f may be expressed as

f = f_{0} + t 1_{X}, (17.13.1)

where f_{0} is a real or complex-valued function on X that vanishes at infinity, as appropriate, 1_{X} is the constant function on X equal to 1 at every point, and t = t(f) ∈ R or C, as appropriate, as before.

If f has a limit at infinity, then f is bounded on X, as before. Remember that c^{lai}(X, R), c^{lai}(X, C) are closed linear subspaces of ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively. It is easy to see that in fact

(17.13.2) c^{lai}(X, R), c^{lai}(X, C) are subalgebras of ℓ^{∞}(X, R), ℓ^{∞}(X, C),

respectively.

If f is as in (17.13.1), then t is called the limit of f at infinity on X, and it is uniquely determined by f. We have seen that

|t(f)| ≤ ∥f∥_{∞}, (17.13.3)

and that f ↦ t(f) (17.13.4) is a linear functional on each of c^{lai}(X, R), c^{lai}(X, C). More precisely, one can check that this is an algebra homomorphism from c^{lai}(X, R), c^{lai}(X, C) into R, C, respectively.

Remember that c_{0}(X, R), c_{0}(X, C) are the spaces of real and complex-valued functions on X that vanish at infinity, respectively, as in Section 1.13. These are commutative and associative algebras over the real and complex numbers, as appropriate, without multiplicative identity elements, because X is an infinite set. Of course, 1_{X} is the multiplicative identity element in each of ℓ^{∞}(X, R), ℓ^{∞}(X, C), and in particular in each of c^{lai}(X, R), c^{lai}(X, C).

Let us take A = c_{0}(X, R) or c_{0}(X, C), and let A_{1} be as in Section 6.15 and the previous section. It is easy to see that

(f_{0}, t) ↦ f_{0} + t 1_{X} (17.13.5)

defines an algebra isomorphism from A_{1} onto c^{lai}(X, R) or c^{lai}(X, C), as appropriate.

Similarly, let X be a metric space or a Hausdorff topological space that is locally compact, as in Section 5.1, and not compact. Let us say that a real or complex-valued function f on X has a limit at infinity on X if f can be expressed as in (17.13.1), where t = t(f) ∈ R or C, as appropriate, and f_{0} vanishes at infinity on X as in Section 5.2. In this case, one can verify that t is unique, and one may call it the limit of f at infinity on X.

This is the same as saying that f has a limit at the point at infinity in the one-point compactification of X, and that this limit is equal to t. The previous remarks for infinite sets correspond to using the discrete metric or topology on X.

One can check that the space of real or complex-valued functions f on X with a limit t at infinity is a subalgebra of the space of all real or complex-valued functions on X, as a commutative algebra over R or C, as appropriate, with respect to pointwise multiplication of functions. The mapping from f to t(f) is an algebra homomorphism from the space of real or complex-valued functions on X with a limit at infinity to R or C, as appropriate. These statements are analogous to standard facts about the limit of a function at a point in a topological space, and one can reduce to that case using the one-point compactification of X, as in the preceding paragraph.

If f is bounded on X, then |t| is less than or equal to the supremum norm of f on X, as before. Note that 1_{X} does not vanish at infinity on X, because X is not compact.

Remember that C_{0}(X, R), C_{0}(X, C) are the spaces of continuous real and complex-valued functions on X that vanish at infinity, as in Section 5.2. These are closed linear subspaces of the spaces C_{b}(X, R), C_{b}(X, C) of bounded continuous real and complex-valued functions on X, as appropriate, as before.

Let C^{lai}(X, R), C^{lai}(X, C) (17.13.6)

be the spaces of continuous real or complex-valued functions f on X with a limit at infinity, as appropriate. This means that f can be expressed as in (17.13.1), where f_{0} is continuous on X and vanishes at infinity. One can check that

(17.13.7) C^{lai}(X, R), C^{lai}(X, C) are closed subalgebras of C_{b}(X, R), C_{b}(X, C),

respectively.

A continuous real or complex-valued function f on X has a limit at infinity if and only if it extends to a continuous real or complex-valued function to the one-point compactification of X, as appropriate, whose value at the point at infinity is the same as the limit of f at infinity. In this case, the supremum norm of f on X is the same as the supremum norm of its continuous extension to the one-point compactification of X.

Let us now take A = C_{0}(X, R) or C_{0}(X, C), which is a commutative associative algebra over the real or complex numbers, as appropriate, with respect to pointwise multiplication of functions. If A_{1} is as in Section 6.15 and the previous section again, then it is easy to see that (17.13.5) defines an algebra isomorphism from A_{1} onto C^{lai}(X, R) or C^{lai}(X, C), as appropriate.

17.14 Some weighted ℓ^{∞} spaces

Let X be a nonempty set, and let w be a positive real-valued function on X. As in Section 15.3, we let ℓ_{w}^{∞}(X, R) and ℓ_{w}^{∞}(X, C) be the spaces of real and complex-valued functions f on X such that

|f(x)| w(x) (17.14.1)

is bounded on X, with the corresponding weighted supremum norm

∥f∥_{∞,w} = ∥f∥_{sup,w} = sup_{x∈X}(|f(x)| w(x)). (17.14.2)

These are the same as the usual ℓ^{∞} spaces and supremum norm when w ≡ 1 on X, as in Section 1.1.

If c is a positive real number, then c w is another positive real-valued function on X, and ℓ_{c w}^{∞}(X, R), ℓ_{c w}^{∞}(X, C) are the same as ℓ_{w}^{∞}(X, R), ℓ_{w}^{∞}(X, C), respectively. If f is an element of one of these spaces, then

∥f∥_{∞,c w} = c ∥f∥_{∞,w}. (17.14.3)

Let w_{1}, w_{2} be positive real-valued functions on X, and suppose for the moment that c w_{1} ≤ w_{2} (17.14.4)

on X. This implies that

ℓ_{w_{2}}^{∞}(X, R) ⊆ ℓ_{w_{1}}^{∞}(X, R), ℓ_{w_{2}}^{∞}(X, C) ⊆ ℓ_{w_{1}}^{∞}(X, R), (17.14.5)

with c ∥f∥_{∞,w_{1}} = ∥f∥_{∞,c w_{1}} ≤ ∥f∥_{∞,w_{2}} (17.14.6)

17.14. SOME WEIGHTED ℓ^{∞} SPACES 399

for all f ∈ ℓ_{w_{2}}^{∞}(X, R) or ℓ_{w_{2}}^{∞}(X, C).

Let w_{1}, w_{2} be any positive real-valued functions on X again, and suppose for the moment that f, g are real or complex-valued functions on X such that (17.14.7) |f| w_{1} and |g| w_{2} are bounded on X. Under these conditions, w_{1} w_{2} is a positive real-valued function on X, and f g ∈ ℓ_{w_{1} w_{2}}^{∞}(X, R) or ℓ_{w_{1} w_{2}}^{∞}(X, C), (17.14.8) as appropriate. We also have that ∥f g∥_{w_{1} w_{2}} ≤ ∥f∥_{w_{1}} ∥g∥_{w_{2}}. (17.14.9)

Suppose now that w ≥ c (17.14.10) on X, so that c w ≤ w^{2} (17.14.11) on X. If f, g ∈ ℓ_{w}^{∞}(x, R) or ℓ_{w}^{∞}(X, C), then f g ∈ ℓ_{w^{2}}^{∞}(X, R) or ℓ_{w^{2}}^{∞}(X, C), (17.14.12) as appropriate, as in (17.14.8). This implies that f g ∈ ℓ_{w}^{∞}(X, R) or ℓ_{w}^{∞}(X, C), (17.14.13) as appropriate, so that ℓ_{w}^{∞}(X, R), ℓ_{w}^{∞}(X, C) are subalgebras of the algebras of bounded real and complex-valued functions on X, respectively. We also get that c ∥f g∥_{∞,w} = ∥f g∥_{∞,c w} ≤ ∥f g∥_{∞,w^{2}} ≤ ∥f∥_{∞,w} ∥g∥_{∞,w}. (17.14.14)

Suppose that c = 1, so that w ≥ 1 (17.14.15) on X. In this case, ℓ_{w}^{∞}(X, R) and ℓ_{w}^{∞}(X, C) are Banach algebras over the real and complex numbers, respectively, with respect to ∥ · ∥_{∞,w}. Of course, the constant function 1_{X} on X equal to 1 at every point is contained in these spaces if and only if w is bounded on X. In fact, ∥1_{X}∥_{∞,w} = 1 (17.14.16) if and only if w ≡ 1 on X, because of (17.14.15).

Let f ∈ ℓ_{w}^{∞}(X, R) or ℓ^{∞}(X, C) be given, and observe that for each integer l ≥ 2, ∥f∥_{∞}^{l} = ∥f^{l}∥_{∞} ≤ ∥f^{l}∥_{∞,w} ≤ ∥f^{l−1}∥_{∞} ∥f∥_{∞,w} = ∥f∥_{∞}^{l−1} ∥f∥_{∞,w}. (17.14.17) This implies that ∥f∥_{∞} ≤ ∥f^{l}∥_{∞,w}^{1/l} ≤ ∥f∥_{∞}^{1−(1/l)} ∥f∥_{∞,w}^{1/l}. (17.14.18) Let us take A = ℓ_{w}^{∞}(X, R) or ℓ_{w}^{∞}(X, C), as appropriate, so that r_{A}(f) may be defined as in Sections 6.13 and 7.2. Using (17.14.18), we obtain that

r_{A}(f) = lim_{l→∞} ∥f^{l}∥_{∞,w}^{1/l} = ∥f∥_{∞}. (17.14.19)

17.15 Continuous weights

Let X be a nonempty metric or topological space, and let w be a continuous positive real-valued function on X. Consider the spaces

(17.15.1)  C_{b,w}(X, R) = C(X, R) ∩ ℓ_{w}^{∞}(X, R),
(17.15.2)  C_{b,w}(X, C) = C(X, C) ∩ ℓ_{w}^{∞}(X, C)

of continuous real and complex-valued functions on X, respectively, such that (17.14.1) is bounded on X. These are linear subspaces of C(X, R), C(X, C) and ℓ_{w}^{∞}(X, R), ℓ_{w}^{∞}(X, C), respectively. Of course, these are the same as the usual spaces C_{b}(X, R), C_{b}(X, C) of bounded continuous real and complex-valued functions on X when w ≡ 1 on X, as in Section 1.4. If X is equipped with the discrete metric or topology, then every function on X is continuous, and C_{b,w}(X, R), C_{b,w}(X, C) are the same as ℓ_{w}^{∞}(X, R), ℓ_{w}^{∞}(X, C), respectively.

Note that f ∈ C_{b,w}(X, R) or C_{b,w}(X, C) (17.15.3)

if and only if w f ∈ C_{b}(X, R) or C_{b,w}(X, C), (17.15.4)

as appropriate. Remember that

f ↦ w f (17.15.5)

is an isometric linear mapping from each of ℓ_{w}^{∞}(X, R), ℓ_{w}^{∞}(X, C) onto ℓ^{∞}(X, R), ℓ^{∞}(X, C), respectively, as in Section 15.3. This mapping sends C_{b,w}(X, R), C_{b,w}(X, C) onto C_{b}(X, R), C_{b}(X, C), respectively. It follows that

(17.15.6) C_{b,w}(X, R), C_{b,w}(X, C) are closed subsets of ℓ_{w}^{∞}(X, R), ℓ_{w}^{∞}(X, C),

respectively, because of the analogous statement when w ≡ 1 on X, as in Section 1.4. We also have that

(17.15.7) C_{b,w}(X, R), C_{b,w}(X, C) are Banach spaces

with respect to the weighted supremum norm ∥f∥_{sup,w} in (17.14.2).

Remember that C_{com}(X, R), C_{com}(X, C) are the spaces of continuous real and complex-valued functions on X with compact support, as in Section 5.1. Observe that

C_{com}(X, R) ⊆ C_{b,w}(X, R), C_{com}(X, C) ⊆ C_{b,w}(X, C), (17.15.8)

because w is bounded on compact subsets of X. We also have that (17.15.5) maps C_{com}(X, R) and C_{com}(X, C) onto themselves.

Let C_{0,w}(X, R), C_{0,w}(X, C) (17.15.9)

be the spaces of continuous real and complex-valued functions on X, respectively, such that (17.15.10) w f vanishes at infinity on X, as in Section 5.2. These are the same as the usual spaces C_{0}(X, R), C_{0}(X, C) of continuous real and complex-valued functions on X that vanish at infinity when w ≡ 1 on X. If X is equipped with the discrete metric or topology, then these are the same as c_{0,w}(X, R), c_{0,w}(X, C) in Section 15.3. Of course, (17.15.5) maps C_{0,w}(X, R), C_{0,w}(X, C) onto C_{0}(X, R), C_{0}(X, C), respectively, This implies that (17.15.11) C_{0,w}(X, R), C_{0,w}(X, C) are closed linear subspaces of

C_{b,w}(X, R), C_{b}(X, C),

respectively, because of the analogous statement when w ≡ 1 on X.

In fact, (17.15.12) C_{0,w}(X, R), C_{0,w}(X, C) are the same as the closures of

C_{com}(X, R), C_{com}(X, C) in C_{b,w}(X, R), C_{b,w}(X, C),

respectively. This follows from the analogous statement when w ≡ 1 on X, as in Section 5.2.

Let w_{1}, w_{2} be continuous positive real-valued functions on X such that (17.14.4) holds for some positive real number c. This implies that C_{b,w_{2}}(X, R) ⊆ C_{b,w_{1}}(X, R), C_{b,w_{2}}(X, C) ⊆ C_{b,w_{1}}(X, C), (17.15.13) as in (17.14.5). Similarly, C_{0,w_{2}}(X, R) ⊆ C_{0,w_{1}}(X, R), C_{0,w_{2}}(X, C) ⊆ C_{0,w_{1}}(X, C). (17.15.14) If we also have that c′ w_{2} ≤ w_{1} (17.15.15) on X for some positive real number c′, then we get that C_{b,w_{1}}(X, R) = C_{b,w_{2}}(X, R), C_{b,w_{1}}(X, C) = C_{b,w_{1}}(X, C) (17.15.16) and C_{0,w_{1}}(X, R) = C_{0,w_{2}}(X, C), C_{0,w_{1}}(X, C) = C_{0,w_{2}}(X, C). (17.15.17)

Let w_{1}, w_{2} be any continuous positive real-valued functions on X, so that (17.15.18) w_{1} w_{2} is a continuous positive real-valued function on X as well. If f, g are continuous real or complex-valued functions on X such that (17.14.7) holds, then f g ∈ C_{b,w_{1} w_{2}}(X, R) or C_{b,w_{1} w_{2}}(X, C), (17.15.19) as appropriate, as in (17.14.8). If we also have that (17.15.20) f w_{1} or g w_{2} vanishes at infinity on X, as in Section 5.2, then f g ∈ C_{0,w_{1} w_{2}}(X, R) or C_{0,w_{1} w_{2}}(X, C), (17.15.21) as appropriate.

Algebras, involutions, and positivity

18.1 One-sided multiplicative identity elements

Let A be an algebra in the strict sense over the real or complex numbers. An element e_{L} of A is said to be a left multiplicative identity element in A if

e_{L} x = x (18.1.1)

for every x ∈ A. Similarly, an element e_{R} of A is said to be a right multiplicative identity element in A if x e_{R} = x (18.1.2)

for every x ∈ A. Thus a multiplicative identity element in A is the same as an element of A that is both a left and right multiplicative identity element. If A has a left multiplicative identity element e_{L} and a right multiplicative element e_{R}, then it is easy to see that e_{L} = e_{R}, (18.1.3)

so that this is a multiplicative identity element of A.

If a ∈ A, then let L_{a}(x) = a x (18.1.4)

be the corresponding left multiplication operator on A, as in Section 6.3. Note that e_{L} ∈ A is a left multiplicative identity element in A if and only if

L_{e_{L}} = I_{A}, (18.1.5)

the identity mapping on A. Similarly, e_{R} ∈ A is a right multiplicative identity element in A if and only if L_{a}(e_{R}) = a (18.1.6) for every a ∈ A. If A has a right multiplicative identity element, then it follows that a ↦ L_{a} (18.1.7)

is a one-to-one mapping from A into the space L(A) of all linear mappings from A into itself.

Suppose for the moment that x ↦ x^{∗} is an algebra involution on A, which may be conjugate-linear in the complex case. Observe that e_{L} ∈ A is a left multiplicative identity element in A if and only if

(18.1.8) e_{L}^{∗} is a right multiplicative identity element in A.

Under these conditions, we get that

e_{L}^{∗} = e_{L}, (18.1.9)

as in (18.1.3), and that e_{L} is a multiplicative identity element in A. Of course, there are analogous statements for right multiplicative identity elements in A.

Of course, (18.1.7) is a linear mapping from A into L(A), as in Section 6.3. The kernel of this mapping is equal to

(18.1.10) A_{L} = {a ∈ A : a y = 0 for every y ∈ A}.

Thus (18.1.7) is one-to-one on A if and only if

A_{L} = {0}. (18.1.11)

Note that (18.1.10) is automatically a right ideal in A.

Suppose now that A is an associative algebra. Remember that (18.1.7) is an algebra homomorphism from A into L(A) in this case, as in Section 6.3. This implies that (18.1.10) is a two-sided ideal in A, as in Section 6.11, which can also be verified directly.

If e_{L} ∈ A is a left multiplicative identity element in A, then

(x e_{L}) y = x (e_{L} y) = x y (18.1.12)

for every x, y ∈ A. This means that

(x e_{L} − x) y = 0 (18.1.13)

for every x, y ∈ A, which is the same as saying that

x e_{L} − x ∈ A_{L} (18.1.14)

for every x ∈ A. If (18.1.11) holds, then we get that

x e_{L} = x (18.1.15)

for every x ∈ A, so that e_{L} is a right multiplicative identity element in A too. There are analogous statements for right multiplicative identity elements in A, as usual.

404 CHAPTER 18. ALGEBRAS, INVOLUTIONS, AND POSITIVITY

Suppose that a_{0} ∈ A has the property that (18.1.16) L_{a_{0}} is a one-to-one mapping from A onto itself. This implies that there is an x_{0} ∈ A such that a_{0} x_{0} = L_{a_{0}}(x_{0}) = a_{0}. (18.1.17) It follows that a_{0} (x_{0} y) = (a_{0} x_{0}) y = a_{0} y (18.1.18) for every y ∈ A. This means that x_{0} y = y (18.1.19) for every y ∈ A, because L_{a_{0}} is injective, so that x_{0} is a left multiplicative identity element in A.

Of course, (18.1.16) also implies that there is a b_{0} ∈ A such that a_{0} b_{0} = L_{a_{0}}(b_{0}) = x_{0}. (18.1.20) This means that L_{a_{0}} ∘ L_{b_{0}} = L_{a_{0} b_{0}} = L_{x_{0}} = I_{A}. (18.1.21) It follows that L_{a_{0}}^{−1} = L_{b_{0}}, (18.1.22) which is to say that we also have that L_{b_{0}} ∘ L_{a_{0}} = I_{A}. (18.1.23) If (18.1.11) holds, then x_{0} is the multiplicative identity element in A, and (18.1.23) implies that b_{0} a_{0} = x_{0}. (18.1.24) Of course, this means that a_{0} is invertible in A, with a_{0}^{−1} = b_{0}.

18.2 Some subalgebras of L(A)

Let A be an associative algebra over the real or complex numbers again, and let us continue to use the notation in (18.1.4) and (18.1.10). Put B = {L_{a} : a ∈ A}. (18.2.1) This is a subalgebra of L(A), as an algebra with respect to composition of linear mappings on A, because (18.1.7) is an algebra homomorphism from A into L(A).

Observe that I_{A} ∈ B (18.2.2)

18.3. ANOTHER CONDITION ON L_{A} 405

if and only if A has a left multiplicative identity element. If (18.1.11) holds, then this is the same as saying that A has a multiplicative identity element, as in the previous section.

Let B_{1} be the subset of L(A) consisting of linear mappings of the form

L_{a} + t I_{A}, (18.2.3)

where a ∈ A and t ∈ R or C, as appropriate. This is the same as the linear subspace of L(A) spanned by B and I_{A}. Thus

B = B_{1} (18.2.4)

if and only if (18.2.2) holds. It is easy to see that

(18.2.5) B_{1} is a subalgebra of L(A)

too. In fact, B is a two-sided ideal in B_{1}, as in Section 6.11.

If L_{a} + t I_{A} = 0 (18.2.6)

for some a ∈ A and t ∈ R or C, as appropriate, with t ≠ 0, then (18.2.2) holds. If (18.2.2) does not hold, then

L_{a} + t I_{A} ↦ t (18.2.7)

is a well-defined linear mapping from B_{1} onto R or C, as appropriate, with kernel B. More precisely, this is an algebra homomorphism from B_{1} onto R or C, as appropriate, in this case.

Let A_{1} = A × R or A × C, as appropriate, be defined as an associative algebra over the real or complex numbers, as appropriate, as in Section 6.15. Consider the mapping from A_{1} into L(A) defined by

(a, t) ↦ L_{a} + t I_{A}. (18.2.8)

One can check that this defines an algebra homomorphism from A_{1} into L(A). Of course, B_{1} is the same as the image of A_{1} under this homomorphism.

Suppose that (18.1.11) holds, so that (18.1.7) is one-to-one on A. If the kernel of (18.2.8) is nontrivial, then (18.2.6) holds with t ≠ 0, so that (18.2.2) holds, as before. Otherwise, if (18.2.2) does not hold, then it follows that (18.2.8) is an algebra isomorphism from A_{1} onto B_{1}.

Suppose that A ≠ {0}, so that I_{A} ≠ {0}. If (18.2.2) holds, then it is easy to see that the kernel of (18.2.8) is nontrivial.

18.3 Another condition on L_{a}

Let A be an algebra in the strict sense over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. If a ∈ A, then let L_{a} be as in (18.1.4), which is a bounded linear mapping from A into itself, with

∥L_{a}∥_{op} ≤ ∥a∥_{A}, (18.3.1) as in Section 6.3. Thus (18.2.1) defines a linear subspace of the space BL(A) of all bounded linear mappings from A into itself.

In this section, we ask that there be a nonnegative real number C such that

∥a∥_{A} ≤ C ∥L_{a}∥_{op} (18.3.2)

for every a ∈ A. Of course, this implies in particular that a ↦ L_{a} is a one-to-one linear mapping from A into BL(A), so that (18.1.11) holds, as before. If A has a right multiplicative identity element e_{R}, then

∥a∥_{A} = ∥L_{a}(e_{R})∥_{A} ≤ ∥L_{a}∥_{op} ∥e_{R}∥_{A} (18.3.3)

for every a ∈ A, as in Section 6.3.

Suppose that A is complete with respect to the metric associated to ∥ · ∥_{A}. This implies that that B is complete with respect to the restriction to B of the metric associated to the operator norm on BL(A), because (18.3.1) and (18.3.2). This means that (18.3.4) B is a closed set in BL(A),

as in Section 1.6.

Suppose that I_{A} ∉ B. (18.3.5)

This implies that there is a positive real number c_{1} such that

∥L_{a} + I_{A}∥_{op} ≥ c_{1} (18.3.6)

for every a ∈ A, because of (18.3.4). It follows that

∥L_{a} + t I_{A}∥_{op} ≥ c_{1} |t| (18.3.7)

for every a ∈ A and t ∈ R or C, as appropriate.

Let B_{1} be the linear subspace of BL(A) spanned by B and I_{A}, which is equivalent to the definition of B_{1} in the previous section in this case. As before, (18.2.7) defines a linear functional on B_{1} with kernel equal to B. Let us check that (18.3.8) B_{1} is a closed set in BL(A)

under these conditions.

Let {a_{j}}_{j=1}^{∞} and {t_{j}}_{j=1}^{∞} be sequences of elements of A and R or C, as appropriate, respectively. Suppose that the corresponding sequence

{L_{a_{j}} + t_{j} I_{A}}_{j=1}^{∞} (18.3.9)

of elements of B_{1} converges to an element of BL(A). In particular, this implies that (18.3.9) is a Cauchy sequence in BL(A). This means that {t_{j}}_{j=1}^{∞} is a Cauchy sequence in R or C, as appropriate, because of (18.3.7). It follows that {t_{j}}_{j=1}^{∞} converges to t ∈ R or C, as appropriate, because R and C are complete with respect to their standard metrics.

18.4. AN INVOLUTION ON B_{1} 407

This implies that {L_{a_{j}}}_{j=1}^{∞} converges in BL(A), because (18.3.9) converges in BL(A). In fact, {L_{a_{j}}}_{j=1}^{∞} converges to an element of B, because of (18.3.4). This means that (18.3.9) converges to an element of B_{1}.

Suppose now that A is associative as an algebra over the real or complex numbers, so that B and B_{1} are subalgebras of BL(A), as in the previous section. In this case, (18.3.6) and thus (18.3.7) hold with c_{1} = 1. Otherwise, there would be an element of B that is invertible as an element of B_{1}, as in Section 6.5. This is not possible, because (18.2.7) is an algebra homomorphism from B_{1} onto R or C, as appropriate.

18.4 An involution on B_{1}

Let A be an associative algebra over the real or complex with an involution x ↦ x^{∗}, which may be conjugate-linear in the complex case. If a ∈ A, then let L_{a} be as in (18.1.4), and remember that a ↦ L_{a} is an algebra homomorphism from A into L(A), as in Section 6.3. Suppose that

(18.4.1) a ↦ L_{a} is one-to-one on A,

which is the same as saying that (18.1.11) holds, as before. Let B, B_{1} be the subalgebras of L(A) defined in Section 18.2.

Suppose from now on in this section that

(18.4.2) A does not have a multiplicative identity element.

This implies that

(18.4.3) A does not have a left multiplicative identity element,

as in Section 18.1. It follows that (18.3.5) holds, because of (18.4.3), as in Section 18.2. This means that (18.2.7) is an algebra homomorphism from B_{1} onto R or C, as appropriate, with kernel B, as before.

If a ∈ A and t ∈ R or C, as appropriate, then put

(L_{a} + t I_{A})^{∗} = L_{a^{∗}} + t I_{A} (18.4.4)

in the real case, and in the complex case when the involution on A is complexlinear, and (L_{a} + t I_{A})^{∗} = L_{a^{∗}} + \overline{t} I_{A} (18.4.5)

in the complex case when the involution on A is conjugate-linear. One can check that this defines an algebra involution on B_{1}, which is conjugate-linear in the complex case when the involution on A is conjugate-linear.

Remember that A_{1} = A × R or A × C, as appropriate, may be defined as an associative algebra over R or C, as appropriate, as in Section 6.15, and that (18.2.8) defines an algebra isomorphism from A_{1} onto B_{1} under these conditions. The involution on B_{1} defined in the preceding paragraph corresponds exactly to the one defined on A_{1} as in Section 17.12 with respect to this isomorphism.

408 CHAPTER 18. ALGEBRAS, INVOLUTIONS, AND POSITIVITY

If x ∈ A, then (L_{a} + t I_{A})(x) = a x + t x. (18.4.6) If y ∈ A, then x^{∗} (L_{a})^{∗}(y) = x^{∗} L_{a^{∗}}(y) = x^{∗} a^{∗} y = (a x)^{∗} y. (18.4.7) Using this, it is easy to see that x^{∗} ((L_{a} + t I_{A})^{∗}(y)) = (a x + t x)^{∗} y. (18.4.8) This implies that x^{∗} ((L_{a} + t I_{A})^{∗}((L_{a} + t I_{A})(x))) = (a x + t x)^{∗} (a x + t x). (18.4.9)

18.5 Multiplication operators on C^{∗} algebras

Let A be an algebra in the strict sense over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}. If a ∈ A, then we let L_{a} be as in (18.1.4) again, which is a bounded linear mapping from A into itself, as before.

Let x ↦ x^{∗} be an algebra involution on A, which may be conjugate-linear in the complex case, and suppose that ∥ · ∥_{A} satisfies the C^{∗} identity ∥x^{∗} x∥_{A} = ∥x∥_{A}^{2} (18.5.1) for every x ∈ A, as in Section 7.7. Remember that this implies that ∥x^{∗}∥_{A} = ∥x∥_{A} (18.5.2) for every x ∈ A, so that ∥x x^{∗}∥_{A} = ∥x^{∗}∥_{A}^{2} = ∥x∥_{A} ∥x^{∗}∥_{A} (18.5.3) for every x ∈ A. If a ∈ A, then ∥L_{a}(a^{∗})∥_{A} = ∥a a^{∗}∥_{A} = ∥a∥_{A} ∥a^{∗}∥_{A}. (18.5.4) It follows that ∥L_{a}∥_{op} = ∥a∥_{A} (18.5.5) for every a ∈ A under these conditions, because of (18.3.1). Note that A is not asked to have a left or right multiplicative identity element here.

Of course, (18.5.5) implies that (18.5.6) a ↦ L_{a} is a one-to-one linear mapping from A into BL(A), so that (18.1.11) holds, as in Section 18.3. Suppose now that A is associative as an algebra over the real or complex numbers, and let B, B_{1} be as in Section 18.2. Suppose also that (18.4.2) holds, so that we can define an involution on B_{1} as in the previous section.

We would like to show that

∥(L_{a} + t I_{A})^{∗} ∘ (L_{a} + t I_{A})∥_{op} = ∥L_{a} + t I_{A}∥_{op}^{2} (18.5.7)

for every a ∈ A and t ∈ R or C, as appropriate. To do this, it suffices to verify that ∥(L_{a} + t I_{A})∥_{op}^{2} ≤ ∥(L_{a} + t I_{A})^{∗} ∘ (L_{a} + t I_{A})∥_{op}, (18.5.8)

as in Section 7.7. Let x ∈ A be given, and let us check that

∥(L_{a} + t I_{A})(x)∥_{A}^{2} ≤ ∥(L_{a} + t I_{A})^{∗} ∘ (L_{a} + t I_{A})∥_{op} ∥x∥_{A}^{2}. (18.5.9)

Observe that

∥(L_{a} + t I_{A})(x)∥_{A}^{2} = ∥a x + t x∥_{A}^{2} = ∥(a x + t x)^{∗} (a x + t x)∥_{A}, (18.5.10)

using (18.4.6) in the first step, and (18.5.1) in the second step. We can use (18.4.9) and (18.5.2) to get that

∥(a x + t x)^{∗} (a x + t x)∥_{A} = ∥x^{∗} ((L_{a} + t I_{A})^{∗}((L_{a} + t a)(x)))∥_{A}
(18.5.11)  ≤ ∥x^{∗}∥_{A} ∥(L_{a} + t I_{A})^{∗}((L_{a} + t I_{A})(x))∥_{A}
≤ ∥(L_{a} + t I_{A})^{∗} ∘ (L_{a} + t I_{A})∥_{op} ∥x∥_{A}^{2}.

This corresponds to Proposition 2.9.2 on p75 of [11], some remarks on p264 of [291], and Exercise 5 on p300 of [291].

18.6 Positive linear functionals

Let A be an associative algebra over the complex numbers with a conjugatelinear involution y ↦ y^{∗}. (18.6.1)

Let us say that a linear functional λ on A, as a vector space over the complex numbers, is positive in the strict sense with respect to this involution if for every x ∈ A, we have that λ(x^{∗} x) ∈ R (18.6.2)

and λ(x^{∗} x) ≥ 0. (18.6.3)

Of course, this is the same as saying that

λ(x x^{∗}) ∈ R (18.6.4)

and λ(x x^{∗}) ≥ 0 (18.6.5)

for every x ∈ A, by replacing x with x^{∗}. One may also say that λ is a positive functional in the strict sense on A in this case. This corresponds to Definition 21.16 on p316 of [151], and some other versions of this are mentioned at the beginning of Section 4.7 on p122 of [11], in Definition 11.30 on p283 of [286], and at the beginning of Section 4 in Chapter VI on p270 of [291].

If λ is any linear functional on A that satisfies (18.6.2) for every x ∈ A, then

λ(x^{∗} y) = \overline{λ(y^{∗} x)} (18.6.6)

for every x, y ∈ A, as in part (i) of Theorem 21.17 on p316 of [151]. Indeed, if x, y ∈ A and t ∈ C, then

λ((x + t y)^{∗} (x + t y)) = λ(x^{∗} x) + t λ(x^{∗} y) + \overline{t} λ(y^{∗} x) + |t|^{2}λ(y^{∗} y). (18.6.7)

This implies that t λ(x^{∗} y) + \overline{t} λ(y^{∗} x) ∈ R, (18.6.8)

because of (18.6.2). One can get (18.6.6) from this by taking t = 1, i. Note that (18.6.6) implies (18.6.2), by taking y = x.

If λ is any linear functional on A, then

b_{λ}(x, y) = λ(y^{∗} x) (18.6.9)

defines a sesquilinear form on A, as in Section 5.15. The condition (18.6.6) is the same as saying that (18.6.9) is Hermitian symmetric on A, as in Section 8.5. In this case, (18.6.3) is the same as saying that (18.6.9) is nonnegative as a Hermitian form on A, as in Section 8.10. This implies that

|b_{λ}(x, y)| ≤ b_{λ}(x, x)^{1/2} b_{λ}(y, y)^{1/2} (18.6.10)

for every x, y ∈ A, as before. Equivalently, this means that

|λ(y^{∗} x)| ≤ λ(x^{∗} x)^{1/2} λ(y^{∗} y)^{1/2} (18.6.11)

for every x, y ∈ A. This corresponds to part (ii) of Theorem 21.17 on p316 of [151]. This also corresponds to part of Proposition 4.7.1 on p123 of [11], part (b) of Theorem 11.31 on p284 of [286], and a remark in the proof of 4.1 on p271 of [291].

Now let A be an associative algebra over the real numbers with an involution as in (18.6.1). Let us say that a linear functional on A, as a vector space over the real numbers, is positive in the strict sense with respect to the involution if

λ(x^{∗} y) = λ(y^{∗} x) (18.6.12)

for every x, y ∈ A, and if (18.6.3) holds for every x ∈ A. As before, we may also say that λ is a positive functional in the strict sense on A under these conditions.

If λ is any linear functional on A, then (18.6.9) defines a bilinear form on A, as in Section 5.15. Of course, (18.6.12) is the same as saying that

(18.6.13) b_{λ} is symmetric on A,

as in Section 8.5. If we have this, then (18.6.3) is the same as saying that b_{λ} is nonnegative as a symmetric bilinear form on A, as in Section 8.10. This implies that (18.6.10) holds, which is the same as saying that (18.6.11) holds, as before.

18.7 Positive functionals and identity elements

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗} that is conjugate-linear in the complex case. If A has a multiplicative identity element e_{A}, then e_{A}^{∗} = e_{A}, as in Section 6.4. If λ is a linear functional on A that satisfies (18.6.6) or (18.6.12), as appropriate, then we can take y = e_{A} to get that

λ(x^{∗}) = \overline{λ(x)} (18.7.1)

or λ(x^{∗}) = λ(x), (18.7.2)

as appropriate, for every x ∈ A. Conversely, it is easy to see that each of these conditions implies (18.6.6) or (18.6.12), as appropriate.

If A has a multiplicative identity element e_{A}, and if λ is a positive linear functional on A in the strict sense, then we can take y = e_{A} in (18.6.11) to get that |λ(x)| ≤ λ(e_{A})^{1/2} λ(x^{∗} x)^{1/2} (18.7.3)

for every x ∈ A. Note that λ(e_{A}) ≥ 0, (18.7.4)

by taking x = e_{A} in (18.6.3).

Suppose that we do not ask that A have a multiplicative identity element, and that λ is a positive linear functional on A in the strict sense with respect to the involution. Let us say that

(18.7.5) λ is positive in the strong sense

with respect to the involution if (18.7.1) or (18.7.2) holds for every x ∈ A, as appropriate, and there is a nonnegative real number c such that

|λ(x)| ≤ c^{1/2} λ(x^{∗} x)^{1/2} (18.7.6)

for every x ∈ A. We may also say that λ is a positive functional in the strong sense on A with respect to the involution in this case. If A has a multiplicative identity element e_{A}, then positivity in the strict sense implies positivity in the strong sense, as before, and we may simply say that λ is a positive linear functional or positive functional on A with respect to the involution. In this case, we can take x = e_{A} in (18.7.6) to get that

λ(e_{A}) ≤ c. (18.7.7)

In [11, 286], positive linear functionals are considered on algebras with multiplicative identity elements. In [291], positivity is defined for bounded linear functionals on complex C^{∗} algebras that may not have a multiplicative identity element, and (18.7.1) is basically included in the definition. If the algebra does not have a multiplicative identity element, then a positive linear functional on the algebra may be extended to a positive linear functional on the unitization of the algebra, as in Corollary 1 on p271 of [291].

In the complex case, it is easy to see that (18.7.1) holds if and only if

λ(x) ∈ R (18.7.8)

for every x ∈ A that is self-adjoint. In the real case, (18.7.2) holds if and only if

λ(x) = 0 (18.7.9)

for every x ∈ A that is anti-self-adjoint, as in Section 7.5.

18.8 Positive linear functionals on A_{1}

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗} that is conjugate-linear in the complex case again, and let

A_{1} = A × R or A × C, (18.8.1)

as appropriate, be defined as an associative algebra over the real or complex numbers, as appropriate, as in Section 6.15. Also let

(x, t) ↦ (x, t)^{∗} (18.8.2)

be the involution on A_{1} defined in Section 17.12. This involution on A_{1} is conjugate-linear in the complex case, because of the correspoinding property of the involution on A.

Remember that e_{A_{1}} = (0, 1) is the multiplicative identity element in A_{1}. If λ_{1} is a positive linear functional on A_{1}, then

λ(x) = λ_{1}((x, 0)) (18.8.3)

defines a positive linear functional on A in the strong sense. More precisely, (18.7.6) holds in this case with

c = λ_{1}(e_{A_{1}}) = λ_{1}((0, 1)) ≥ 0. (18.8.4)

This corresponds to the “only if” part of Theorem 21.18 on p317 of [151].

Conversely, suppose that λ is a positive linear functional on A in the strong sense, and let c be a nonnegative real number such that (18.7.6) holds for every x ∈ A. Put λ_{1}((x, t)) = λ(x) + t c (18.8.5)

for every (x, t) ∈ A_{1}, which defines a linear functional on A_{1}. Clearly

λ_{1}((x, t)^{∗}) = \overline{λ_{1}((x, t))} (18.8.6)

or λ_{1}((x, t)^{∗}) = λ_{1}((x, t)), (18.8.7)

18.9. ANOTHER NONNEGATIVITY CONDITION 413 as appropriate, for each (x, t) ∈ A_{1}, because of (18.7.1) or (18.7.2), as appropriate.

If (x, t) ∈ A_{1}, then (x, t)^{∗} (x, t) = (x^{∗} x + t x^{∗} + \overline{t} x, |t|^{2}) (18.8.8) in the complex case, and (x, t)^{∗} (x, t) = (x^{∗} x + t x^{∗} + t x, t^{2}) (18.8.9) in the real case. This implies that λ_{1}((x, t)^{∗} (x, t)) = λ(x^{∗} x) + 2 Re(\overline{t} λ(x)) + |t|^{2} c (18.8.10) in the complex case, and λ((x, t)^{∗} (x, t)) = λ(x^{∗} x) + 2 t λ(x) + t^{2}c (18.8.11) in the real case, using (18.7.1) and (18.7.2), as appropriate. It follows that

(18.8.12)  λ((x, t)^{∗} (x, t)) ≥ λ(x^{∗} x) − 2 |t| |λ(x)| + |t|^{2} c
≥ λ(x^{∗} x) − 2 |t| c^{1/2} + |t|^{2} c
= (λ(x^{∗} x)^{1/2} − |t| c^{1/2})^{2} ≥ 0

in both cases, using (18.7.6) in the second step. This corresponds to the “if” part of Theorem 21.18 on p317 of [151].

18.9 Another nonnegativity condition

Let X be a nonempty metric or topological space, and remember that C_{com}(X, R), C_{com}(X, C) (18.9.1) are the spaces of continuous real and complex-valued functions on X with compact support, respectively, as in Section 5.1. A linear functional λ on either of these spaces is said to be nonnegative if λ(f) ≥ 0 (18.9.2) for every nonnegative real-valued continuous function f on X with compact support. In the complex case, this includes the condition that λ(f) ∈ R (18.9.3) for all such functions f.

Of course, C_{com}(X, R) and C_{com}(X, C) are commutative associative algebras over the real and complex numbers, respectively, with respect to pointwise multiplication of functions on X. We may also consider these algebras to be equipped with involutions, where we take the involution on C_{com}(X, R) to be the identity mapping, and the involution on C_{com}(X, C) to be defined by taking the complex conjugate \overline{f} (18.9.4)

of f ∈ C_{com}(X, C).

In both cases, a nonnegative linear functional in the sense considered here is a positive linear functional in the strict sense as in Section 18.6, with respect to the corresponding involution. Conversely, every positive linear functional in the strict sense on these algebras is nonnegative in the sense considered here. This uses the fact that if f is a nonnegative real-valued continuous function on X with compact support, then its square root

f^{1/2} (18.9.5)

is a nonnegative real-valued continuous function on X with the same support as f.

Note that (18.7.2) holds trivially for any linear functional on C_{com}(X, R), because the involution is taken to be the identity mapping on C_{com}(X, R). In the complex case, (18.7.1) is equivalent to the condition that (18.9.3) hold when

(18.9.6) f is a continuous real-valued function on X with compact support,

as in (18.7.8).

If f is any continuous real-valued function on X, then

f_{+} = max(f, 0) (18.9.7)

and f_{−} = max(−f, 0) (18.9.8)

are nonnegative continuous real-valued functions on X. Note that

f = f_{+} − f_{−} (18.9.9)

on X. If f has compact support in X, then

(18.9.10) f_{+}, f_{−} have compact supports as well.

This uses the well-known fact that a closed set in X that is contained in a compact set is compact too. If λ is a linear functional on C_{com}(X, C) that satisfies (18.9.3) when f is a nonnegative real-valued continuous function on X with compact support, then it follows that (18.9.3) holds for every continuous real-valued function f on X with compact support.

18.10 Nonnegativity and local compactness

Let X be a nonempty metric or topological space again, and let λ be a nonnegative linear functional on C_{com}(X, R) or C_{com}(X, C). Thus λ is a positive linear functional in the strict sense on C_{com}(X, R) or C_{com}(X, C), as appropriate, as a commutative associative algebra over the real or complex numbers, as appropriate, with respect to the involution mentioned in the previous section. We have also seen that λ satisfies one of the conditions needed to be a positive linear functional in the strong sense, as in Section 18.7. The other condition may be restated as saying that there is a nonnegative real number c such that

|λ(f)| ≤ c^{1/2} λ(|f|^{2})^{1/2} (18.10.1)

for every f ∈ C_{com}(X, R) or C_{com}(X, C), as appropriate.

Suppose for the moment that this condition holds, and let a be a continuous real-valued function on X with compact support such that

0 ≤ a ≤ 1 (18.10.2)

on X. This implies that a^{2} ≤ a (18.10.3)

on X, so that a − a^{2} ≥ 0 on X, and thus

λ(a − a^{2}) ≥ 0. (18.10.4)

This means that λ(a^{2}) ≤ λ(a), which we can use in (18.10.1) to get that

λ(a) ≤ c. (18.10.5)

Suppose now that X is a nonempty metric space or Hausdorff topological space that is locally compact, as in Section 5.1. If f is a continuous real or complex-valued function on X with compact support, then the version of Urysohn’s lemma mentioned in Section 5.1 implies that there is a continuous real-valued function b on X with compact support such that

0 ≤ b ≤ 1 (18.10.6)

on X and b = 1 on supp f. (18.10.7)

This means that f = b f (18.10.8)

on X, so that |λ(f)| = |λ(b f)| ≤ λ(b^{2})^{1/2} λ(|f|^{2})^{1/2}, (18.10.9)

where the second step is as in Section 18.6. If (18.10.5) holds for some c ≥ 0 and all continuous real-valued functions a on X with compact support that satisfy (18.10.2) on X, then we can take a = b^{2} to get that (18.10.1) holds. This implies that λ is positive in the strong sense on C_{com}(X, R) or C_{com}(X, C), as appropriate, as in Section 18.7.

Let μ be a nonnegative Borel measure on X such that

μ(K) < +∞ (18.10.10) for all compact subsets K of X. This implies that continuous real and complexvalued functions on X with compact support are integrable on X with respect to μ, so that

λ_{μ}(f) = ∫_{X} f dμ (18.10.11)

defines a nonnegative linear functional on C_{com}(X, R) and C_{com}(X, C).

If μ(X) < ∞, (18.10.12)

then

|∫_{X} fdμ| ≤ ∫_{X} |f| dμ ≤ μ(X)^{1/2} ( ∫_{X} |f|^{2} dμ)^{1/2} (18.10.13)

for all square-integrable functions f on X with respect to μ, by the Cauchy– Schwarz inequality. In this case, we get that λ_{μ} is positive in the strong sense on C_{com}(X, R) and C_{com}(X, C), as in Section 18.7. Of course, if X is compact, then (18.10.12) follows from (18.10.10).

If λ is any nonnegative linear functional on C_{com}(X, R) or C_{com}(X, C), then a version of another type of Riesz representation theorem states that there is a unique nonnegative Borel measure μ on X with suitable regularity properties such that (18.10.10) holds, and λ can be expressed as in (18.10.11). If there is a nonnegative real number c such that (18.10.5) holds for all continuous realvalued functions a on X that satisfy (18.10.2), then

μ(X) ≤ c. (18.10.14)

This is all much simpler when X is equipped with the discrete metric or topology, so that C_{com}(C, R) and C_{com}(X, C) are the same as c_{00}(X, R) and c_{00}(X, C), respectively, as mentioned in Section 5.1. If ρ is a nonnegative realvalued function on X, then

λ_{ρ}(f) = ∑_{x∈X} f(x)ρ(x) (18.10.15)

defines a nonnegative linear functional on c_{00}(X, R) and c_{00}(X, C). If ρ is summable on X, then one can use the Cauchy–Schwarz inequality for sums to get that λ_{ρ} is positive in the strong sense on c_{00}(X, R) and c_{00}(X, C), as before. Conversely, if λ_{ρ} is positive in the strong sense on c_{00}(X, R) or c_{00}(X, C), then ρ is summable on X, with sum less than or equal to the corresponding constant c. Any nonnegative linear functional on c_{00}(X, R) or c_{00}(X, C) can be expressed as (18.10.15) for a unique nonnegative real-valued function ρ on X.

18.11 Positivity on Banach algebras

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A}, ∥e_{A}∥_{A} = 1, and an involution y ↦ y^{∗} that is conjugate linear in the complex case. Also let λ be a positive linear functional on A, as in Section 18.7. Suppose that w ∈ A has the property that

e_{A} − w = v^{∗} v (18.11.1)

for some v ∈ A. This implies that

λ(e_{A}) − λ(w) = λ(e_{A} − w) = λ(v^{∗} v) ≥ 0, (18.11.2)

so that λ(w) ≤ λ(e_{A}). (18.11.3)

Note that w^{∗} = w (18.11.4)

in this case, because e_{A} and v v^{∗} are self-adjoint.

If ∥w∥_{A} ≤ 1, (18.11.5)

then b_{1/2}(−w) ∈ A may be defined as in Section 17.2, and

b_{1/2}(−w)^{2} = e_{A} − w. (18.11.6)

Suppose that ∥y^{∗}∥_{A} ≤ C ∥y∥_{A} (18.11.7)

for some C ≥ 1 and all y ∈ A, so that the involution is continuous on A. If w satisfies (18.11.4) too, then

b_{1/2}(−w)^{∗} = b_{1/2}(−w), (18.11.8)

so that (18.11.1) holds, with v = b_{1/2}(−w). It follows that (18.11.3) holds, as before. If w is any self-adjoint element of A, then we get that

|λ(w)| ≤ λ(e_{A}) ∥w∥_{A}, (18.11.9)

by multiplying w by ±1/∥w∥_{A} when w ≠ 0.

If x is any element of A, then

|λ(x)| ≤ λ(e_{A})^{1/2} λ(x^{∗} x)^{1/2} ≤ λ(e_{A}) ∥x^{∗} x∥_{A}^{1/2}, (18.11.10)

using (18.7.3) in the first step, and (18.11.9) in the second step. This implies that |λ(x)| ≤ C^{1/2} λ(e_{A}) ∥x∥_{A}, (18.11.11)

because of (18.11.7). This corresponds to part of Proposition 4.7.1 on p123 of [11], and to Theorem 21.19 on p317 of [151], when C = 1. This also corresponds to part of part (e) of Theorem 11.31 on p284 of [286], and it is related to 4.1 and Corollary 1 on p271 of [291].

If z ∈ A is self-adjoint, then

|λ(z)| ≤ λ(e_{A})^{1/2} λ(z^{2})^{1/2}, (18.11.12)

418 CHAPTER 18. ALGEBRAS, INVOLUTIONS, AND POSITIVITY

because of (18.7.3). If x ∈ A and l is a positive integer, then we can take z = (x^{∗} x)^{l}, to get that λ((x^{∗} x)^{l}) ≤ λ(e_{A})^{1/2} λ((x^{∗} x)^{2 l})^{1/2}. (18.11.13) It is well known that

∑_{j=1}^{n} 2^{−j} = 1 − 2^{−n} (18.11.14)

for every positive integer n. One can use (18.11.13) repeatedly to get that λ(x^{∗} x) ≤ λ(e_{A})^{1−2^{−n}} λ((x^{∗} x)^{2^{n}})^{2^{−n}} (18.11.15) for every nonnegative integer n. Note that λ((x^{∗} x)^{2^{n}}) ≤ C^{1/2} λ(e_{A}) ∥(x^{∗} x)^{2^{n}}∥_{A}, (18.11.16) as in (18.11.11).

It follows that λ(x^{∗} x) ≤ C^{2^{−n−1}} λ(e_{A}) ∥(x^{∗} x)^{2^{n}}∥_{A}^{2^{−n}} (18.11.17) for every nonnegative integer n. Taking the limit as n → ∞, we get that λ(x^{∗} x) ≤ λ(e_{A}) r_{A}(x^{∗} x), (18.11.18) where r_{A}(·) is as in Section 6.13. This implies that |λ(x)| ≤ λ(e_{A})^{1/2} λ(x^{∗} x)^{1/2} ≤ λ(e_{A}) r_{A}(x^{∗} x)^{1/2}, (18.11.19) using (18.7.3) in the first step. This corresponds to Corollary 21.21 on p319 of [151]. This also corresponds to part (c) of Theorem 11.31 on p284 of [286], and we shall say more about this in the next section.

18.12 Another approach to estimating λ

Let (A, ∥·∥_{A}) be a real or complex Banach algebra with a multiplicative identity element e_{A} and ∥e_{A}∥_{A} = 1. Suppose that w ∈ A satisfies r_{A}(w) < 1, (18.12.1) where r_{A}(w) is as in Section 6.13. Remember that r_{A}(t w) = |t| r_{A}(w) (18.12.2) for every t ∈ R or C, as appropriate, as in Section 10.14. In particular, r_{A}(−w) < 1, so that b_{1/2,−w}(1) ∈ A may be defined as in Section 17.10, and satisfies b_{1/2,−w}(1)^{2} = e_{A} − w. (18.12.3)

18.13. SOME RELATED PROPERTIES OF λ 419

Let y ↦ y^{∗} be an involution on A that is conjugate-linear in the complex case, and suppose that w is self-adjoint. If this involution is continuous on A, then b_{1/2,−w}(1)^{∗} = b_{1/2,−w}(1), (18.12.4)

as in Section 17.10. This also holds when A is complex, without asking that the involution be continuous, as before.

Let λ be a positive linear functional on A again. Under the conditions mentioned in the previous paragraphs, (18.11.3) holds, as before. If w is any self-adjoint element of A, t ∈ R, and

|t| r_{A}(w) < 1, (18.12.5)

then t w is self-adjoint in A, and

λ(t w) ≤ λ(e_{A}). (18.12.6)

This implies that |λ(w)| ≤ λ(e_{A}) r_{A}(w) (18.12.7)

for every self-adjoint w ∈ A.

If x is any element of A, then x^{∗} x is self-adjoint, and we can take w = x^{∗} x in (18.12.7) to get (18.11.18), and thus (18.11.19). This corresponds to part (c) of Theorem 11.31 on p284 of [286].

18.13 Some related properties of λ

Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers with a multiplicative identity element e_{A}, ∥e_{A}∥_{A} = 1, and an involution y ↦ y^{∗} which is conjugate-linear in the complex case. Also let λ be a positive linear functional on A, and suppose that

(18.13.1) the involution on A is continuous, or A is complex.

Remember that (18.11.19) holds for every x ∈ A under these conditions, as in the previous two sections. If x is normal, then we can combine this with (17.11.5) to get that |λ(x)| ≤ λ(e_{A}) r_{A}(x). (18.13.2)

This corresponds to part (d) of Theorem 11.31 on p284 of [286].

Note that (18.13.2) implies that

|λ(x)| ≤ λ(e_{A}) ∥x∥_{A}. (18.13.3)

This also follows from (18.11.11) when (18.11.7) holds with C = 1. If A is commutative, then every element of A is normal. This means that (18.13.3) holds for every x ∈ A when A is commutative, and (18.13.1) holds. This corresponds to part of part (e) of Theorem 11.31 on p284 of [286].

If A is complex, then the first part of part (e) of Theorem 11.31 on p284 of [286] states that

(18.13.4) λ is a bounded linear functional on A.

In this statement, A is not asked to be commutative, nor is the involution on A asked to be continuous.

18.14 Positivity and homomorphisms

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗} that is conjugate-linear in the complex case. Also let h be an algebra homomorphism from A into R or C, as appropriate, as in Section 6.9. Suppose that for every x ∈ A,

h(x^{∗}) = \overline{h(x)} (18.14.1)

in the complex case, or h(x^{∗}) = h(x) (18.14.2)

in the real case. In both cases, we get that

h(x^{∗} x) = h(x^{∗}) h(x) = |h(x)|^{2}, (18.14.3)

so that h is positive in the strong sense with respect to the involution on A, as in Section 18.7.

Let U be a nonempty open subset of the complex plane, and suppose that

\overline{z} ∈ U (18.14.4)

for every z ∈ U. Remember that the space H(U) of holomorphic functions on U is a commutative associative algebra over the complex numbers with respect to pointwise multiplication of functions, as in Section 13.12. If f ∈ H(U), then let f^{∗} be the complex-valued function defined on U by

f^{∗}(z) = \overline{f(\overline{z})}. (18.14.5)

It is well known that f^{∗} is holomorphic on U as well. This defines a conjugatelinear involution on H(U).

If w ∈ U, then h_{w}(f) = f(w) (18.14.6)

defines an algebra homomorphism from H(U) onto C, as in Section 13.12. Observe that h_{w}(f^{∗}) = \overline{f(\overline{w})} = \overline{h\overline{w}(f)}. (18.14.7)

Thus h_{w} satisfies (18.14.1) if and only if

\overline{w} = w, (18.14.8) which is to say that w ∈ R.

Let U_{1} be the open unit disk in the complex plane, and let A(U_{1}) be the disk algebra, consisting of continuous complex-valued functions on the closed unit disk that are holomorphic on U_{1}, as in Section 13.13. This is a Banach algebra with respect to the supremum norm, as before, and (18.14.5) defines an involution on A(U_{1}) that preserves the supremum norm. Note that (18.14.6) defines an algebra homomorphism from A(U_{1}) onto C for every w in the closed unit disk, as before. This example is mentioned in Exercise (3) on p126 of [11], and in Exercise 10 on p289 of [286].

18.15 Positivity and commutativity

Let (A, ∥ · ∥_{A}) be a commutative Banach algebra over the complex numbers with a multiplicative identity element e_{A}, ∥e_{A}∥_{A} = 1, and a conjugate-linear involution y ↦ y^{∗}, and let λ be a positive linear functional on A. If x ∈ A, then

|λ(x)| ≤ λ(e_{A}) r_{A}(x) (18.15.1)

as in Section 18.13. Remember that Sp(A) is the space of all nonzero algebra homomorphisms from A into C, as in Section 12.10, and that

x̂(h) = h(x) (18.15.2)

is called the Gelfand transform of x. We also have that

∥x̂∥_{sup} = ∥x̂∥_{sup,Sp(A)} = r_{A}(x), (18.15.3)

where the left side is the supremum norm of x̂ on Sp(A), as before. Combining this with (18.15.1), we get that

|λ(x)| ≤ λ(e_{A}) ∥x̂∥_{sup}. (18.15.4)

Remember too that Â = {x̂ : x ∈ A} (18.15.5)

is a subalgebra of the algebra C(Sp(A), C) of continuous complex-valued functions on Sp(A) that contains the constant functions and separates points in Sp(A), as in Section 12.11. We would like to define a linear functional λ̂ on Â by λ̂(x̂) = λ(x) (18.15.6)

for every x ∈ A. Note that λ(x) = 0 when x̂ = 0, because of (18.15.4). One can use this to check that λ̂ is well defined on Â in this way. Of course,

|λ̂(x̂)| ≤ λ(e_{A}) ∥x̂∥_{sup} (18.15.7)

for every x ∈ A, by (18.15.4).

Suppose that y ↦ y^{∗} is symmetric on A, in the sense that

h(y^{∗}) = \overline{h(y)} (18.15.8)

for every y ∈ A, as on p285 of [286]. This implies that Â is invariant under complex conjugation, so that

(18.15.9)  Â is dense in C(Sp(A), C)

with respect to the supremum metric, by the Stone–Weierstrass theorem. It follows that

λ̂ has a unique extension to a bounded

(18.15.10)

linear functional on C(Sp(A), C),

with respect to the supremum norm, and it is easy to see that this extension is nonnegative on C(Sp(A), C). This corresponds to part of the proof of Theorem 11.32 on p285 of [286]. This is related to a famous theorem of Bochner, as in [286].

In fact, the argument in [286] shows more than this. One can use the Hahn– Banach theorem to get an extension of λ̂ to a bounded linear functional on C(Sp(A), C), with dual norm equal to λ(e_{A}) with respect to the supremum norm on C(Sp(A), C), without using (18.15.8). A version of the Riesz representation theorem implies that this extension corresponds to a regular complex Borel measure μ on Sp(A), with total variation norm equal to λ(e_{A}). In particular,

μ(Sp(A)) = λ̂(ê_{A}) = λ(e_{A}), (18.15.11)

and one can use this to get that μ is a nonnegative real measure on Sp(A). If (18.15.8) holds, then μ is uniquely determined by λ, because of (18.15.9).

Conversely, if (18.15.8) holds, then any nonnegative linear functional on C(Sp(A), C) leads to a positive linear functional on A, using composition with the Gelfand transform. This corresponds to another part of the proof of Theorem 11.32 on p285 of [286].

Algebras over fields

19.1 Algebras and homomorphisms

Let k be a field, such as the real or complex numbers, and let V, W, and Z be vector spaces over k. A mapping B from V × W into Z is said to be bilinear if B(v, w) is linear as a function of v ∈ V for each w ∈ W, and linear as a function of w ∈ W for each v ∈ V, as in Section 5.13.

We can define the notion of an algebra in the strict sense over k in the same way as in Section 6.1, for algebras over R or C. Thus A is an algebra in the strict sense over k if A is a vector space over k equipped with a mapping from A × A into A that is bilinear over k. The properties of associativity and commutativity may be defined as before, as well as multiplicative identity elements.

If X is a nonempty set, then the space of all k-valued functions on X is a commutative associative algebra over k, with respect to pointwise multiplication of functions. If 1 = 1_{k} is the multiplicative identity element in k, then the function 1_{X} equal to 1_{k} at every point in X is the multiplicative identity in this algebra.

If V is a vector space over k, then the space L(V) of linear mappings from V into itself is an associative algebra over k with respect to composition of mappings. The identity mapping I = I_{V} on V is the multiplicative identity element in L(V), as before.

If A is an algebra in the strict sense over k and A_{0} is a linear subspace of A such that a b ∈ A_{0} (19.1.1)

for every a, b ∈ A_{0}, then A_{0} is an algebra in the strict sense over k with respect to the restriction of multiplication on A to A_{0}. In this case, A_{0} is said to be a subalgebra of A, as in Section 6.1. If A is associative or commutative, then A_{0} has the same property, as before.

Let A, B be algebras in the strict sense over k. A linear mapping ϕ from A into B is said to be an algebra homomorphism if

ϕ(x y) = ϕ(x) ϕ(y) (19.1.2) for every x, y ∈ A, as in Section 6.3. If A and B have multiplicative identity elements e_{A} and e_{B}, respectively, then one may also wish to ask that

ϕ(e_{A}) = e_{B}, (19.1.3)

as before. If A has a multiplicative identity element e_{A} and ϕ(A) = B, then (19.1.2) implies that ϕ(e_{A}) is the multiplicative identity element in B. A one-to-one homomorphism ϕ from A onto B is called as algebra isomorphism, in which case ϕ^{−1} is an isomorphism from B onto A.

A linear mapping ϕ from A into B is said to be an opposite algebra homomorphism if ϕ(x y) = ϕ(y) ϕ(x) (19.1.4)

for every x, y ∈ A, as in Section 6.4. If A and B have multiplicative identity elements e_{A} and e_{B}, respectively, then one may wish to ask that (19.1.3) hold, as usual. If A has a multiplicative identity element e_{A} and ϕ(A) = B, then (19.1.4) implies that ϕ(e_{A}) is the multiplicative identity element in B. A oneto-one opposite algebra homomorphism ϕ from A onto B is called an opposite algebra isomorphism, which means that ϕ^{−1} is an opposite algebra isomorphism from B onto A. If ϕ is an opposite algebra homomorphism from A onto itself, and ϕ ∘ ϕ = I_{A}, (19.1.5)

then ϕ is said to be an algebra involution on A, as before.

Let C be another algebra in the strict sense over k. If ϕ is a homomorphism from A into B, and if ψ is a homomorphism from B into C, then ψ ∘ ϕ is a homomorphism from A into C. If ϕ and ψ are opposite algebra homomorphisms, then ψ ∘ ϕ is a homomorphism. If one of ϕ and ψ is a homomorphism and the other is an opposite algebra homomorphism, then ψ ∘ ϕ is an opposite algebra homomorphism. This was mentioned in Section 12.7 when k = R or C.

19.2 Ideals and identity elements

Let k be a field, let A be an algebra in the strict sense over k, and consider

A_{1} = A × k (19.2.1)

initially as a vector space over k, with respect to coordinatewise addition and scalar multiplication. We can use multiplication on A to define multiplication on A_{1} by (a, t) (b, z) = (a b + z a + t b, t z) (19.2.2)

for every a, b ∈ A and t, z ∈ k, as in Section 6.15. This makes A_{1} into an algebra in the strict sense over k, for which a ↦ (a, 0_{k}) is an injective algebra homomorphism from A into A_{1}. By construction,

e_{A_{1}} = (0, 1_{k}) (19.2.3) is the multiplicative identity element in A_{1}. If A is associative or commutative, then A_{1} has the same property, as before.

Let A be an algebra over k in the struct sense again, and let I be a linear subspace of A. If a x ∈ I (19.2.4)

for every a ∈ A and x ∈ I, then I is said to be a left ideal in A. Similarly, if

x a ∈ I (19.2.5)

for every a ∈ A and x ∈ I, then I is said to be a right ideal in A. Note that left and right ideals in A are both subalgebras of A. If A is commutative, then left and right ideals in A are the same.

If I is both a left and right ideal in A, then I is said to be a two-sided ideal in A, as in Section 6.11. If B is another algebra in the strict sense over k, then the kernel of any homomorphism from A into B is a two-sided ideal in A. The kernel of an opposite algebra homomorphism from A into B is a two-sided ideal in A as well.

If I is a linear subspace of A, then the quotient space A/I may be defined as a vector space over k in a standard way. Let q be the natural quotient mapping from A onto A/I, which is a linear mapping with kernel equal to I. If I is a two-sided ideal in A, then one can define multiplication on A/I in such a way that q(a) q(b) = q(a b) (19.2.6)

for every a, b ∈ A, as mentioned in Section 6.11. In this case, A/I may be considered as an algebra in the strict sense over k, and q is an algebra homomorphism from A onto A/I. Of course, if A is commutative or associative, then A/I has the same property, as before.

Let V be a vector space over k. If v ∈ V, then

{T ∈ L(V) : T (v) = 0} (19.2.7)

is a left ideal in L(V). If λ is a linear functional on V, then

{T ∈ L(V) : λ ∘ T = 0} (19.2.8)

is a right ideal in L(V).

19.3 Invertible elements and homomorphisms

Let k be a field, and let A be an associative algebra over k with a multiplicative identity element e_{A}. An element x of A is said to be invertible in A if there is an element x^{−1} of A such that

x x^{−1} = x^{−1} x = e_{A}, (19.3.1) as in Section 6.5. The collection G(A) of invertible elements of A is a group with respect to multiplication, as before. If x ∈ A is nilpotent, so that x^{l} = 0 for some positive integer l, then

e_{A} − x ∈ G(A), (19.3.2)

as in Section 6.13. If x is any element of A, then the spectrum of x with respect to A may be defined by

σ_{A}(x) = {λ ∈ k : λ e_{A} − x ∉ G(A)}, (19.3.3)

as in Section 6.8.

Let h be a homomorphism from A into k, as an algebra into itself, and let 0_{k} and 1_{k} be the additive and multiplicative identity elements in k. It is easy to see that h(e_{A}) is either 0_{k} or 1_{k}, as in Section 6.9, and that h ≡ 0_{k} on A when h(e_{A}) = 0_{k}. Suppose that h(e_{A}) = 1_{k}, and observe that

h(a) ≠ 0_{k} when a ∈ G(A), (19.3.4)

as before. If a is any element of A, then it follows that

h(a) ∈ σ_{A}(a), (19.3.5)

as before. We may use the notation Sp(A) = Sp_{k}(A) for the set of all nonzero algebra homomorphisms from A into k, as in Sections 12.10 and 12.14, particularly when A is commutative.

Let B be another associative algebra over k with a multiplicative identity element e_{B}, and let ϕ be a homomorphism or an opposite algebra homomorphism from A into B such that ϕ(e_{A}) = e_{B}. If x ∈ A is invertible, then ϕ(x) is invertible in B, with ϕ(x^{−1}) = ϕ(x)^{−1}, (19.3.6)

as in Section 10.11. This implies that

ϕ(G(A)) ⊆ G(B), (19.3.7)

as before. It follows that σ_{B}(ϕ(y)) ⊆ σ_{A}(y) (19.3.8)

for every y ∈ A. If β ∈ Sp(B), then

ϕ̂(β) = β ∘ ϕ (19.3.9)

is an element of Sp(A), as in Section 12.14.

Let V ≠ {0} be a vector space over k, and let T be a linear mapping from V into itself. Remember that λ ∈ k is said to be an eigenvalue of T if there is a nonzero eigenvector v ∈ V corresponding to λ, so that

T (v) = λ v. (19.3.10)

The set σ_{p}(T) of eigenvalues of T is called the point spectrum of T, as in Section 9.1. It is easy to see that σ_{p}(T) ⊆ σ_{L(V)}(T), (19.3.11)

and that σ_{p}(T) = σ_{L(V)}(T) (19.3.12)

when V has finite dimension, as before. If k is algebraically closed, then

σ_{L(V)}(T) ≠ ∅ (19.3.13)

when V has finite dimension.

19.4 Left and right multiplication operators

Let k be a field, and let A be an algebra over k in the strict sense. An element e_{L} of A is said to be a left multiplicative identity element in A if

e_{L} x = x (19.4.1)

for every x ∈ A, as in Section 18.1. Similarly, e_{R} ∈ A is said to be a right multiplicative identity element in A if

x e_{R} = x (19.4.2)

for every x ∈ A. Of course, a multiplicative identity element in A is the same as an element of A that is both a left and right multiplicative identity element. If A has a left multiplicative identity element e_{L} and a right multiplicative identity element e_{R}, then e_{L} = e_{R}, so that this is a multiplicative identity element in A, as before.

If a ∈ A, then let L_{a}(x) = a x (19.4.3)

and R_{a}(x) = x a (19.4.4)

be the corresponding left and right multiplication operators on A, as in Sections 6.3 and 6.4. Of course, a ↦ L_{a} (19.4.5)

and a ↦ R_{a} (19.4.6)

are linear mappings from A into the space L(A) of linear mappings from A into itself, as before. Remember that e_{L} ∈ A is a left multiplicative identity element in A if and only if L_{e_{L}} = I_{A}, (19.4.7)

as in Section 18.1. This happens if and only if

R_{a}(e_{L}) = a (19.4.8) for every a ∈ A. Similarly, e_{R} ∈ A is a right multiplicative identity element in A if and only if R_{e_{R}} = I_{A}, (19.4.9)

which is the same as saying that

L_{a}(e_{R}) = a (19.4.10)

for every a ∈ A.

Suppose now that A is an associative algebra over k, so that (19.4.5) is an algebra homomorphism from A into L(A), as in Section 6.3. Similarly, (19.4.6) is an opposite algebra homomorphism from A into L(A), as in Section 6.4. Suppose also that A has a multiplicative identity element e_{A}, so that (19.4.5) and (19.4.6) are injective, as in (19.4.8) and (19.4.10).

If A has finite dimension as a vector space over k, and k is algebraically closed, then σ_{L(A)}(L_{a}) ≠ ∅ (19.4.11)

for every a ∈ A, as in (19.3.13). This implies that

σ_{A}(a) ≠ ∅ (19.4.12)

for every a ∈ A, as in (19.3.8).

19.5 Centralizers and invertibility

Let k be a field, and let A be an associative algebra over k. If E is a nonempty subset of A, then the centralizer of E in A is defined by

(19.5.1) C(E) = C_{A}(E) = {a ∈ A : a x = x a for every x ∈ E},

as in Section 14.5. One can check that

(19.5.2) C_{A}(E) is a subalgebra of A,

as before. Of course, if A has a multiplicative identity element e_{A}, then

e_{A} ∈ C_{A}(E). (19.5.3)

Note that E ⊆ C_{A}(C_{A}(E)), (19.5.4)

as before.

If A has a multiplicative identity element e_{A} and a ∈ C_{A}(E) is invertible in A, then one can check that a^{−1} ∈ C_{A}(E), (19.5.5)

as in Section 14.5. This implies that

σ_{A}(y) = σ_{C_{A}(E)}(y) (19.5.6) for every y ∈ C_{A}(E), as before.

If E_{1}, E_{2} are nonempty subsets of A and

E_{1} ⊆ E_{2}, (19.5.7)

then C_{A}(E_{2}) ⊆ C_{A}(E_{1}), (19.5.8)

as in Section 14.5. The elements of E commute with each other if and only if

E ⊆ C_{A}(E), (19.5.9)

in which case C_{A}(C_{A}(E)) ⊆ C_{A}(E), (19.5.10)

by (19.5.8). Observe that

(19.5.11) C_{A}(E_{1}) is a commutative subalgebra of A

when C_{A}(E_{1}) ⊆ E_{1}. (19.5.12)

This implies that

(19.5.13) C_{A}(C_{A}(E)) is a commutative subalgebra of A

when the elements of E commute with each other, as in Section 14.5. If E is any nonempty subset of A, then

B = C_{A}(C_{A}(E)) (19.5.14)

is a subalgebra of A that contains E. If A has a multiplicative identity element e_{A}, then e_{A} ∈ B, and σ_{A}(b) = σ_{B}(b) (19.5.15)

for every b ∈ B, as in (19.5.6). This holds in particular when b ∈ E, as in Section 14.5.

If a ∈ A, then let L_{a} and R_{a} be as in (19.4.3) and (19.4.4), respectively, and put L_{A} = {L_{a} : a ∈ A}, R_{A} = {R_{a} : a ∈ A}, (19.5.16)

as in Section 14.7. These are subalgebras of L(A), and

L_{a} ∘ R_{b} = R_{b} ∘ L_{a} (19.5.17)

for every a, b ∈ A, because A is associative, as before. If A has a multiplicative identity element e_{A}, then

C_{L(A)}(L_{A}) = R_{A}, C_{L(A)}(R_{A}) = L_{A}, (19.5.18)

as in Section 14.7. One can use this to get that L_{a} is invertible in L(A) exactly when a is invertible in A, and similarly for R_{a}, as before. One can use this to get that σ_{L(A)}(L_{b}), σ_{L(A)}(R_{b}) = σ_{A}(b) (19.5.19)

for every b ∈ A, as in Section 14.7 again.

430 CHAPTER 19. ALGEBRAS OVER FIELDS

19.6 Some algebras of polynomials

Let k be a field, let n be a positive integer, and let X_{1}, …, X_{n} be n commuting indeterminates. One can define formal polynomials in X_{1}, …, X_{n} with coefficients in k in the same way as in Section 13.3. The space k[X_{1}, …, X_{n}] of these formal polynomials is a commutative associative algebra over k with a multiplicative identity element, as before.

Let a = (a_{1}, …, a_{n}) be an n-tuple of commuting elements of A. If p(X) = p(X_{1}, …, X_{n}) is an element of k[X_{1}, …, X_{n}], then one can define p(a) = p(a_{1}, …, a_{n}) ∈ A (19.6.1) as in Section 13.4. One may also use the notation p_{A}(a) = p_{A}(a_{1}, …, a_{n}) to indicate the role of A here, as before. Note that p(X) ↦ p(a) (19.6.2) defines an algebra homomorphism from k[X_{1}, …, X_{n}] into A, as before.

Let k^{n} be the space of n-tuples of elements of k, as usual. If w ∈ k^{n}, then h_{w}(p(X)) = p(w) (19.6.3) defines an algebra homomorphism from k[X_{1}, …, X_{n}] onto k, as in Section 13.5. Of course, h_{w}(X_{j}) = w_{j} (19.6.4) for each j = 1, …, n, by construction. In fact, w ↦ h_{w} (19.6.5) is a one-to-one mapping from k^{n} onto Sp(k[X_{1}, …, X_{n}]), as before.

Let us now take n = 1, and let X be a single indeterminate. If p(X) is an element of k[X] and a ∈ A, then p(a) = p_{A}(a) ∈ A may be defined as before, which is essentially the same as in Section 8.13. In particular, p(X) determines a polynomial function p_{k} on k. One can check that p_{k}(σ_{A}(a)) ⊆ σ_{A}(p_{A}(a)), (19.6.6) as in Section 8.13. This may be considered as at least part of a version of the spectral mapping theorem, as before.

One might also like to have that σ_{A}(p_{A}(a)) ⊆ p_{k}(σ_{A}(a)) (19.6.7) under suitable conditions, as before. If p(X) is a constant polynomial, then this holds when σ_{A}(a) ≠ ∅. If p(X) is not a constant polynomial, and k is algebraically closed, then one can verify that (19.6.7) holds, as in Section 8.13.

19.7. SUBALGEBRAS OF FINITE DIMENSION 431

19.7 Subalgebras of finite dimension

Let k be a field, let B be an associative algebra over k with a multiplicative identity element e = e_{B}, and let A be a subalgebra of B that contains e. Note that G(A) ⊆ G(B), (19.7.1) as in Section 7.3. If x ∈ A, then σ_{B}(x) ⊆ σ_{A}(x), (19.7.2) as before. This corresponds to (19.3.8) as well, with ϕ taken to be the obvious inclusion mapping from A into B.

Suppose that A has finite dimension as a vector space over k, and that e ≠ 0, to avoid trivialities. If a ∈ A, then one can get nontrivial formal polynomial p(X) with coefficients in k such that p_{A}(a) = 0. (19.7.3) Note that this implies that σ_{A}(a) ⊆ {t ∈ k : p_{k}(t) = 0}, (19.7.4) as in (19.6.6).

If a is invertible in B, then one can take p(X) to have degree which is as small as possible, to get that the constant term is not zero. One can use this to get a formal polynomial q(X) with coefficients in k such that a q_{A}(a) = e. (19.7.5) This means that a^{−1} = q_{A}(a) ∈ A. (19.7.6)

It follows that G(A) = A ∩ G(B) (19.7.7) in this case. This implies that σ_{A}(x) = σ_{B}(x) (19.7.8) for every x ∈ A.

19.8 An invariant linear subspace

Let k be a field, let V be a vector space over k, and let W be a linear subspace of V. Put L_{W} (V) = {T ∈ L(V) : T (W) ⊆ W }, (19.8.1) which is a subalgebra of L(V) that contains I_{V}.

If T ∈ L_{W} (V), then let T_{W} be the restriction of T to W. Note that

T ↦ T_{W} (19.8.2)

defines a homomorphism from L_{W} (V) into L(W), as algebras over k. The kernel of this homomorphism is equal to

{T ∈ L(V) : T = 0 on W }. (19.8.3)

Thus this a two-sided ideal in L_{W} (V), which is a left ideal in L(V) too.

The quotient space V/W may be defined as a vector space over k in a natural way. Let q_{V/W} be the natural quotient mapping from V onto V/W, which is a linear mapping with kernel equal to W. If T is any linear mapping from V into itself, then q_{V/W} ∘ T (19.8.4)

is a linear mapping from V into V/W. Clearly T ∈ L_{W} (V) if and only if

q_{V/W} ∘ T = 0 on W. (19.8.5)

In this case, there is a unique linear mapping T^{V/W} from V/W into itself such that T^{V/W} ∘ q_{V/W} = q_{V/W} ∘ T, (19.8.6)

by standard arguments.

One can check that T ↦ T^{V/W} (19.8.7)

is a homomorphism from L_{W} (V) into L(V/W), as algebras over k. The kernel of this homomorphism is equal to

{T ∈ L(V) : T (V) ⊆ W }. (19.8.8)

This is another two-sided ideal in L_{W} (V), which is also a right ideal in L(V).

Of course, the group G(L(V)) of invertible elements of L(V) consists of the one-to-one linear mappings T from V onto itself. This is also known as the general linear group of V, which may be denoted GL(V) as well.

It is easy to see that

G(L_{W} (V)) = {T ∈ GL(V) : T (W) = W }. (19.8.9)

If W has finite dimension, as a vector space over k, then

G(L_{W} (V)) = GL(V) ∩ L_{W} (V). (19.8.10)

Indeed, if W has finite dimention, T ∈ L_{W} (V), and T is injective on W, then T (W) = W.

The codimension of W as a linear subspace of V may be defined as the dimension of V/W, as a vector space over k. If W has finite codimension in V, then (19.8.10) holds. More precisely, if T ∈ L_{W} (V) and T (V) = V, then

T^{V/W} (V/W) = V/W, (19.8.11) by (19.8.6). If dim(V/W) < ∞, then it follows that

ker T^{V/W} = {0}. (19.8.12)

This means that T^{−1}(W) = W. (19.8.13)

Suppose that V has finite dimension, so that W and V/W have finite dimension too, with (19.8.14) dim V = dim W + dim(V/W).

In this case, L(V) has finite dimension as well, with

dim L(V) = (dim V)^{2}, (19.8.15)

by standard arguments. One can use a basis for V that includes a basis for W to get that

(19.8.16) dim L_{W} (V) = (dim W)^{2} + (dim V − dim W) (dim V).

19.9 Algebraic duals

Let k be a field, and let V be a vector space over k. As in Section 3.1, a linear functional on V is a linear mapping from V into k, considered as a onedimensional vector space over itself. The space V^{alg} of all linear functionals on V may be called the algebraic dual of V, as before. This is a linear subspace of the space of all k-valued functions on V, as usual.

Let W be another vector space over k, and let T be a linear mapping from V into W. If μ is a linear functional on W, then

T^{alg}(μ) = μ ∘ T (19.9.1)

is a linear functional on V, as in Section 3.13. This defines a linear mapping from W^{alg} into V^{alg}, which is the dual linear mapping associated to T, as before.

The space L(V, W) of all linear mappings from V into W is a linear subspace of the space of all W -valued functions on V, as in Section 2.2. Note that

T ↦ T^{alg} (19.9.2)

is a linear mapping from L(V, W) into L(W^{alg}, V^{alg}), as in Section 3.13.

If T ≠ 0, then T (v) ≠ 0 for some v ∈ V, which implies that μ(T (v)) ≠ 0 for some μ ∈ W^{alg}. This means that

(T^{alg}(μ))(v) = (μ ∘ T)(v) = μ(T (v)) ≠ 0, (19.9.3)

so that T^{alg} ≠ 0. Thus (19.9.2) is one-to-one on L(V, W).

Let Z be another vector space over k. If T_{1} is a linear mapping from V into W, and T_{2} is a linear mapping from W into Z, then

(19.9.4) (T_{2} ∘ T_{1})^{alg} = T_{1}^{alg} ∘ T_{2}^{alg}, as linear mappings from Z^{alg} into W^{alg}, as in Section 3.13.

In particular, we may consider linear mappings from V into itself, and the corresponding dual linear mappings on V^{alg}. The remarks in the previous paragraphs imply that (19.9.2) defines an injective opposite algebra homomorphism from L(V) into L(V^{alg}). Note that

(I_{V})^{alg} = I_{V^{alg}}. (19.9.5)

Suppose for the moment that V has finite dimension, so that (19.8.15) holds. In this case, we also have that

(19.9.6) dim V^{alg} = dim V,

and (19.9.7) dim L(V^{alg}) = (dim V^{alg})^{2} = (dim V)^{2}.

It follows that (19.9.2) maps L(V) onto L(V^{alg}) under these conditions.

If v ∈ V, then L̂_{v}(λ) = L̂_{v}^{V} (λ) = λ(v) (19.9.8)

defines an element of the algebraic dual (V^{alg})^{alg} of V^{alg}, as in Section 3.14. More precisely, v ↦ L̂_{v} (19.9.9)

defines a one-to-one linear mapping from V into (V^{alg})^{alg}, as before. If V has finite dimension, then

(19.9.10) dim V = dim V^{alg} = dim(V^{alg})^{alg},

as before. This implies that (19.9.9) maps V onto (V^{alg})^{alg} in this case.

If T is a linear mappng from V into W, then the second dual mapping (T^{alg})^{alg} may be defined as a linear mapping from (V^{alg})^{alg} into (W^{alg})^{alg} by taking the dual linear mapping associated to T^{alg}, as in Section 3.15. If w ∈ W, then let L̂_{w}^{W} ∈ (W^{alg})^{alg} be as in (19.9.8). One can check that

(T^{alg})^{alg}(L̂_{v}^{V}) = L̂_{T (v)}^{W} (19.9.11)

for every v ∈ V, as in Section 3.15.

19.10 Algebraic duals and linear subspaces

Let k be a field, let V be a vector space over k, and let V_{0} be a linear subspace over V. Also let T_{0} be the natural inclusion mapping from V_{0} into V, which sends every element of V_{0} to itself, considered as an element of V. If μ is a linear functional on V, then (T_{0})^{alg}(μ) = μ ∘ T_{0} (19.10.1)

is the same as the restriction of μ to V_{0}, considered as an element of (V_{0})^{alg}. This defines a linear mapping from V^{alg} into (V_{0})^{alg}, as in the previous section.

In fact, (19.10.2) (T_{0})^{alg}(V^{alg}) = (V_{0})^{alg}.

This is the same as saying that every linear functional on V_{0} can be extended to a linear functional on V. This can be seen using elementary arguments when V_{0} has finite codimension in V, and in particular when V has finite dimension. Otherwise, there are standard arguments based on the axiom of choice using Zorn’s lemma or Hausdorff’s maximality principle.

Put (19.10.3) (V_{0})^{⊥_{alg}} = ker(T_{0})^{alg} = {μ ∈ V^{alg} : μ ≡ 0 on V_{0}},

which is a linear subspace of V^{alg}. If V_{0} has finite dimension, then the codimension of (V_{0})^{⊥_{alg}} in V^{alg} is equal to

(19.10.4) dim(V_{0})^{alg} = dim V_{0},

because of (19.10.2). If V has finite dimension, then we get that

(19.10.5)  dim V = dim V^{alg} = dim(V_{0})^{⊥_{alg}} + dim(V_{0})^{alg}
= dim(V_{0})^{⊥_{alg}} + dim V_{0}.

If V_{1} is a linear subspace of V^{alg}, then put

(19.10.6) ^{⊥_{alg}}V_{1} = ⋂_{λ∈V_{1}} ker λ = {v ∈ V : λ(v) = 0 for every λ ∈ V_{1}},

which is a linear subspace of V. Clearly

V_{0} ⊆ ^{⊥_{alg}}((V_{0})^{⊥_{alg}}). (19.10.7)

In fact, V_{0} = ^{⊥_{alg}}((V_{0})^{⊥_{alg}}). (19.10.8)

This means that if v ∈ V and v is not an element of V_{0}, then there is a linear functional μ on V such that μ ≡ 0 on V_{0}, and

μ(v) ≠ 0. (19.10.9)

One can first get such a linear functional on the linear subspace of V spanned by V_{0} and v, and then use an extension of that to a linear functional on V.

Let T be a linear mapping from V into itself. If

T (V_{0}) ⊆ V_{0}, (19.10.10)

then it is easy to see that

(19.10.11) T^{alg}(μ) = μ ∘ T ≡ 0 on V_{0} for every μ ∈ (V_{0})^{⊥_{alg}}.

Equivalently, this means that

(19.10.12) T^{alg}((V_{0})^{⊥_{alg}}) ⊆ (V_{0})^{⊥_{alg}}.

This is also the same as saying that

(19.10.13) T (V_{0}) ⊆ ker μ for every μ ∈ (V_{0})^{⊥_{alg}},

which means that T (V_{0}) ⊆ ^{⊥_{alg}}((V_{0})^{⊥_{alg}}). (19.10.14)

This shows that (19.10.12) is equivalent to (19.10.10), because of (19.10.8).

19.11 Bilinear functionals

Let k be a field, and let V, W be vector spaces over k. A bilinear mapping b from V × W into k may be called a bilinear functional on V × W, as in Section 5.15. This is called a bilinear function on p251 of [37], and a bilinear form on p337 of [206] and p338 of [226]. The term “bilinear form” is defined another way on p281 of [37], as a type of formal polynomial with coefficients in k. This corresponds to a bilinear functional on the Cartesian product of two finite-dimensional vector spaces over k with the appropriate dimensions, using bases for these vector spaces, as on p282 of [37].

Let b be a bilinear functional on V × W, and put

b_{1,w}(v) = b(v, w) (19.11.1)

and b_{2,v}(w) = b(v, w) (19.11.2)

for all v ∈ V and w ∈ W, as in Sections 5.13 and 5.15. Note that b_{1,w} is a linear functional on V for each w ∈ W, and that w ↦ b_{1,w} is a linear mapping from W into V^{alg}, as before. Similarly, b_{2,v} is a linear functional on W for every v ∈ V, and v ↦ b_{2,v} is a linear mapping from V into W^{alg}. Any linear mapping from W into V^{alg} or from V into W^{alg} corresponds to a bilinear functional on V × W in this way, as before.

Let 0_{k} and 1_{k} be the additive and multiplicative identity elements in k, as before. If 1_{k} + 1_{k} = 0_{k}, then k is said to have characteristic two, as usual. This is the same as saying that −1_{k} = 1_{k}, and otherwise 1_{k} + 1_{k} has a multiplicative inverse in k.

Suppose now that V = W. In this case, one may refer to b as a bilinear form on V, as in Sections 5.15 and 8.5. If

b(v, w) = b(w, v) (19.11.3)

for all v, w ∈ V, then we say that b is symmetric on V, as in Section 8.5. If

b(v, w) = −b(w, v) (19.11.4)

for all v, w ∈ V, then b is said to be antisymmetric on V, as before, or skewsymmetric, as on p341 of [226]. If

b(v, v) = 0 (19.11.5) for all v ∈ V, then b is said to be alternating on V, as on p329, 354 of [206], and p341 of [226].

It is easy to see that b is alternating when b is antisymmetric and the characteristic of k is not equal to 2. If the characteristic of k is equal to 2, then (19.11.3) and (19.11.4) are the same.

Note that

b(v + w, v + w) = b(v, v) + b(v, w) + b(w, v) + b(w, w) (19.11.6)

for all v, w ∈ V, as in Section 8.5. If b is alternating on V, then it follows that b is antisymmetic on V, as before. If b is symmetric on V, and if the characteristic of k is not equal to 2, then one can use (19.11.6) to get that b is determined by the values of b(u, u), u ∈ V, as in Section 8.5 again.

If b is any bilinear form on V, then

b(v, w) + b(w, v) (19.11.7)

is a symmetric bilinear form on V. If the characteristic of k is equal to 2, then (19.11.7) is alternating on V.

19.12 Nondegenerate bilinear forms

Let k be a field, and let V be a finite-dimensional vector space over k. A bilinear form b on V is said to be nondegenerate if for every v ∈ V with v ≠ 0 there is a w ∈ V such that b(v, w) ≠ 0. (19.12.1)

If b_{2,v} is as in (19.11.2), then b is nondegenerate on V if and only if

(19.12.2) v ↦ b_{2,v} is one-to-one on V,

as a linear mapping from V into V^{alg}. Equivalently, this means that

(19.12.3) v ↦ b_{2,v} maps V onto V^{alg},

because dim V = dim V^{alg}.

Similarly, if b_{1,w} is as in (19.11.1), then

(19.12.4) w ↦ b_{1,w} is one-to-one on V,

as a linear mapping from V into V^{alg}, if and only if for every w ∈ V with w ≠ 0 there is a v ∈ V such that (19.12.1) holds. As before, (19.12.4) holds if and only if (19.12.5) w ↦ b_{1,w} maps V onto V^{alg},

because dim V = dim V^{alg}. It is easy to see that (19.12.5) implies that b is nondegenerate on V, in the formulation used earlier. Conversely, (19.12.3) implies the version of nondegeneracy that is equivalent to (19.12.4).

Suppose that b is nondegenerate on V, and let T be a linear mapping from V into itself. If w ∈ V, then put

μ_{w}(v) = μ_{T,w}(v) = b(T (v), w) (19.12.6)

for each v ∈ V, which defines a linear functional on V. It follows that there is a unique element T^{∗_{b}}(w) of V such that

b(T (v), w) = μ_{w}(v) = b(v, T^{∗_{b}}(w)) (19.12.7)

for every v ∈ V, as in the preceding paragraph. It is easy to see that T^{∗_{b}} is a linear mapping from V into itself, which may be called the adjoint of T with respect to b. If k = R and b is an inner product on V, then this corresponds to the adjoint of T in the sense of Section 3.5.

We also have that T ↦ T^{∗_{b}} (19.12.8)

is a linear mapping from L(V) into itself. If T ≠ 0, then there is a v ∈ V such that T (v) ≠ 0, which implies that

b(T (v), w) ≠ 0 (19.12.9)

for some w ∈ V. This means that

b(v, T^{∗_{b}}(w)) ≠ 0, (19.12.10)

so that T^{∗_{b}} ≠ 0. Thus (19.12.8) is injective on L(V). It follows that (19.12.8) maps L(V) onto itself, because V has finite dimension, by hypothesis.

If T_{1}, T_{2} are linear mappings from V into itself, then

b(T_{2}(T_{1}(v)), w) = b(T_{1}(v), T_{2}^{∗_{b}}(w)) = b(v, T_{1}^{∗_{b}}(T_{2}^{∗}(w))) (19.12.11)

for all v, w ∈ V. This implies that

(T_{2} ∘ T_{1})^{∗_{b}} = T_{1}^{∗_{b}} ∘ T_{2}^{∗_{b}}, (19.12.12)

so that (19.12.8) is an opposite algebra isomorphism from L(V) onto itself.

19.13 More on bilinear forms

Let us continue with the same notation and hypotheses as in the previous section. If r ∈ k and r ≠ 0, then it is easy to see that

(19.13.1) r b is nondegenerate as a bilinear form on V.

If T is a linear mapping from V into itself, then T^{∗_{b}} and T^{∗_{r b}} may be defined as linear mappings from V into itself, as in the previous section, and one can check that T^{∗_{r b}} = T^{∗_{b}}. (19.13.2)

Note that c(v, w) = b(w, v) (19.13.3)

is another bilinear form on V, and that the nondegeneracy of b on V is equivalent to the nondegeneracy of c on V, as before. If T is a linear mapping from V into itself, then there is a unique linear mapping T^{∗_{c}} from V into itself such that

c(T (v), w) = c(v, T^{∗_{c}}(w)) (19.13.4)

for every v, w ∈ V, as before. Equivalently, this means that

b(w, T (v)) = b(T^{∗_{c}}(w), v) (19.13.5)

for every v, w ∈ V. Of course, this is the same as saying that

b(v, T (w)) = b(T^{∗_{c}}(v), w) (19.13.6)

for every v, w ∈ V, so that T = (T^{∗_{c}})^{∗_{b}}. (19.13.7)

Similarly, T = (T^{∗_{b}})^{∗_{c}}. (19.13.8)

Thus T ↦ T^{∗_{c}} (19.13.9)

is the inverse of (19.12.8).

Of course, b = c if and only if b is symmetric on V, and b = −c if and only if b is antisymmetric on V. In both cases, we get that

T^{∗_{b}} = T^{∗_{c}} (19.13.10)

for every T ∈ L(V), as in (19.13.2).

It follows that (T^{∗_{b}})^{∗_{b}} = T (19.13.11)

for every T ∈ L(V) when b is symmetric or antisymmetric on V, because of (19.13.7). This means that (19.12.8) defines an involution on L(V) in these two cases.

Note that T ∈ L(V) satisfies

T^{∗_{b}} = T (19.13.12)

if and only if b(T (v), w) = b(v, T (w)) (19.13.13)

for every v, w ∈ V. Similarly, T^{∗_{b}} = −T (19.13.14)

if and only if b(T (v), w) = −b(v, T (w)) (19.13.15)

440 CHAPTER 19. ALGEBRAS OVER FIELDS

for every v, w ∈ V.

If A is a linear mapping from V into itself, then b_{A}(v, w) = b(A(v), w) (19.13.16) defines another bilinear form on V. One can check that every bilinear form on V corresponds to a unique A ∈ L(V) in this way, because b is nondegenerate on V. We also have that (19.13.17) b_{A} is nondegenerate on V exactly when A is injective on V, which is the same as saying that A is invertible on V, because V has finite dimension. In this case, if T ∈ L(V), then

b_{A}(T (v), w) = b(A(T (v)), w) = b(A(v), (A^{−1})^{∗_{b}}(T^{∗_{b}}(A^{∗_{b}})(w)))
(19.13.18)  = b_{A}(v, (A^{−1})^{∗_{b}}(T^{∗_{b}}(A^{∗_{b}}(w))))

for every v, w ∈ V. This means that if T^{∗_{b_{A}}} is defined on V using b_{A} as in the previous section, then T^{∗_{b_{A}}} = (A^{−1})^{∗_{b}} ∘ T^{∗_{b}} ∘ A^{∗_{b}}. (19.13.19)

Suppose for the moment that b is symmetric on V, so that b_{A}(w, v) = b(A(w), v) = b(v, A(w)) (19.13.20) for every v, w ∈ V. Observe that b_{A}(v, w) = b_{A}(w, v) (19.13.21) for every v, w ∈ V if and only if A^{∗_{b}} = A, and that b_{A}(v, w) = −b_{A}(w, v) (19.13.22) for all v, w ∈ V if and only if A^{∗_{b}} = −A. Similarly, suppose for the moment that b is antisymmetric on V, so that b_{A}(w, v) = b(A(w), v) = −b(v, A(w)) (19.13.23) for all v, w ∈ V. In this case, (19.13.21) holds if and only if A^{∗_{b}} = −A, and (19.13.22) holds if and only if A^{∗_{b}} = A.

19.14 Involutions and conjugations

Let k be a field, and let A be an associative algebra over k with a multiplicative identity element e_{A}. Also let x ↦ x^{∗} be an opposite algebra isomorphism from A onto itself, and note that e_{A}^{∗} = e_{A}. Of course, if x is an invertible element of A, then x^{∗} is invertible too, with (x^{∗})^{−1} = (x^{−1})^{∗}. (19.14.1)

19.14. INVOLUTIONS AND CONJUGATIONS 441

Put U(A) = {x ∈ A : x x^{∗} = x^{∗} x = e_{A}}, (19.14.2) as in Section 7.5. One can check that this is a subgroup of the group G(A) of invertible elements of A, as before. Equivalently, U(A) = {x ∈ G(A) : x^{−1} = x^{∗}}. (19.14.3) If x ∈ U(A), then (x^{∗})^{∗} = (x^{−1})^{∗} = (x^{∗})^{−1} = (x^{−1})^{−1} = x. (19.14.4)

Let us say that y ∈ A is self-adjoint with respect to x ↦ x^{∗} if y^{∗} = y, (19.14.5) and that y is anti-self-adjoint if y^{∗} = −y, (19.14.6) as in Section 7.5. If x ↦ x^{∗} is an involution on A, then y + y^{∗} (19.14.7) is self-adjoint, and y − y^{∗} (19.14.8) is anti-self-adjoint. If k has characteristic 2, then (19.14.5) and (19.14.6) are the same, and (19.14.7) is equal to (19.14.8). If k does not have characteristic 2, and if x ↦ x^{∗} is an involution on A, then every element of A can be expressed in a unique way as a sum of self-adjoint and anti-self-adjoint elements, as in Section 7.5.

Let a be an invertible element of A, so that x ↦ a x a^{−1} (19.14.9) is an automorphism of A. This implies that x^{#_{a}} = a x^{∗} a^{−1} (19.14.10) also defines an opposite algebra isomorphism from A onto itself. If x ∈ A, then (x^{#_{a}})^{#_{a}} = a (a x^{∗} a^{−1})^{∗} a^{−1} = a (a^{−1})^{∗} (x^{∗})^{∗} a^{∗} a^{−1}. (19.14.11) This is similar to a remark in Section 12.7, when k = R or C.

If a^{∗} = t a (19.14.12) for some t ∈ k with t ≠ 0, then (a^{−1})^{∗} = (a^{∗})^{−1} = t^{−1} a^{−1}, (19.14.13) as in Section 12.7. This implies that

(x^{#_{a}})^{#_{a}} = (x^{∗})^{∗} (19.14.14)

for all x ∈ A, because of (19.14.11), as before. If x ↦ x^{∗} is an involution on A, then it follows that (19.14.10) defines an involution on A, as before.

Of course, (a x a^{−1})^{∗} = (a^{−1})^{∗} x^{∗} a^{∗} = (a^{∗})^{−1} x^{∗} a^{∗} (19.14.15)

for every x ∈ A. If a^{∗} = τ a^{−1} (19.14.16)

for some τ ∈ k with τ ≠ 0, then

(a x a^{−1})^{∗} = a x^{∗} a^{−1} (19.14.17)

for every x ∈ A, as in Section 12.7 again.

Let V be a finite-dimensional vector space over k, and let b be a nondegenerate bilinear form on V. Thus T ↦ T^{∗_{b}} defines an opposite algebra isomorphism from L(V) onto itself, as in Section 19.12. If T ∈ L(V), then

b(T (v), T (w)) = b(v, T^{∗_{b}}(T (w))) (19.14.18)

for every v, w ∈ V. This means that

b(T (v), T (w)) = b(v, w) (19.14.19)

for all v, w ∈ V if and only if T^{∗_{b}} ∘ T = I (19.14.20)

on V.

Note that (19.14.20) implies that T is injective on V. This means that T is invertible on V, because V has finite dimension. Thus (19.14.20) is the same as saying that T is invertible on V, with

T^{−1} = T^{∗_{b}}. (19.14.21)

This is related to remarks in Sections 2.10, 3.5, and 7.5 when k = R and b is an inner product on V.

19.15 Nondegenerate sesquilinear forms

Let V be a finite-dimensional vector space over the complex numbers, and let b be a sesquilinear form on V, as in Section 5.15. We say that b is nondegenerate on V if for every v ∈ V with v ≠ 0 there is a w ∈ V such that

b(v, w) ≠ 0. (19.15.1)

Remember that V_{R} is the same as V, considered as a vector space over the real numbers, as in Section 1.1. The dimension of V_{R} is equal to twice the dimension of V. It is easy to see that the real part of b defines a bilinear form on V_{R}. One can check that b is nondegenerate on V if and only if

(19.15.2) Re b is nondegenerate on V_{R}.

As in Section 19.12, (19.15.2) holds if and only if for every w ∈ V with w ≠ 0 there is a v ∈ V such that Re b(v, w) ≠ 0. (19.15.3)

One can use this to get that b is nondegenerate on V if and only if for every w ∈ V with w ≠ 0 there is a v ∈ V such that (19.15.1) holds.

If w ∈ V, then b_{1,w}(v) = b(v, w) defines a linear functional on V, as before. In this case, w ↦ b_{1,w} defines a conjugate-linear mapping from V into V^{alg}. Conversely, any conjugate-linear mapping from V into V^{alg} corresponds to a sesquilinear form on V in this way.

Similarly, if v ∈ V, then

b̃_{2,v}(w) = \overline{b(v, w)} (19.15.4)

is complex-linear in w, as in Section 5.15. We also have that

v ↦ b̃_{2,v} (19.15.5)

defines a conjugate-linear mapping from V into V^{alg}, and that every such mapping corresponds to a sesquilinear form on V in this way.

Observe that b is nondegenerate on V if and only if

(19.15.6) v ↦ b̃_{2,v} is one-to-one on V.

This is equivalent to the condition that

(19.15.7) v ↦ b̃_{2,v} maps V onto V^{alg},

because V and V^{alg} have the same dimension. One can consider (19.15.5) as a real-linear mapping here, in order to use standard results from linear algebra.

Similarly, (19.15.8) w ↦ b_{1,w} is one-to-one on V,

as a conjugate-linear mapping from V into V^{alg}, if and only if for every w ∈ V with w ≠ 0 there is a v ∈ V such that (19.15.1) holds. This condition holds if and only if (19.15.9) w ↦ b_{1,w} maps V onto V^{alg},

as in the preceding paragraph. It is easy to see that (19.15.9) implies the earlier formulation of nondegeneracy of b on V, as in Section 19.12. Conversely, (19.15.7) implies the version of nondegeneracy that is equivalent to (19.15.8), as before. This is another way to look at the equivalence of these two formuations of nondegeneracy of b on V.

Suppose that b is nondegenerate on V, let T be a linear mapping from V into itself, and put μ_{w}(v) = μ_{T,w}(v) = b(T (v), w) (19.15.10)

for every v, w ∈ V, as in Section 19.12. If w ∈ V, then (19.15.10) defines a linear functional on V, and thus there is a unique element T^{∗_{b}}(w) ov V such that b(T (v), w) = μ_{w}(v) = b(v, T^{∗_{b}}(w)) (19.15.11)

for all v ∈ V, as in the previous paragraph. This defines a linear mapping T^{∗_{b}} from V into itself, as in Section 19.12, which may be called the adjoint of T with respect to b. This corresponds to the adjoint of T in the sense of Section 3.5 when b is an inner product on V.

In this case, T ↦ T^{∗_{b}} (19.15.12)

is a conjugate-linear mapping from L(V) into itself. If T_{1}, T_{2} are linear mappings from V into itself, then

(T_{2} ∘ T_{1})^{∗_{b}} = T_{1}^{∗_{b}} ∘ T_{2}^{∗_{b}}, (19.15.13)

as in Section 19.12.

Observe that c(w, v) = \overline{b(v, w)} (19.15.14)

is another sesquilinear form on V, and that the nondegeneracy of b on V is equivalent to the nondegeneracy of c on V, by the earlier remarks. If T is a linear mapping from V into itself, then there is a unique linear mapping T^{∗_{c}} from V into itself such that

c(T (v), w) = c(v, T^{∗_{c}}(w)) (19.15.15)

for every v, w ∈ V, as before, which is the same as saying that

b(w, T (v)) = b(T^{∗_{c}}(w), v) (19.15.16)

for all v, w ∈ V. This implies that

T = (T^{∗_{c}})^{∗_{b}}, (19.15.17)

as in Section 19.13. Similarly,

T = (T^{∗_{b}})^{∗_{c}}, (19.15.18)

so that T ↦ T^{∗_{c}} is the inverse of (19.15.12), as before.

Note that b = c exactly when b is Hermitian symmetric on V, as in Section 8.5. In this case, (T^{∗_{b}})^{∗_{b}} = T (19.15.19)

for every T ∈ L(V), so that (19.15.12) defines a conjugate-linear algebra involution on L(V).

If A is a linear mapping from V into itself, then

b_{A}(v, w) = b(A(v), w) (19.15.20)

defines another sesquilinear form on V, and one can check that every sesquilinear form on V corresponds to a unique A ∈ L(V) in this way, because b is nondegenerate on V, as in Section 19.13. Note that

(19.15.21) b_{A} is nondegenerate on V

exactly when A is injective on V, which means that A is invertible on V, because V has finite dimension, by hypothesis, as before. In this case, if T ∈ L(V), and T^{∗_{ba}} is defined using b_{A} in the same way as before, then one can check that

T^{∗_{b_{A}}} = (A^{−1})^{∗_{b}} ∘ T^{∗_{b}} ∘ A^{∗_{b}}, (19.15.22)

as in Section 19.13.

If b is Hermitian symmetric, then one can verify that b_{A} is Hermitian symmetric on V exactly when A^{∗_{b}} = A.

If T ∈ L(V), then b(T (v), T (w)) = b(v, w) (19.15.23)

for every v, w ∈ V if and only if

T^{∗_{b}} ∘ T = I (19.15.24)

on V, as in the previous section. This happens exactly when T is invertible on V, with T^{−1} = T^{∗_{b}}, (19.15.25)

as before. This is related to remarks in Sections 2.10, 3.5, and 7.5 when b is an inner product on V, as before.

Representations of associative algebras

20.1 Representations and modules

Let k be a field, and let A be an associative algebra over k. A representation ρ = ρ^{V} of A on a vector space V over k is an algebra homomorphism from A into L(V). If x ∈ A, then we may use ρ_{x} = ρ_{x}^{V} for the corresponding linear mapping on V. Thus ρ_{x y} = ρ_{x} ∘ ρ_{y} (20.1.1)

for every x, y ∈ A, as in (19.1.2). If A has a multiplicative identity element e_{A}, then we may wish to ask that ρ_{e_{A}} = I_{V} , (20.1.2)

as in (19.1.3).

This corresponds to Definition 21.4 on p312 of [151], and to the definition on p384 of [206]. More precisely, if A has a multiplicative identity element e_{A}, then one can look at the behavior of ρ_{e_{A}} on A, as in Note 21.5 on p313 of [151], and we shall return to this in Section 20.9. After those remarks, it is stated in [151] that representations of an associative algebra with a multiplicative identity element will be asked to satisfy (20.1.2). In [206], A is asked to have a multiplicative identity element, and (20.1.2) is included in the definition of a representation, because the analogous condition is included in the definition of a ring homomorphism.

If x ∈ A and v ∈ V, then we may put

x · v = ρ_{x}(v), (20.1.3)

to indicate the action of ρ_{x} on V. Using this notation, (20.1.1) is the same as saying that (x y) · v = x · (y · v) (20.1.4) for every x, y ∈ A and v ∈ V. Similarly, if A has a multiplicative identity element e_{A}, then (20.1.2) is the same as saying that

e_{A} · v = v (20.1.5)

for every v ∈ V. We may also refer to the representation of A on V by saying that V is a left module over A.

A vector space V over k is said to be a right module over A if it is equipped with an action of A on the right that corresponds to an opposite algebra homomorphism from A into L(V). This means that if x ∈ A and v ∈ V, then

v · x (20.1.6)

is defined as an element of V, with the following properties. If x ∈ A, then

v ↦ v · x (20.1.7)

is a linear mapping from V into itself. If v ∈ V, then

x ↦ v · x (20.1.8)

is a linear mapping from A into V. This means that the mapping from x ∈ A to (20.1.7) is linear as a mapping from A into L(V). The condition that this mapping be an opposite algebra homomorphism is the same as saying that

v · (x y) = (v · x) · y (20.1.9)

for every x, y ∈ A and v ∈ V. If A has a multiplicative identity element e_{A}, then we may wish to ask that v · e_{A} = v (20.1.10)

for every v ∈ V, as before.

A left module over A is sometimes simply called a module over A. Of course, if A is commutative, then left and right modules over A are the same.

20.2 Some examples and remarks

Let k be a field, and let V be a vector space over k. If A is a subalgebra of L(V), then the obvious inclusion mapping from A into L(V) defines a representation of A on V.

Let A be any associative algebra over k, and if a, x ∈ A, then put

L_{a}(x) = a x, (20.2.1)

as in Sections 6.3 and 19.4. This defines L_{a} as a linear mapping from A into itself, which is the left multiplication operator associated to a. In fact,

448 CHAPTER 20. REPRESENTATIONS OF ASSOCIATIVE ALGEBRAS

a ↦ L_{a} (20.2.2) defines a homomorphism from A into the algebra L(A) of linear mappings on A, as before, and thus a representation of A on itself, as a vector space over k. Equivalently, A may be considered as a left module over itself, as a vector space over k, where the action of a ∈ A on A on the left is defined using multiplication on A.

Similarly, A may be considered as a right module over itself, as a vector space over k, where the action of a ∈ A on A on the right is defined using multiplication in A again. Equivalently, if a, x ∈ A, then put

R_{a}(x) = x a, (20.2.3)

as in Sections 6.4 and 19.4. This defines a linear mapping from A into itself, which is the right multiplication operator associated to a. We also have that

a ↦ R_{a} (20.2.4)

defines an opposite algebra homomorphism from A into L(A), as before. Let V be a vector space over k again. If V is a left module over A, then

(a, v) ↦ a · v (20.2.5)

is a bilinear mapping from A × V into V. Similarly, if V is a right module over A, then (v, a) ↦ v · a (20.2.6)

is a bilinear mapping from V × A into V.

Suppose now that k = R or C, and that V is a vector space over k with a norm ∥ · ∥_{V}. In this case, we may be interested in representations of associative algebras A over k corresponding to homomorphisms from A into the algebra BL(V) of bounded linear mappings on V. Similarly, we may be interested in actions of A on V on the right, corresponding to opposite algebra homomorphisms from A into BL(V).

If ∥ · ∥_{A} is a norm on A, then we may be interested in representations of A on V corresponding to homomorphisms from A into BL(V) that are bounded linear mappings, with respect to the operator norm on BL(A) associated to ∥ · ∥_{V}. This means that there is a nonnegative real number C such that

∥a · v∥_{V} ≤ C∥a∥_{A} ∥v∥_{V} (20.2.7)

for every a ∈ A and v ∈ V. This is the same as saying that (20.2.5) is bounded as a bilinear mapping from A × V into V, as in Section 5.13.

Similarly, we may be interested in actions of A on V on the right that correspond to opposite algebra homomorphisms from A into BL(V) that are bounded as linear mappings, so that

∥v · a∥_{V} ≤ C ∥a∥_{A} ∥v∥_{V} (20.2.8)

for some C ≥ 0 and all a ∈ A, v ∈ V. This is the same as saying that (20.2.6) is bounded as a bilinear mapping from V × A into V. If ∥ · ∥_{A} is submultiplicative on A, then the actions of A on itself on the left and right satisfy these conditions with C = 1.

20.3 Opposite algebras

Let k be a field, and let A be an algebra in the strict sense over k, as in Section 19.1. The corresponding opposite algebra A^{op} is defined to be the same as A as a vector space over k, with the order of multiplication exchanged. More precisely, if x ∈ A, then we may use x^{op} to refer to x as an element of A^{op}. With this notation, multiplication in A^{op} is defined by

x^{op} y^{op} = (y x)^{op}. (20.3.1)

Of course, A^{op} is an algebra in the strict sense over k too.

Thus multiplication in A^{op} is the same as in A exactly when A is commutative. If A is associative, then A^{op} is associative as well. This corresponds to the notion of an opposite ring, as on p109 of [61], and on p185 of [226].

If A has a multiplicative identity element e_{A}, then e_{A}^{op} is the multiplicative identity element in A^{op}. Clearly A and A^{op} have the same subalgebras, and the same two-sided ideals. Left ideals in A correspond to right ideals in A^{op}, and right ideals in A correspond to left ideals in A^{op}.

Let B be another algebra over k in the strict sense. An opposite algebra homomorphism from A into B is the same as an algebra homomorphism from A^{op} into B. This is also the same as an algebra homomorphism from A into B^{op}. Similarly, an opposite algebra isomorphism from A onto B corresponds exactly to an algebra isomorphism from A^{op} onto B, and to an algebra isomorphism from A onto B^{op}.

Suppose that A is an associative algebra over k, and let V be a vector space over k. If V is a left module over A, then V may be considered as a right module over A^{op}, with v · x^{op} = x · v (20.3.2)

for every x ∈ A and v ∈ V. Similarly, if V is a right module over A, then V may be considered as a left module over A^{op}, with

x^{op} · v = v · x (20.3.3)

for every x ∈ A and v ∈ V. This corresponds to Proposition 16 on p185 of [226], for modules over rings, and it is also mentioned on p109 of [61].

20.4 Invariant linear subspaces

Let k be a field, let A be an associative algebra over k, and let ρ^{V} be a representation of A on a vector space V over k. A linear subspace W of V is said to be invariant under ρ^{V} if ρ_{x}^{V} (W) ⊆ W (20.4.1)

for every x ∈ A. In this case, for each x ∈ A,

(20.4.2) let ρ_{x}^{W} be the restriction of ρ_{x}^{V} to W.

It is easy to see that this defines a representation of A on W. This may be called a subrepresentation of ρ^{V}.

Equivalently, if V is a left module over A, then a linear subspace W of V is said to be a submodule of V if

x · w ∈ W (20.4.3)

for every x ∈ A and w ∈ W. This means that W may be considered as a left module over A as well, using the restriction of the action of A on V to W.

Similarly, if V is a right module over A, then a linear subspace W of V is said to be a submodule of V if

w · x ∈ W (20.4.4)

for every x ∈ A and w ∈ W. This implies that W may be considered as a right module over A, with respect to the restriction of the action of A on V to W.

Remember that A may be considered as a left and right module over itself, as in Section 20.2. A linear subspace I of A is a submodule of A, as a left or right module over itself, exactly when I is a left or right ideal in A, as appropriate, as in Section 19.2.

Suppose for the moment that k = R or C, and that V is a vector space over k with a norm ∥ · ∥_{V}. Suppose also that V is a left or right module over A, and that the action of each element of A on V is a bounded linear mapping on V. If W is a submodule of V, as a module over A, then it is easy to see that

(20.4.5) \overline{W} is a submodule of V,

as a module over A. Here \overline{W} is the closure of W in V with respect to the metric associated to ∥ · ∥_{V}, as usual.

Let k be any field again, let X_{1}, …, X_{n} be commuting indeterminates, and let k[X_{1}, …, X_{n}] be the algebra of formal polynomials in X_{1}, …, X_{n} with coefficients in k, as in Section 19.6. Also let V be a vector space over k, and let T_{1}, …, T_{n} be n commuting linear mappings from V into itself, so that

T_{j} ∘ T_{l} = T_{l} ∘ T_{j} (20.4.6)

for all j, l = 1, …, n. This leads to a representation of k[X_{1}, …, X_{n}] on V, as in Section 19.6 again. This representation is characterized by the properties that

1_{k} · v = v (20.4.7)

for every v ∈ V, and X_{j} · v = T_{j}(v) (20.4.8)

for each j = 1, …, n and v ∈ V. If W is a linear subspace of V such that

T_{j}(W) ⊆ W (20.4.9)

for each j = 1, …, n, then W is invariant under this representation.

In particular, we can take n = 1 here. Equivalently, if X is an indeterminate, then we let k[X] be the algebra of formal polynomials in X with coefficients in k, as before. If T is a linear mapping from V into itself, then there is a unique representation of k[X] on V that satisfies (20.4.7) and

X · v = T (v) (20.4.10)

for every v ∈ V. If W is a linear subspace of V such that

T (W) ⊆ W, (20.4.11)

then W is invariant under this representation.

20.5 More on invariant linear subspaces

Let k be a field, let V be a vector space over k, and let W be a linear subspace of V. Remember that L_{W} (V) be the subalgebra of L(V) of linear mappings T from V into itself that map W into itself, as in Section 19.8. One may consider V as a left module over L_{W} (V), as mentioned at the beginning of Section 20.2. Of course, W is a submodule of V, as a left module over L_{W} (V), by construction.

Suppose now that k = R or C, and that ∥ · ∥_{V} is a norm on V. Consider the space BL_{W} (V) = BL(V) ∩ L_{W} (V) (20.5.1)

of bounded linear mappings T from V into itself such that T (W) ⊆ W. This is a subalgebra of BL(V) that contains the identity mapping on V. If W is also a closed set in V, with respect to the metric associated to ∥ · ∥_{V}, then it is easy to see that (20.5.2) BL_{W} (V) is a closed set in BL(V),

with respect to the metric associated to the operator norm. Note that

BL_{W} (V) ⊆ BL\overline{W} (V). (20.5.3)

If T ∈ L_{W} (V), then let T_{W} be the restriction of T to W, as in Section 19.8. Remember that T ↦ T_{W} defines a homomorphism from L_{W} (V) into L(W), as algebras over k, as before. The restriction of this homomorphism to BL_{W} (V) defines a homomorphism into BL(W). More precisely, if T ∈ BL_{W} (V), then T_{W} ∈ BL(W), and the operator norm of T_{W} on W is less than or equal to the operator norm of T on V.

The kernel of T ↦ T_{W} on BL_{W} (V) is equal to

{T ∈ BL(V) : T = 0 on W }. (20.5.4)

This is a two-sided ideal in BL_{W} (V), and a left ideal in BL(V). Note that (20.5.4) is a closed set in BL(V), with respect to the metric associated to the operator norm. It is easy to see that (20.5.4) is the same as

\{T ∈ BL(V) : T = 0 on \overline{W} \}. (20.5.5)

Let V/W be the usual quotient space, and let q_{V/W} be the corresponding quotient mapping from V onto V/W. Suppose from now on in this section that W is a closed linear subspace of V, and let ∥ · ∥_{V/W} be the corresponding quotient norm on V/W, as in Section 6.10. If T ∈ BL_{W} (V), then there is a unique linear mapping T^{V/W} from V/W into itself such that

T^{V/W} ∘ q_{V/W} = q_{V/W} ∘ T, (20.5.6)

as in Section 19.8. Let us check that T^{V/W} is bounded, with respect to the quotient norm.

If v ∈ V and w ∈ W, then

∥T^{V/W} (q_{V/W} (v))∥_{V/W} = ∥T^{V/W} (q_{V/W} (v − w))∥_{V/W}
(20.5.7)  = ∥q_{V/W} (T (v − w))∥_{V/W}
≤ ∥T (v − w)∥_{V} ≤ ∥T ∥_{op,V V} ∥v − w∥_{V} ,

using the definition of the quotient norm in the third step. This implies that

∥T^{V/W} (q_{V/W} (v))∥_{V/W} ≤ ∥T ∥_{op,V V} ∥q_{V/W} (v)∥_{V/W} , (20.5.8)

by the definition of the quotient norm. This shows that T^{V/W} is bounded on V/W, with operator norm less than or equal to the operator norm of T on V.

As in Section 19.8, T ↦ T^{V/W} (20.5.9)

is a homomorphism from BL_{W} (V) into BL(V/W), as algebras over k. The kernel of this homomorphism is equal to

{T ∈ BL(V) : T (V) ⊆ W }, (20.5.10)

as before. This is another two-sided ideal in BL_{W} (V), and a right ideal in BL(V). We also have that (20.5.10) is a closed set in BL(V), with respect to the metric associated to the operator norm, because W is a closed set in V, by hypothesis.

20.6 Quotient representations

Let k be a field, let A be an associative algebra over k, and let ρ^{V} be a representation of A on a vector space V over k. Also let W be a linear subspace of V that is invariant under ρ, as in Section 20.4. If x ∈ A, and q_{V/W} is the natural quotient mapping from V onto V/W, then there is a unique linear mapping ρ_{x}^{V/W} from V/W into itself such that

ρ_{x}^{V/W} ∘ q_{V/W} = q_{V/W} ∘ ρ_{x}^{V} , (20.6.1)

as in Section 19.8. One can check that

(20.6.2) ρ^{V/W} is a representation of A on V/W.

This may be called a quotient representation of ρ^{V}.

Equivalently, if V is a left module over A, and W is a submodule of V, then there is a unique action of A on V/W such that

x · q_{V/W} (v) = q_{V/W} (x · v) (20.6.3)

for every x ∈ A and v ∈ V. This makes V/W a left module over A, which may be called a quotient module. Similarly, if V is a right module over A, and W is a submodule of V, then there is a unique action of A on V/W such that

q_{V/W} (v) · x = q_{V/W} (v · x) (20.6.4)

for every x ∈ A and v ∈ V. This makes V/W a right module over A, which may also be called a quotient module.

In particular, if I is a left or right ideal in A, then

(20.6.5) A/I may be considered as a left or right module over A,

as appropriate. If I is a two-sided ideal in A, then A may be considered as both a left and right module over A, and as an associative algebra over k, as in Section 19.2. If q_{A/I} is the natural quotient homomorphism from A onto A/I, as in Section 19.2, then the action of x ∈ A on A/I on the left and the right corresponds to multiplication by q_{A/I}(x) on the left and the right in A/I.

Suppose now that k = R or C, and that V is a vector space over k with a norm ∥ · ∥_{V}. Let W be a closed linear subspace of V, and let ∥ · ∥_{V/W} be the corresponding quotient norm on V/W, as in Section 6.10. Suppose that V is a left or right module over A, where the action of elements of A on V are bounded linear mappings on V. In this case, the action of elements of A on V/W are bounded linear mappings on V/W, as in the previous section. More precisely, the operator norm of the action of a ∈ A on V/W is less than or equal to the operator norm of the action of a on V, as before.

Let ∥ · ∥_{A} be a norm on A, and suppose that there is a nonnegative real number C such that the operator norm of the action of a ∈ A on V is less than or equal C ∥a∥_{A}. This implies that the operator norm of the action of a on V/W is less than or equal to C ∥a∥_{A} as well.

20.7 Homomorphisms between representations

Let k be a field, let A be an associative algebra over k, and let V_{1}, V_{2} be vector spaces over k. Also let ρ^{V_{1}}, ρ^{V_{2}} be representations of A on V_{1}, V_{2}, respectively. A linear mapping T from V_{1} into V_{2} is said to intertwine these two representations if T ∘ ρ_{x}^{V_{1}} = ρ_{x}^{V_{2}} ∘ T, (20.7.1)

as linear mappings from V_{1} into V_{2}, for every x ∈ A. We may also say that T is a homomorphism between these two representations in this case.

Equivalently, if V_{1} and V_{2} are left modules over A, then a linear mapping T from V_{1} into V_{2} is said to be a homomorphism from V_{1} into V_{2}, as modules over A, if T (x · v_{1}) = x · T (v_{1}) (20.7.2)

for every x ∈ A and v_{1} ∈ V_{1}. Similarly, if V_{1} and V_{2} are right modules over A, then a linear mapping T from V_{1} into V_{2} is said to be a homomorphism from V_{1} into V_{2}, as modules over A, if

T (v_{1} · x) = T (v_{1}) · x (20.7.3)

for every x ∈ A and v_{1} ∈ V_{!}.

Let V be a left or right module over A, and let W be a submodule of V. The obvious inclusion mapping from W into V may be considered as a homomorphism from W into V, as modules over A. The corresponding quotient mapping q_{V/W} from V onto V/W is a module homomorphism too, as in the previous section.

Let V_{1}, V_{2} be both left or both right modules over A again, and let T be a homomorphism from V_{1} into V_{2}, as modules over A. It is easy to see that

(20.7.4) ker T is a submodule of V_{1}

and (20.7.5) T (V_{1}) is a submodule of V_{2}.

If T is also a one-to-one mapping from V_{1} onto V_{2}, then T^{−1} is a homomorphism from V_{2} onto V_{1}, as modules over A. In this case, we may say that T is an isomorphism from V_{1} onto V_{2}, as modules over A. If V_{1} and V_{2} are left modules over A, then we may say that T is an isomorphism between these two representations, or that these two representations of A are equivalent.

Suppose now that k = R or C, and that ∥ · ∥_{V_{1}}, ∥ · ∥_{V_{2}} are norms on V_{1}, V_{2}, respectively. If V_{1}, V_{2} are both left or both right modules over A, then one may be interested in module homomorphisms that are also bounded as linear mappings from V_{1} into V_{2}.

Similarly, one may be interested in isomorphisms T from V_{1} onto V_{2} such that T and T^{−1} are both bounded linear mappings. One may also be interested in such an isomorphism T that is an isometric linear mapping from V_{1} onto V_{2}.

In particular, if V_{1} and V_{2} are Hilbert spaces, then this means that T is a unitary mapping, as in Section 2.10. One may say that V_{1} and V_{2} are unitarily equivalent in this case, or that the corresponding representations are unitarily equivalent when V_{1} and V_{2} are left modules over A. This corresponds to a definition on p58 of [11], part of Definition 21.8 on p314 of [151], and a definition on p273 of [291], at least for suitable classes of representations.

20.8 Direct sums of representations

Let k be a field, and let A be an associative algebra over k. Also let V_{1} and V_{2} be vector spaces over k that are both left or both right modules over k. Remember that the Cartesian product V_{1} × V_{2} of V_{1} and V_{2} may be considered as a vector space over k with respect to coordinatewise addition and scalar multiplication, which is the direct sum of V_{1} and V_{2}, as in Section 5.12. We may consider V = V_{1} × V_{2} as a left or right module over A, as appropriate, with

x · (v_{1}, v_{2}) = (x · v_{1}, x · v_{2}) (20.8.1)

or (v_{1}, v_{2}) · x = (v_{1} · x, v_{2} · x) (20.8.2)

for all x ∈ A, v_{1} ∈ V_{1}, and v_{2} ∈ V_{2}, as appropriate. This is the direct sum of V_{1} and V_{2}, as modules over A.

If V_{1} and V_{2} are left modules over A, then we can look at this in terms of the corresponding representations ρ^{V_{1}} and ρ^{V_{2}} of A as well. Namely,

ρ_{x}^{V} ((v_{1}, v_{2})) = (ρ_{x}^{V_{1}}(v_{1}), ρ_{x}^{V_{2}}(v_{2})) (20.8.3)

defines a representation ρ^{V} of A on V, which is the direct sum of ρ^{V_{1}} and ρ^{V_{2}}. Note that

(20.8.4) V_{1} × {0} and {0} × V_{2} are submodules of V.

Of course, v_{1} ↦ (v_{1}, 0) and v_{2} ↦ (0, v_{2}) are isomorphisms from V_{1} and V_{2} onto these submodules of V, respectively. One could also consider these mappings as injective homomorphisms from V_{1} and V_{2} into V, respectively, as modules over A.

Similarly, the coordinate mappings (v_{1}, v_{2}) ↦ v_{1} and (v_{1}, v_{2}) ↦ v_{2} are homomorphisms from V onto V_{1} and V_{2}, respectively, as modules over A. The kernels of these homomorphisms are {0} × V_{2} and V_{1} × {0}, respectively.

Suppose now that k = R or C. If ⟨·, ·⟩_{V_{1}} and ⟨·, ·⟩_{V_{2}} are inner products on V_{1} and V_{2}, respectively, then we can get an inner product ⟨·, ·⟩_{V} on V as in Section 5.12. This is used in the definition of a direct sum on p57 of [11], under suitable conditions.

If ∥ · ∥_{V_{1}}, ∥ · ∥_{V_{2}} are norms on V_{1}, V_{2}, respectively, then we can define a norm ∥ · ∥_{V,p} on V for 1 ≤ p ≤ ∞ as in Section 5.12. If T_{1}, T_{2} are bounded linear mappings on V_{1}, V_{2}, respectively, then it is easy to see that

T ((v_{1}, v_{2})) = (T_{1}(v_{1}), T_{2}(v_{2})) (20.8.5)

defines a bounded linear mapping on V with respect to ∥ · ∥_{V,p} for each p. More precisely, the operator norm of T with respect to ∥ · ∥_{V,p} is equal to

max(∥T_{1}∥_{op,V_{1}V_{1}}, ∥T_{2}∥_{op,V_{2}V_{2}}) (20.8.6)

for each p.

If the actions of x ∈ A on V_{1} and V_{2} are bounded linear mappings, then the action of x on V is a bounded linear mapping with respect to ∥ · ∥_{V,p} for each p. Let ∥ · ∥_{A} be a norm on A, and suppose that there are nonnegative real numbers C_{1}, C_{2} such that the operator norms of the actions of x on V_{1} and V_{2} are less than or equal to C_{1} ∥x∥_{A} and C_{2} ∥x∥_{A}, respectively. This implies that the operator norm of the action of x on V with respect to ∥ · ∥_{V,p} is less than or equal to max(C_{1}, C_{2}) ∥x∥_{A} (20.8.7)

for each p.

20.9 Nondegenerate representations

Let k be a field, let A be an associative algebra over k, and let V be a vector space over k that is a left or right module over A. If V is a left module over A, then it is easy to see that

(20.9.1) {v ∈ V : x · v = 0 for every x ∈ A}

is a submodule of V. Equivalently, if ρ^{V} is the corresponding representation on V, then this is the same as

⋂_{x∈A} ker ρ_{x}^{V} . (20.9.2)

Similarly, if V is a right module over A, then

(20.9.3) {v ∈ V : v · x = 0 for every x ∈ A}

is a submodule of V.

Let us say that V is nondegenerate as a left or right module over A if (20.9.1) or (20.9.3) is equal to {0}, as appropriate. If V is a left module over A, corresponding to the representation ρ^{V}, then this is the same as saying that (20.9.2) is equal to {0}. In this case, we may say that ρ^{V} is nondegenerate, as a representation on V. This corresponds to the definition of nondegeneracy mentioned on p57 of [11], for a certain type of representation.

Suppose for the moment that A has a multiplicative identity element e_{A}. If V is a left module over A, then one can check that (20.9.1) is the same as

{v ∈ V : e_{A} · v = 0}. (20.9.4)

If ρ^{V} is the corresponding representation of A on V, then this is the same as the kernel of ρ_{e_{A}}^{V}. Similarly, if V is a right module over A, then (20.9.3) is the same as {v ∈ V : v · e_{A} = 0}. (20.9.5)

In both cases, if the action of e_{A} on V is the identity mapping on V, then V is nondegenerate as a module over A, as in Exercise (2) on p59 of [11].

Let x be an idempotent element of A, so that x^{2} = x, as in Section 7.6. If V is a left module over A, then it follows that

v ↦ x · v (20.9.6) is idempotent as an element of L(V). This means that (20.9.6) is a projection on V, as in Section 8.2. Similarly, if V is a right module over A, then

v ↦ v · x (20.9.7)

is a projection on V.

Suppose that A has a multiplicative identity element e_{A} again. If V is a left module over A, then v ↦ e_{A} · v (20.9.8)

is a projection on V, as in the preceding paragraph. Note that

{e_{A} · v : v ∈ V } (20.9.9)

is a submodule of V in this case. Similarly, if V is a right module over A, then

v ↦ v · e_{A} (20.9.10)

is a projection on V, and {v · e_{A} : v ∈ V } (20.9.11)

is a submodule of V.

If P is any projection on V, then V corresponds to the direct sum of the kernel of P and P (V), as a vector space over k, as in Section 8.2. If V is a left module over A, then V corresponds to the direct sum of the submodules (20.9.4) and (20.9.9), as a module over A. The action of e_{A} on (20.9.9) is the same as the identity mapping on (20.9.9). This corresponds to a remark in Note 21.5 on p313 of [151]. If V is nondegenerate as a module over A, then it follows that the action of e_{A} on V is the identity mapping on V, as in Exercise (2) on p59 of [11].

Similarly, if V is a right module over A, then V corresponds to the direct sum of the submodules (20.9.5) and (20.9.11), as a module over A, and the action of e_{A} on (20.9.11) is the identity mapping on (20.9.11). If V is nondegenerate as a module over A, then the action of e_{A} on V is the identity mapping on V.

Suppose now that k = R or C, and let ∥ · ∥_{V} be a norm on V. If V is a left module over A, and the actions of the elements of A on V are bounded linear mappings on V, then (20.9.1) is a closed linear subspace of V. If A has a multiplicative identity element e_{A}, then (20.9.9) is a closed linear subspace of V as well, as in Section 8.2. This corresponds to another remark in Note 21.5 on p313 of [151]. There are analogous statements when V is a right module over A, as usual.

20.10 Representations and identity elements

Let k be a field, let A be an associative algebra over k, and let A_{1} = A × k be as in Section 19.2. Thus A_{1} is an associative algebra over k, a ↦ (a, 0_{k}) is an algebra homomorphism from A into A_{1}, and e_{A_{1}} = (0, 1_{k}) is the multilpicative identity element in A_{1}. Also let B be another associative algebra over k with a multiplicativc identity element e_{B}, and let ϕ be an algebra homomorphism from A into B.

Consider the mapping ϕ_{1} from A_{1} into B defined by

ϕ_{1}((a, t)) = ϕ(a) + t e_{B} (20.10.1)

for every a ∈ A and t ∈ k. This is clearly a linear mapping from A_{1} into B, and one can check that ϕ_{1} is an algebra homomorphism too. Of course,

ϕ_{1}(e_{A_{1}}) = e_{B}, (20.10.2)

by construction. Similarly, if ϕ is an opposite algebra homomorphism from A into B, then ϕ_{1} is an opposite algebra homomorphism from A_{1} into B.

Suppose for the moment that k = R or C, and that ∥·∥_{A} and ∥·∥_{B} are norms on A and B, respectively. As in Section 6.15, ∥(a, t)∥_{A_{1}} = ∥a∥_{A} + |t| defines a norm on A_{1}. If ϕ is a bounded linear mapping from A into B, then it is easy to see that ϕ_{1} is a bounded linear mapping from A_{1} into A. More precisely, the operator norm of ϕ_{1} is equal to the maximum of the operator norm of ϕ and ∥e_{B}∥_{B}.

Let k be any field again, and let V be a vector space over k. A representation of A on V is the same as an algebra homomorphism from A into L(V), which leads to an algebra homomorphism from A_{1} into L(V), as before. This is the same as a representation of A_{1} on V, for which the action of e_{A_{1}} is the identity operator on V. This corresponds to a remark in the proof of Theorem 2.5.5 on p58 of [11].

Equivalently, if V is a left module over A, then V may be considered as a left module over A_{1}, with

(a, t) · v = a · v + t v (20.10.3)

for every a ∈ A, t ∈ k, and v ∈ V. Similarly, if V is a right module over A, then V may be considered as a right module over A_{1}, with

v · (a, t) = v · a + t v (20.10.4)

for every a ∈ A, t ∈ k, and v ∈ V. If V is a left or right module over A, then a submodule of V, as a module over A, is the same as a submodule of V, as a module over A_{1}. A homomorphism between left or right modules over A may be considered as a homomorphism between modules over A_{1}.

20.11 Dual modules

Let k be a field, and let A be an associative algebra over k. Also let V be a vector space over k, and let V^{alg} be the algebraic dual of V, as in Section 19.9. If V is a left module over A, then V^{alg} may be considered as a right module over A, as follows. If x ∈ A and μ ∈ V^{alg}, then μ · x ∈ V^{alg} is defined by

(μ · x)(v) = μ(x · v) (20.11.1) for every v ∈ V.

Equivalently, μ ↦ μ · x (20.11.2)

is the linear mapping from V^{alg} into itself that is dual to v ↦ x · v, as a linear mapping from V into itself, as in Section 19.9. If y ∈ A, then

(μ · x) · y = μ · (x y), (20.11.3)

because

(20.11.4)  ((μ · x) · y)(v) = (μ · x)(y · v) = μ(x · (y · v))
= μ((x y) · v) = (μ · (x y))(v)

for every v ∈ V. One can also look at this in terms of the usual property of duals of compositions of linear mappings, as in Section 19.9. If A has a multiplicative identity element e_{A}, and if e_{A} · v = v for all v ∈ V, then

μ · e_{A} = μ (20.11.5)

for all μ ∈ V^{alg}.

Similarly, if V is a right module over A, then V^{alg} may be considered as a left module over A, with (x · μ)(v) = μ(v · x) (20.11.6)

for every x ∈ A, v ∈ V, and μ ∈ V^{alg}. If A has a multiplicative identity element e_{A}, and if v · e_{A} = v for every v ∈ V, then

e_{A} · μ = μ (20.11.7)

for every μ ∈ V^{alg}. This corresponds to some remarks on p186 of [226]. In both cases, we may refer to V^{alg} as the dual of V, as modules over A. More precisely, we may refer to V^{alg} as the algebraic dual of V, as modules over A.

Suppose for the moment that k = R or C, and that ∥ · ∥_{V} is a norm on V. Thus the dual space V′ of bounded linear functionals on V may be defined as in Section 3.1, with the corresponding dual norm ∥ · ∥_{V′}. If V is a left or right module over A, and the action of every x ∈ A on V is a bounded linear mapping on V, then it is easy to see that

(20.11.8) V′ is a submodule of V^{alg},

as a module over A. In fact, the action of x on V′ is the dual of the action of x on V, as in Section 3.13. In particular, the operator norm of the action of x on V′ is equal to the operator norm of the action of x on V, as before.

Let k be any field again, and suppose that x ↦ x^{∗} is an opposite algebra isomorphism from A onto itself. If V is a left module over A, then V^{alg} may be considered as a left module over A too, with

x · μ = μ · x^{∗} (20.11.9) for every x ∈ A and μ ∈ V^{alg}. The right side is defined as before, so that

(x · μ)(v) = (μ · x^{∗})(v) = μ(x^{∗} · v) (20.11.10)

for every v ∈ V. Similarly, if V is a right module over A, then V^{alg} may be considered as a right module over A as well, with

μ · x = x^{∗} · μ (20.11.11)

for every x ∈ A and μ ∈ V^{alg}. This means that

(μ · x)(v) = (x^{∗} · μ)(v) = μ(v · x^{∗}) (20.11.12)

for every v ∈ V.

20.12 More on quotient spaces

Let k be a field, let V_{1}, V_{2} be vector spaces over k, and let T be a linear mapping from V_{1} into V_{2}. Also let W_{1} be a linear subspace of V_{1}, and suppose that

W_{1} ⊆ ker T. (20.12.1)

Consider the quotient space V_{1}/W_{1}, with the corresponding quotient mapping q_{V_{1}/W_{1}} from V_{1} onto V_{1}/W_{1}. It is well known and not difficult to see that there is a unique linear mapping T̃ from V_{1}/W_{1} into V_{2} such that

T = T̃ ∘ q_{V_{1}/W_{1}}. (20.12.2)

Equivalently, T̃(q_{V_{1}/W_{1}}(v_{1})) = T (v_{1}) (20.12.3)

for every v_{1} ∈ V_{1}.

Note that ker T̃ = q_{V_{1}/W_{1}}(ker T). (20.12.4)

If W_{1} = ker T_{1}, then we get that ker T̃ = {0}, so that T̃ is one-to-one on V_{1}/W_{1}.

Suppose for the moment that k = R or C, and let ∥ · ∥_{V_{1}}, ∥ · ∥_{V_{2}} be norms on V_{1}, V_{2}, respectively. Suppose also that T is bounded as a linear mapping from V_{1} into V_{2}, and that W_{1} is a closed set in V_{1}, with respect to the metric associated to ∥ · ∥_{V_{1}}. Let ∥ · ∥_{V_{1}/W_{1}} be the quotient norm on V_{1}/W_{1} associated to ∥ · ∥_{V_{1}}, as in Section 6.10. If v_{1} ∈ V_{1} and w_{1} ∈ W_{1}, then

(20.12.5) ∥T̃(q_{V_{1}/W_{1}}(v_{1}))∥_{V_{2}} = ∥T (v_{1})∥_{V_{2}} = ∥T (v_{1} − w_{1})∥_{V_{2}}
≤ ∥T ∥_{op,V_{1}V_{2}} ∥v_{1} − w_{1}∥_{V_{1}}.

This implies that

∥T̃(q_{V_{1}/W_{1}}(v_{1}))∥_{V_{2}} ≤ ∥T ∥_{op,V_{1}V_{2}} ∥q_{V_{1}/W_{1}}(v_{1})∥_{V_{1}/W_{1}}. (20.12.6)

This means that T̃ is a bounded linear mapping from V_{1}/W_{1} into V_{2}, with

∥T̃∥_{op,(V_{1}/W_{1})V_{2}} ≤ ∥T ∥_{op,V_{1}V_{2}}. (20.12.7)

The operator norm of T is less than or equal to the operator norm of T̃, because of (20.12.2). It follows that

∥T̃∥_{op,(V_{1}/W_{1})V_{2}} = ∥T ∥_{op,V_{1}V_{2}}. (20.12.8)

Let k be any field again, let A be an associative algebra over k, and let ρ^{V_{1}}, ρ^{V_{2}} be representations of A on V_{1}, V_{2}. Suppose that T is a homomorphism between these representations, and that W_{1} is invariant under ρ^{V_{1}}. Let ρ^{V_{1}/W_{1}} be the corresponding quotient representation of A on V_{1}/W_{1}, as in Section 20.6. Under these conditions, it is easy to see that T̃ is a homomorphism between ρ^{V_{1}/W_{1}} and ρ^{V_{2}}.

Equivalently, suppose that V_{1}, V_{2} are left modules over A, and that T is a homomorphism from V_{1} into V_{2}, as modules over A. Suppose too that W_{1} is a submodule of V_{1}, so that V_{1}/W_{1} may be considered as a left module over A as well. In this case, T̃ is a homomorphism from V_{1}/W_{1} into V_{2}, as modules over A. There is an analogous statement for right modules, as usual.

Let V be a vector space over k that is a left module over A, and let v_{0} ∈ V be given. Observe that x ↦ x · v_{0} (20.12.9)

defines a homomorphism from A into V, as left modules over A. The kernel of this homomorphism is

I_{L}(v_{0}) = {x ∈ A : x · v_{0} = 0}, (20.12.10)

which is a left ideal in A, as in Sections 20.4 and 20.7. This means that (20.12.9) corresponds to an injective homomorphism from A/I_{L}(v_{0}) into V, as left modules over A, as before.

Similarly, if V is a right module over A, then

x ↦ v_{0} · x (20.12.11)

is a homomorphism from A into V, as right modules over A. The kernel

I_{R}(v_{0}) = {x ∈ A : v_{0} · x = 0} (20.12.12)

of this homomorphism is a right ideal in A. This leads to an injective homomorphism from A/I_{R}(v_{0}) into V, as right modules over A, as before.

Suppose that k = R or C again, and let ∥ · ∥_{A} and ∥ · ∥_{V} be norms on A and V, respectively. If V is a left module over A, and (20.12.9) is a bounded linear mapping from A into V, then (20.12.10) is a closed set in A, with respect to the metric associated to ∥ · ∥_{A}. Similarly, if V is a right module over A, and (20.12.11) is a bounded linear mapping from A into V, then (20.12.12) is a closed set in A, with respect to the metric associated to ∥ · ∥_{A}.

20.13 Cyclic representations

Let k be a field, let A be an associative algebra over k, let V be a vector space over k, and let v_{0} ∈ V be given. If V is a left module over A, then

{x · v_{0} : x ∈ A} (20.13.1)

is a submodule of V. If this submodule is equal to V, then we say that V is cyclic as a left module over A. In this case, v_{0} may be called a cyclic vector in V, as a left module over A.

If v_{0} is an element of (20.13.1), then

(20.13.2) (20.13.1) is cyclic as a left module over A.

Note that this happens when A has an element whose action on V is the identity operator on V.

Equivalently, if ρ^{V} is a representation of A on V, then

{ρ_{x}^{V} (v_{0}) : x ∈ A} (20.13.3)

is a linear subspace of V that is invariant under ρ^{V}. If this subspace is equal to V, then ρ^{V} is said to be cyclic as a representation of A, and v_{0} may be called a cyclic vector of ρ^{V}. Note that this terminology is not quite the same as in Exercise (3) on p59 of [11], Definition 21.12 on p315 of [151], or on p273 of [291], and we shall say more about this soon. We may say that V is algebraically cyclic in this case, to be more precise.

Similarly, if V is a right module over A, then

{v_{0} · x : x ∈ A} (20.13.4)

is a submodule of V. If this submodule is equal to V, then we say that V is cyclic as a right module over A, and v_{0} may be called a cyclic vector in V. More precisely, we may say that V is algebraically cyclic as a right module over A, as before. If v_{0} is an element of (20.13.4), then

(20.13.5) (20.13.4) is cyclic as a right module over A.

This happens when A has an element whose action on V is the identity operator on V, as before.

Remember that (20.12.10) is a left ideal in A when V is a left module over A. If v_{0} is a cyclic vector in V, then we get an isomorphism from A/I_{L}(v_{0}) onto V, as left modules over A, as before. Similarly, (20.12.12) is a right ideal in A when V is a right module over A. If v_{0} is a cyclic vector in V, then we obtain an isomorphism from A/I_{R}(v_{0}) onto V, as right modules over A.

If A has a right multiplicative identity element e_{R}, as in Section 19.4, then

(20.13.6) A is cyclic as a left module over itself, with cyclic vector e_{R}.

In this case, if I is a left ideal in A, then

(20.13.7) A/I is cyclic as a left module over A,

with the image of e_{R} in A/I under the natural quotient mapping as the cyclic vector. Similarly, if e_{L} is a left multiplicative identity element in A, then

(20.13.8) A is cyclic as a right module over itself, with cyclic vector e_{L}.

If I is a right ideal in A, then

(20.13.9) A/I is cyclic as a right module over A,

with the image of e_{L} in A/I as the cyclic vector.

Suppose now that k = R or C, and that ∥ · ∥_{V} is a norm on V. If V is a left module over A, and (20.13.1) is dense in V with respect to the metric associated to ∥ · ∥, then let us say that V is norm-cyclic or topologically cyclic. Equivalently, if ρ^{V} is a representation on V, then let us say that ρ^{V} is normcyclic or topologically cyclic if (20.13.3) is dense in V, with respect to the metric associated to ∥·∥_{V}. This is the type of condition used in [11, 151, 291]. We may say that v_{0} is a norm-cyclic or topologically cyclic vector in V as a left module over A, or of ρ^{V}, under these conditions. Similarly, if V is a right module over A, and (20.13.4) is dense in V with respect to the metric associated to ∥ · ∥_{V}, then we say that V is norm-cyclic or topologically cyclic, and that v_{0} is a norm-cyclic or topologically cyclic vector in V. Of course,

(20.13.10) algebraically cyclic modules are topologically cyclic.

If V is a left or right module over A that is topologically cyclic with topologically cyclic vector v_{0}, and if V has finite dimension as a vector space over k, then

(20.13.11) V is algebraically cyclic as a module over A,

with cyclic vector v_{0}.

If V is a left module over A, and the action of each element of A on V is a bounded linear mapping on V, then

\overline{\{x · v_{0} : x ∈ A\}} (20.13.12)

is a submodule of V, as in Section 20.4. If v_{0} is an element of (20.13.12), then (20.13.12) is topologically cyclic as a left module over A. Similarly, if V is a right module over A, and the action of each element of A is a bounded linear mapping on V, then \overline{\{v_{0} · x : x ∈ A\}} (20.13.13)

is a submodule of V. If v_{0} is an element of (20.13.13), then (20.13.13) is topologically cyclic as a right module over A.

20.14 Irreducibility

Let k be a field, let V be a vector space over k, and let E be a nonempty collection of linear mappings from V into itself. A linear subspace W of V is said to be invariant under E if

T (W) ⊆ W (20.14.1)

for every T ∈ E. If there is a nonzero proper linear subspace W of V that is invariant under E, then E is said to be reducible on V. Otherwise, E is said to be irreducible on V.

This terminology is not quite the same as in Definition 21.26 on p323 of [151], and we shall say more about this soon. This is also related to Exercise (4) on p59 of [11], and to Exercise 12 on p302 of [291]. To be more precise, we may say that E is algebraically reducible or algebraically irreducible on V, as appropriate.

Let A be an associative algebra over k. If V is a left module over A, then V is said to be irreducible if

(20.14.2) V has no nonzero proper submodules.

Equivalently, if ρ^{V} is a representation of A on V, then ρ^{V} is said to be irreducible if there are no nonzero proper linear subspaces of V that are invariant under V. This is the same as saying that the collection

{ρ_{x}^{V} : x ∈ A} (20.14.3)

of linear mappings on V is irreducible, as before. Similarly, if V is a right module over A, then V is said to be irreducible if (20.14.2) holds.

The definition of irreducibility on p384 of [206] includes the additional condition that V ≠ {0}. (20.14.4)

This condition is not included in [11, 151, 291], and we shall not include it here. A representation or module is also said to be simple on p104, 384 of [206], if it is irreducible in the sense used there, so that it is irreducible in the sense used here, and (20.14.4) holds. We may use this terminology here as well. We may say that a representation or module is algebraically irreducible or algebraically simple, to be more precise, as before.

One can check that a left or right module V over A is irreducible if and only if for every v_{0} ∈ V with v_{0} ≠ 0, we have that

(20.14.5) V is cyclic as a module over A, with cyclic vector v_{0}.

If V = {0}, then V may be considered as a cyclic module with cyclic vector 0. Thus irreducible modules are cyclic.

Suppose that k = R or C, and let ∥ · ∥_{V} be a norm on V. Also let E be a nonempty collection of bounded linear mappings from V into itself. Let us say that E is norm-reducible or topologically reducible if there is a nonzero proper closed linear subspace of V that is invariant under E. Otherwise, we say that E is norm-irreducible or topologically irreducible on V. This corresponds to part of Definition 21.26 on p323 of [151]. Note that

(20.14.6) algebraic irreducibility implies topological irreducibility.

If E is topologically irreducible on V, and V has finite dimension over k, then

(20.14.7) E is algebraically irreducible on V.

If V is a left or right module over A, then we say that V is norm-irreducible or topologically irreducible if

(20.14.8) V has no nonzero proper submodules that are closed sets,

with respect to the metric associated to ∥ · ∥_{V}. If ρ^{V} is a representation on V, then ρ^{V} is said to be norm-irreducible or topologically irreducible if (20.14.3) is topologically irreducible on V, which is the same as saying that V is topologically irreducible as a left module over V. This corresponds to another part of Definition 21.26 on p323 of [151], as well as Exercise (4) on p59 of [11], and Exercise 12 on p302 of [291].

Suppose that V is a left or right module over A, and that the action of each element of A on V is a bounded linear mapping. One can verify that V is topologically irreducible if and only if for every v_{0} ∈ V with v_{0} ≠ 0,

(20.14.9) V is topologically cyclic as a module over A, with topologically cyclic vector v_{0}.

This is the way that irreducibility is defined in Exercise 12 on p302 of [291].

20.15 Schur’s lemma

Let k be a field, and let A be an associative algebra over k. Also let V_{1}, V_{2} be vector spaces over k, and suppose that they are both left or both right modules over A. If T is a linear mapping from V_{1} into V_{2} that is a homomorphism between them, as modules over A, then ker T is a submodule of V_{1}, and T (V_{1}) is a submodule of V_{2}, as in Section 20.7.

If V_{1} is irreducible as a module over A, then it follows that

ker T = {0} or V_{1}. (20.15.1)

This means that T = 0 (20.15.2)

on V_{1}, or that (20.15.3) T is one-to-one on V_{1}.

This is part of a version of Schur’s lemma.

Similarly, if V_{2} is irreducible as a module over A, then

T (V_{1}) = {0} or V_{2}. (20.15.4)

Of course, the first possibility is the same as saying that (20.15.2) holds. This is another part of this version of Schur’s lemma. If V_{1} and V_{2} are both irreducible as modules over A, then we get that (20.15.2) holds, or

(20.15.5) T is an isomorphism from V_{1} onto V_{2},

as modules over A.

Suppose that k = R or C, and that ∥ · ∥_{V_{1}}, ∥ · ∥_{V_{2}} are norms on V_{1}, V_{2}, respectively. Suppose also for the moment T is bounded as a linear mapping from V_{1} into V_{2}, so that ker T is a closed set in V_{1}, with respect to the metric associated to ∥ · ∥_{V_{1}}. If V_{1} is topologically irreducible as a module over A, then we get that (20.15.1) holds again, to that (20.15.2) or (20.15.3) holds. This is part of another version of Schur’s lemma.

Suppose for the moment again that the action of every element of A on V_{2} is a bounded linear mapping, with respect to ∥ · ∥_{V_{2}}. This implies that

(20.15.6) \overline{T (V_{1})} is a submodule of V_{2},

as a module over A, as in Section 20.4, because T (V_{1}) is a submodule of V_{2}. If V_{2} is topologically irreducible, as a module over A, then it follows that

\overline{T (V_{1})} = \{0\} or V_{2}. (20.15.7)

The first possibility is the same as saying that (20.15.2) holds, as before. This is another part of this second version of Schur’s lemma.

Let k be any field again, let V be a vector space over k, and let T be a linear mapping from V into itself. If λ ∈ k, then

E_{λ}(T) = {v ∈ V : T (v) = λ v} (20.15.8)

is a linear subspace of V. Equivalently,

E_{λ}(T) = ker(T − λ I), (20.15.9)

where I = I_{V} is the identity mapping on V. Let R be another linear mapping from V into itself, and suppose that R commutes with T, so that

R ∘ T = T ∘ R (20.15.10)

on V. Under these conditions, it is easy to see that

R(E_{λ}(T)) ⊆ E_{λ}(T). (20.15.11)

If k = R or C, ∥ · ∥_{V} is a norm onV, and T is bounded as a linear mapping on V with respect to ∥ · ∥_{V}, then one can check that

(20.15.12) E_{λ}(T) is a closed set in V, with respect to the metric associated to ∥ · ∥_{V}. This can be obtained from (20.15.9), because T − λ I is bounded as a linear mapping on V as well.

Suppose now that V is a left or right module over A, and that T is a homomorphism from V into itself, as a module over A. Observe that

(20.15.13) E_{λ}(T) is a submodule of V

for every λ ∈ k, as in the preceding paragraph. If λ is an eigenvalue of T, so that E_{λ}(T) ≠ {0}, and if V is irreducible as a module over A, then

E_{λ}(T) = V, (20.15.14)

so that T = λ I (20.15.15)

on V. The same conclusions hold when k = R or C, ∥·∥_{V} is a norm on V, T is a bounded linear mapping on V with respect to this norm, and V is topologically irreducible as a module over A.

If V has positive finite dimension, as a vector space over k, then λ ∈ k is an eigenvalue of T if and only if T − λ I is not invertble on V, which means that

det(T − λ I) = 0. (20.15.16)

In particular, if k is algebraically closed, then there is a λ ∈ k such that this holds. In this case, if V is irreducible as a module over A, then (20.15.15) holds on V, as in the previous paragraph. This is another part of Schur’s lemma.

Part V

Algebras, norms, and operators, 4 Some more representations and modules

21.1 Spaces of linear mappings

Let k be a field, and let V_{1}, V_{2} be vector spaces over k. Remember that the space L(V_{1}, V_{2}) of all linear mappings from V_{1} into V_{2} is a linear subspace of the space of all functions on V_{1} with values in V_{2}, as in Section 19.9. This space may also be denoted L_{k}(V_{1}, V_{2}), (21.1.1)

to indicate the role of k.

Let A be an associative algebra over k. If V_{2} is a left module over A, then L(V_{1}, V_{2}) may be considered as a left module over A in a simple way too. Namely, if T is a linear mapping from V_{1} into V_{2} and x ∈ A, then

(x ·_{2} T)(v_{1}) = x · (T (v_{1})) (21.1.2)

defines another linear mapping x ·_{2} T from V_{1} into V_{2}. Similarly, if V_{2} is a right module over A, then L(V_{1}, V_{2}) may be considered as a right module over A. More precisely, if T is a linear mapping from V_{1} into V_{2} and x ∈ A, then

(T ·_{2} x)(v_{1}) = T (v_{1}) · x (21.1.3)

defines a linear mapping T ·_{2} x from V_{1} into V_{2}.

If V_{1} is a left module over A, then L(V_{1}, V_{2}) may be considered as a right module over A. In this case, if T is a linear mapping from V_{1} into V_{2} and x ∈ A, then (T ·_{1} x)(v_{1}) = T (x · v_{1}) (21.1.4)

defines a linear mapping T ·_{1} x from V_{1} into V_{2}. Similarly, if V_{1} is a right module over A, then L(V_{1}, V_{2}) may be considered as a left module over A. Indeed, if T is a linear mapping from V_{1} into V_{2} and x ∈ A, then

470 CHAPTER 21. SOME MORE REPRESENTATIONS AND MODULES

(x ·_{1} T)(v_{1}) = T (v_{1} · x) (21.1.5) defines a linear mapping x ·_{1} T from V_{1} into V_{2}. This corresponds to some of the remarks in Section 20.11 when V_{2} = k.

Suppose now that k = R or C, and that ∥ · ∥_{V 1}, ∥ · ∥_{V_{2}} are norms on V_{1}, V_{2}. Of course, the space BL(V_{1}, V_{2}) of bounded linear mappings from V_{1} into V_{2} is a linear subspace of L(V_{1}, V_{2}), as in Section 2.2. Suppose that V_{2} is a left or right module over A, and that the action of every x ∈ A on V_{2} is a bounded linear mapping on V_{2}. Under these conditions,

(21.1.6) BL(V_{1}, V_{2}) is a submodule of L(V_{1}, V_{2}),

as a left or right module over A with respect to (21.1.2) or (21.1.3), as appropriate. In fact, if x ∈ A and T is a bounded linear mapping from V_{1} into V_{2}, then the action of x on T is the same as

(21.1.7) the composition of T with the action of x on V_{2},

and this composition is also a bounded linear mapping from V_{1} into V_{2}.

Similarly, suppose that V_{1} is a left or right module over A, and that the action of every x ∈ A is a bounded linear mapping from V_{1} into itself. In this case, (21.1.6) holds when L(V_{1}, V_{2}) is considered as a right or left module over A with respect to (21.1.4) or (21.1.5), as appropriate. This is because the action of x ∈ A on a bounded linear mapping T from V_{1} into V_{2} is the same as

(21.1.8) the composition of the action of x on V_{1} with T,

and this composition is another bounded linear mapping from V_{1} into V_{2}.

In each of these cases, the action of x ∈ A is a bounded linear mapping on BL(V_{1}, V_{2}), with respect to the operator norm associated to ∥ · ∥_{V_{1}}, ∥ · ∥_{V_{2}}. Indeed, if T is a bounded linear mapping from V_{1} into V_{2}, then the operator norm of the action of x on T is less than or equal to the operator norm ∥T ∥_{op,V_{1}V_{2}} of T times the operator norm of the action of x on V_{1} or V_{2}, as appropriate.

If λ_{1} is a bounded linear functional on V_{1} and v_{2} ∈ V_{2}, then

v_{1} ↦ λ_{1}(v_{1}) v_{2} (21.1.9)

defines a bounded linear mapping from V_{1} into V_{2}. The operator norm of this bounded linear mapping is equal to

∥λ_{1}∥_{V′_{1}} ∥v_{2}∥_{V_{2}}, (21.1.10)

where ∥ · ∥_{V′_{1}} is the dual norm on the space V′_{1} of bounded linear functionals on V_{1} associated to ∥ · ∥_{V_{1}}, as in Section 3.1.

Suppose that V_{2} is a left or right module over A, the action of every x ∈ A on V_{2} is a bounded linear mapping, and that

V_{1} ≠ {0}. (21.1.11)

Under these conditions, one can check that the operator norm of the action of every x ∈ A on BL(V_{1}, V_{2}) is equal to the operator norm of the action of x on V_{2}, using the remarks in the previous paragraph. This uses the fact that there is a nonzero bounded linear functional on V_{1}, because of (21.1.11), by the Hahn–Banach theorem, as in Sections 3.10 and 3.12.

Similarly, suppose that V_{1} is a left or right module over A, the action of every x ∈ A on V_{1} is a bounded linear mapping, and that

V_{2} ≠ {0}. (21.1.12)

In this case, one can verify that the operator norm of the action of every x ∈ A on BL(V_{1}, V_{2}) is equal to the operator norm of the action of x on V_{1}. This uses consequences of the Hahn–Banach theorem mentioned in Section 3.10 and 3.12 again. This also corresponds to a remark in Section 20.11 when V_{2} = k.

21.2 More on module homomorphisms

Let k be a field, and let A be an associative algebra over k. Also let V_{1}, V_{2} be vector spaces over k that are both left or both right modules over A. Consider the space Hom_{A}(V_{1}, V_{2}) (21.2.1)

of all linear mappings from V_{1} into V_{2} that are homomorphisms from V_{1} into V_{2}, as modules over A. It is easy to see that

(21.2.2) Hom_{A}(V_{1}, V_{2}) is a linear subspace of L(V_{1}, V_{2}),

as a vector space over k.

Let V_{3} be another vector space over k that is a left or right module over A, as appropriate. Suppose that T_{1}, T_{2} are linear mappings from V_{1}, V_{2} into V_{2}, V_{3}, respectively, that are module homomorphisms. One can check that

(21.2.3) T_{2} ∘ T_{1} is a module homomorphism from V_{1} into V_{3}.

If V_{1}, V_{2} are left modules over A, then L(V_{1}, V_{2}) is a left module over A with respect to (21.1.2), and a right module over A with respect to (21.1.4). In this case, a linear mapping T from V_{1} into V_{2} is a module homomorphism if and only if x ·_{2} T = T ·_{1} x (21.2.4)

for every x ∈ A.

Similarly, if V_{1}, V_{2} are right modules over A, then L(V_{1}, V_{2}) is a right module over A with respect to (21.1.3), and a left module over A with respect to (21.1.5). In this situation, a linear mapping T from V_{1} into V_{2} is a module homomorphism if and only if T ·_{2} x = x ·_{1} T (21.2.5)

for every x ∈ A.

Suppose for the moment that A is commutative, so that left and right modules over A are the same. Of course, L(V_{1}, V_{2}) is still a module over A with respect to the actions of A corresponding to the actions of A on V_{1} and on V_{2}, as in the previous section. It is easy to see that

(21.2.6) Hom_{A}(V_{1}, V_{2}) is a submodule of L(V_{1}, V_{2})

with respect to both actions of A on L(V_{1}, V_{2}) in this case. The restrictions of these two actions on L(V_{1}, V_{2}) to Hom_{A}(V_{1}, V_{2}) are the same, as in the previous two paragraphs. This means that

(21.2.7) Hom_{A}(V_{1}, V_{2}) is a module over A

in a natural way under these conditions.

Suppose for the moment again that k = R or C, and that ∥ · ∥_{V_{1}}, ∥ · ∥_{V_{2}} are norms on V_{1}, V_{2}, respectively. Note that the space

BL(V_{1}, V_{2}) ∩ Hom_{A}(V_{1}, V_{2}) (21.2.8)

of bounded linear mappings from V_{1} into V_{2} that are homomorphisms from V_{1} into V_{2}, as modules over A, is a linear subspace of the space BL(V_{1}, V_{2}) of all bounded linear mappings from V_{1} into V_{2}. If the action of every element of A on V_{2} is a bounded linear mapping from V_{2} into itself, then one can check that

(21.2.9) BL(V_{1}, V_{2}) ∩ Hom_{A}(V_{1}, V_{2}) is a closed set in BL(V_{1}, V_{2}),

with respect to the metric associated to the operator norm.

Let k be any field again, and let V be a vector space over k that is a left or right module over A. Observe that

(21.2.10) Hom_{A}(V, V) is a subalgebra of L(V),

as an associative algebra over k, because of (21.2.3). Of course, the identity mapping I = I_{V} on V is a homomorphism from V into itself, as a module over A.

If k = R or C, and ∥ · ∥_{V} is a norm on V, then

(21.2.11) BL(V) ∩ Hom_{A}(V, V) is a subalgebra of BL(V),

as an associaitve algebra over k. If the action of every element of V is bounded on V, then (21.2.12) BL(V) ∩ Hom_{A}(V, V) is a closed set in BL(V),

with respect to the metric associated to the operator norm, as before.

21.3 Division algebras

Let k be a field, and let A be an associative algebra over k. Suppose that A has a multiplicative identity element e_{A}, and that e_{A} ≠ 0. In this case,

t ↦ t e_{A} (21.3.1)

21.4. T -CYCLIC VECTORS 473

defines an injective homomorphism from k into A, as algebras over k.

We say that A is a division algebra if every nonzero element of A has a multiplicative inverse in A. If A is commutative, then this means that A is a field, that contains an isomorphic copy of k as a subfield, as in the preceding paragraph. The algebra H of quaternions is a division algebra over the real numbers that is not commutative, as in Section 7.1. A famous theorem of Frobenius states that the only finite-dimensional division algebras over the real numbers are R, C, and H, as in [149, 233, 258].

One can also consider division algebras that are not necessarily associative, normally using a more complicated formulation that is equivalent to the previous one when the algebra is associative. See [6, 7, 244, 260], for instance.

An (associative) ring with a nonzero multiplicative identity element is said to be a division ring if every nonzero element of the ring has a multiplicative inverse in the ring. An associative algebra over k that is a division algebra is a division ring in particular.

The center of a ring is defined as usual as the set of elements of the ring that commute with all other elements of the ring, and it is easy to see that this is a subring of the ring. One can check that the center of a division ring is a division ring, and thus a field. A division ring may be considered as an associative division algebra over any field contained in the center of the ring.

Sometimes division rings are called skew-fields, and sometimes fields are not required to be commutative. Here we always take fields to be commutative.

Modules over division rings have many of the same properties as vector spaces over fields, except that one should be careful about whether one is considering a left or right module. This corresponds to remarks on p438 of [206], and p193 of [226]. The brief review of linear algebra beginning on p9 of [291] deals with left and right vector spaces over not necessarily commutative fields. Left and right vector spaces over division rings are discussed in some detail in [163].

Let k be a (commutative) field again, and let V be a vector space over k, with V ≠ {0}. Also let A be an associative algebra over k, and suppose that V is a left or right module over A. If V is irreducible as a module over A, then

(21.3.2) Hom_{A}(V, V) is a division algebra over k,

because of Schur’s lemma, as in Section 20.15.

21.4 T -Cyclic vectors

Let k be a field, let V be a vector space over k, and let T be a linear mapping from V into itself. If v_{0} ∈ V, then the linear subspace of V spanned by

v_{0}, T (v_{0}), T^{2}(v_{0}), T^{3}(v_{0}), … (21.4.1)

may be called the T -cyclic subspace of V generated by v_{0}, as on p344 of [37], and p389 of [226]. Of course, T maps this subspace into itself.

Let X be an indeterminate, and let k[X] be the algebra of formal polynomials in X with coefficients in k, as in Section 19.6. Remember that there is a unique representation of k[X] on V such that 1_{k} · v = v and

X · v = T (v) (21.4.2)

for every v ∈ V, as in Section 20.4. If v_{0} ∈ V, then

{f(X) · v_{0} : f(X) ∈ k[X]} (21.4.3)

is a submodule of V, as a module over k[X], as in Section 20.13. This is the same as the T -cyclic subspace of V generated by v_{0}. In particular, V is cyclic as a module over k[X], with cyclic vector v_{0}, if and only if V is spanned by (21.4.1).

Suppose that k = R or C, ∥ · ∥_{V} is a norm on V, and that T is a bounded linear mapping on V. The closure of the linear span of (21.4.1) in V, with respect to the metric associated to ∥ · ∥_{V}, may be called the norm T -cyclic closed subspace or topologically T -cyclic closed subspace of V generated by v_{0}. Note that T also maps this subspace into itself, because T is bounded on V, by hypothesis. This subspace is the same as the closure of (21.4.3) in V, as in the preceding paragraph. Thus V is topologically cyclic as a module over k[X], with topologically cyclic vector v_{0}, if and only if V is the topologically T -cyclic closed subspace generated by v_{0}.

21.5 Representations and ∗-homomorphisms

Let k be a field, and let A_{1}, A_{2} be algebras in the strict sense over k. Also let a_{1} ↦ a_{1}^{∗_{1}} be an opposite algebra isomorphism from A_{1} onto itself, and let a_{2} ↦ a_{2}^{∗_{2}} be an opposite algebra isomorphism from A_{2} onto itself. An algebra homomorphism ϕ from A_{1} into A_{2} is said to be a ∗-homomorphism if

ϕ(a_{1}^{∗_{1}}) = ϕ(a_{1})^{∗_{2}} (21.5.1)

for every a_{1} ∈ A, as in Section 12.7. Similarly, we may use the terms ∗- isomorphisms and ∗-automorphisms for algebra isomorphisms and automorphisms that are ∗-homomorphisms, as before.

Let A be an associative algebra over k, let x ↦ x^{∗_{A}} be an opposite algebra isomorphism from A onto itself, and let V be a vector space over k. If V is a left module over A, then V may be considered as a right module over A, with

v · x = x^{∗_{A}} · v (21.5.2)

for every x ∈ A and v ∈ V. Similarly, if V is a right module over A, then V may be considered as a left module over A, with

x · v = v · x^{∗_{A}} (21.5.3)

for every x ∈ A and v ∈ V.

21.5. REPRESENTATIONS AND ∗-HOMOMORPHISMS 475

Let b be a bilinear form on V, as in Section 19.11. If V is a left module over A, then an interesting compatibility condition between this and b is that

b(x · v, w) = b(v, x^{∗_{A}} · w) (21.5.4)

for every x ∈ A and v, w ∈ V. Similarly, if V is a right module over A, then an interesting compatibility condition between this and b is that

b(v · x, w) = b(v, w · x^{∗_{A}}) (21.5.5)

for every x ∈ A and v, w ∈ V.

Suppose for the moment that V has finite dimension as a vector space over k, and that b is nondegenerate on V, as in Section 19.12. If T is a linear mapping from V into itself, then its adjoint T^{∗_{b}} with respect to b may be defined as a linear mapping from V into itself, and T ↦ T^{∗_{b}} defines an opposite algebra isomorphism from L(V) onto itself, as before. If ρ^{V} is a representation of A on V, then (21.5.4) is the same as saying that

ρ_{x^{∗_{A}}}^{V} = (ρ_{x}^{V})^{∗_{b}} (21.5.6)

for every x ∈ A. Equivalently, this means that the representation corresponds to a ∗-homomorphism from A into L(V), with respect to x ↦ x^{∗_{A}} on A, and T ↦ T^{∗_{b}} on L(V). One may say that ρ^{V} is a ∗-representation with respect to b on V in this case.

Suppose for the moment again that k = C, that x ↦ x^{∗_{A}} is a conjugatelinear opposite algebra isomorphism from A onto itself, and that b is a sesquilinear form on V, as in Section 5.15. If V is a left module over A, then an interesting compatibility condition between this and b is that (21.5.4) hold for every x ∈ A and v, w ∈ V. Similarly, if V is a right module over A, then an interesting compatibility condition between this and b is that (21.5.5) hold for every x ∈ A and v, w ∈ V.

Let us also ask that V have finite dimension as a vector space over C, and that b be nondegenerate on V, as in Section 19.15. If T is a linear mapping from V into itself, then its adjoint T^{∗_{b}} with respect to b may be defined as a linear mapping from V into itself, and T ↦ T^{∗_{b}} defines a conjugate-linear opposite algebra isomorphism from L(V) onto itself, as before. If ρ^{V} is a representation of A on V, then (21.5.4) is the same as saying that (21.5.6) holds for every x ∈ A. This means that the representation corresponds to a ∗-homomorphism from A into L(V), with respect to x ↦ x^{∗_{A}} on A and T ↦ T^{∗_{b}} on L(V), as before. One may say that ρ^{V} is a ∗-representation with respect to b on V in this case too.

Suppose now that k = R or C, and that x ↦ x^{∗_{A}} is an opposite algebra isomorphism from A onto itself, that is conjugate-linear in the complex case. Also let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers, as appropriate. If T is a bounded linear mapping from V into itself, then let T^{∗_{V}} be the adjoint of T with respect to ⟨·, ·⟩_{V}, as in Section 3.5.

Let ρ^{V} be a representation of A on V, and suppose that ρ_{x}^{V} is a bounded linear mapping from V into itself, with respect to the norm ∥ · ∥_{V} associated to the inner product, for every x ∈ A. If

ρ_{x^{∗_{A}}}^{V} = (ρ_{x}^{V})^{∗_{V}} (21.5.7)

for every x ∈ A, then ρ^{V} is said to be a ∗-representation of A on V, as in Definition 21.8 on p314 of [151]. This is related to Definition 2.5.2 on p57 of [11]. Note that the definition of a representation of a C^{∗} algebra on a Hilbert space on p273 of [291] includes (21.5.7), because the definition of a homomorphism between C^{∗} algebras on p262 of [291] includes (21.5.1). As before, (21.5.7) is the same as saying that

⟨x · v, w⟩_{V} = ⟨v, x^{∗_{A}} · w⟩_{V} (21.5.8)

for every x ∈ A and v, w ∈ V under these conditions.

21.6 Involutions and bounded linear mappings

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers, and let T_{1}, T_{2} be linear mappings from V into itself such that

⟨T_{1}(v), w⟩_{V} = ⟨v, T_{2}(w)⟩_{V} (21.6.1)

for every v, w ∈ V. Of course, this is the same as saying that

⟨T_{2}(v), w⟩_{V} = ⟨v, T_{1}(w)⟩_{V} (21.6.2)

for every v, w ∈ V. Under these conditions, we have that

(21.6.3) T_{1} and T_{2} are bounded linear mappings from V into itself,

with respect to the norm ∥ · ∥_{V} associated to the inner product. This was mentioned in Section 8.6 for linear mappings from V into itself that are symmetric or antisymmetric with respect to the inner product. One can reduce to those cases by considering T_{1} ± T_{2}. Alternatively, one can show this using essentially the same arguments as before. It follows that T_{2} is the usual Hilbert space adjoint of T_{1} on V.

Let A be an associative algebra over the real or complex numbers, as appropriate, with an involution x ↦ x^{∗_{A}} that is conjugate-linear in the complex case. Also let ρ^{V} be a representation of A on V that is compatible with the involution on A and the inner product on V, in the sense that

⟨ρ_{x}^{V} (v), w⟩_{V} = ⟨v, ρ_{x^{∗_{A}}}^{V} (w)⟩_{V} (21.6.4)

for every x ∈ A and v, w ∈ V, as in the previous section. Observe that

(21.6.5) ρ_{x}^{V} is a bounded linear mapping from V into itself

for each x ∈ A, as in the preceding paragraph. This implies that ρ_{x^{∗_{A}}}^{V} is the Hilbert space adjoint of ρ_{x}^{V} on V, as before. This means that ρ^{V} is a ∗-representation of A on V, as in the previous section.

21.7 Invariant subspaces and Hilbert spaces

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers. Remember that the orthogonal complement V_{0}^{⊥} of a linear subspace V_{0} of V is defined to be the set of u ∈ V that are orthogonal to every element of V_{0}, as in Section 2.15. This is a closed linear subspace of V, and if V_{0} is a closed set in V with respect to the metric associated to the norm ∥ · ∥_{V} corresponding to the inner product, then every element of V can be expressed as a sum of elements of V_{0} and V_{0}^{⊥} in a unique way, as before. This means that V corresponds to the direct sum of V_{0} and V_{0}^{⊥} as a Hilbert space, as in Section 5.12. Of course, this is much simpler when V has finite dimension.

Let T be a bounded linear mapping from V into itself, and let T^{∗} = T^{∗_{V}} be the adjoint of T with respect to ⟨·, ·⟩_{V}, as in Section 3.5. Also let V_{0} be a linear subspace of V again, and suppose for the moment that

T (V_{0}) ⊆ V_{0}. (21.7.1)

If v_{0} ∈ V_{0} and u ∈ V_{0}^{⊥}, then

⟨v_{0}, T^{∗_{V}} (u)⟩_{V} = ⟨T (v_{0}), u⟩_{V} = 0. (21.7.2)

This means that T^{∗_{V}} (u) ∈ V_{0}^{⊥}, so that

T^{∗_{V}} (V_{0}^{⊥}) ⊆ V_{0}^{⊥}. (21.7.3)

Note that (21.7.1) implies that

T (\overline{V_{0}}) ⊆ \overline{V_{0}}, (21.7.4)

where \overline{V_{0}} is the closure of V_{0} in V with respect to the metric associated to ∥·∥_{V}.

Let E be a nonempty collection of bounded linear mappings from V into itself, and put E^{∗_{V}} = {T^{∗_{V}} : T ∈ E}. (21.7.5)

Suppose that (21.7.6) V_{0} is invariant under E,

so that (21.7.1) holds for every T ∈ E, as in Section 20.14. This implies that (21.7.3) holds for every T ∈ E, so that

(21.7.7) V_{0}^{⊥} is invariant under E^{∗_{V}} .

In particular, if E is self-adjoint, in the sense that

E^{∗_{V}} = E, (21.7.8)

then (21.7.9) V_{0}^{⊥} is invariant under E.

Let A be an associative algebra over the real or complex numbers, as appropriate, and let x ↦ x^{∗_{A}} be an opposite algebra isomorphism from A onto itself, which is conjugate-linear in the complex case. Also let ρ^{V} be a ∗-representation of A on V, which includes the condition that ρ_{x}^{V} be a bounded linear mapping on V for each x ∈ A, as in Section 21.5. If

(21.7.10) V_{0} is invariant under ρ^{V} ,

then (21.7.11) V_{0}^{⊥} is invariant under ρ^{V} ,

as in the preceding paragraph. If V_{0} is also a closed set in V, then ρ^{V} corresponds to the direct sum of the subrepresentations associated to V_{0} and V_{0}^{⊥}. Of course, one can replace V_{0} with its closure in V, to get a closed linear subspace of V that is invariant under ρ^{V}.

Suppose now that V has positive finite dimension. One can repeat this argument finitely many times, to get finitely many nonzero linear subspaces V_{1}, …, V_{n} of V with the following properties. First, V corresponds to the orthogonal direct sum of V_{1}, …, V_{n}, so that every element of V can be expressed as a sum of elements of V_{1}, …, V_{n} in a unique way, and the elements of V_{j} are orthogonal to the elements of V_{l} when j ≠ l. Second, V_{j} is invariant under ρ^{V} for each j, so that ρ^{V} corresponds to the direct sum of the subrepresentations associated to V_{1}, …, V_{n}. Third, each of these subrepresentations is irreducible.

21.8 Some closed invariant linear subspaces

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers again, and let T be a bounded linear mapping from V into itself. Suppose that T is self-adjoint with respect to this inner product, so that

T^{∗_{V}} = T. (21.8.1)

Also let σ(T) = σ_{BL(V)}(T) be the spectrum of T with respect to the algebra of bounded linear mappings from V into itself, as in Section 6.8. This is automatically a subset of R in the real case, and we have seen in Section 8.11 that this also holds in the complex case.

Remember that σ(T) is closed and bounded, as in Section 6.8. Note that σ(T) ≠ ∅, as in Section 8.12, or as in Section 6.8 in the complex case. If

σ(T) = {r} (21.8.2)

for some real number r, then T = r I_{V} , (21.8.3)

where I_{V} is the identity operator on V. This can be obtained from the remarks in Section 8.12, although it is easier to first reduce to the case when r = 0.

Suppose that (21.8.4) σ(T) has at least two elements.

This implies that there are continuous real-valued functions f_{1}, f_{2} on σ(T) such that neither f_{1} nor f_{2} is identically zero on σ(T), and

f_{1} f_{1} ≡ 0 (21.8.5)

on σ(T). This leads to bounded self-adjoint linear operators f̃_{1}(T), f̃_{2}(T) on V, as in Section 8.14. Note that

f̃_{1}(T), f̃_{2}(T) ≠ 0, (21.8.6)

because the operator norms of f̃_{1}(T), f̃_{2}(T) are equal to the supremum norms of f_{1}, f_{2} on σ, respectively. We also have that

f̃_{1}(T) ∘ f̃_{2}(T) = 0 (21.8.7)

on V, because of (21.8.5).

Note that ker f̃_{1}(T) ≠ V and (f̃_{2}(T))(V) ≠ {0}, (21.8.8)

by (21.8.6). We also have that

(f̃_{2}(T))(V) ⊆ ker f̃_{1}(T), (21.8.9)

by (21.8.7). Thus ker f̃_{1}(T) ≠ {0}. (21.8.10)

Of course, the kernel of f̃_{1}(T) is a closed linear subspace of V, because f̃_{1}(T) is a bounded linear mapping on V.

Let R be a bounded linear mapping from V into itself that commutes with T. This implies that R commutes with powers of T, so that R commutes with polynomials of T. One can use this to get that

R ∘ f̃_{1}(T) = f̃_{1}(T) ∘ R (21.8.11)

on V. More precisely, f̃_{1}(T) can be approximated by polynomials of T with real coefficients with respect to the metric associated to the operator norm, essentially by construction, as in Section 8.14. This permits one to reduce (21.8.11) to the analogous statement for polynomials of T.

It follows from (21.8.11) that

R(ker f̃_{1}(T)) ⊆ ker f̃_{1}(T). (21.8.12)

This shows that the collection of bounded linear mappings on V that commute with T is topologically reducible, as in Section 20.14.

Let E be a nonempty collection of bounded linear mappings on V, and suppose that E is topologically irreducible on V. If T is a bounded self-adjoint linear mapping from V into itself that commutes with every element of E, then the previous argument shows that (21.8.3) holds for some r ∈ R.

21.9 Self-adjoint collections of operators

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers, and let E be a nonempty collection of bounded linear mappings from V into itself. Consider the collection

(21.9.1) B = {T ∈ BL(V) : T ∘ R = R ∘ T on V for every R ∈ E}

of bounded linear mappings from V into itself that commute with every element of E. It is easy to see that B is a subalgebra of BL(V) that contains the identity operator I_{V} on V, and that B is a closed set in BL(V) with respect to the metric associated to the operator norm. If T ∈ B, and if T is invertible as an element of BL(V), then one can check that

T^{−1} ∈ B. (21.9.2)

Suppose that E is self-adjoint, in the sense that E^{∗_{V}} = E, as in Section 21.7. One can verify that this happens when

E^{∗_{V}} ⊆ E, (21.9.3)

which is to say that this implies that E ⊆ E^{∗_{V}}. Under these conditions, it is easy to see that B^{∗_{V}} ⊆ B, (21.9.4)

so that B is self-adjoint as well.

Suppose from now on in this section that

(21.9.5) E is topologically irreducible on V,

as in Section 20.14. This implies that the only self-adjoint elements of B are of the form r I_{V} for some real number r, as in the previous section.

If T is any bounded linear mapping from V into itself, then

(1/2) (T + T^{∗_{V}}) (21.9.6)

is self-adjoint on V, and (1/2) (T − T^{∗_{V}}) (21.9.7)

is anti-self-adjoint on V. Of course, T is equal to the sum of (21.9.6) and (21.9.7). If T ∈ B, than (21.9.6) and (21.9.7) are elements of B as well, because of (21.9.4). It follows that (21.9.6) is a real multiple of the identity operator on V, as in the preceding paragraph.

Suppose for the moment that V is a complex Hilbert space. In this case, if a bounded linear mapping from V into itself is self-adjoint, then it is i times a self-adjoint operator on V. This implies that an anti-self-adjoint element of B is i times a self-adjoint element of B, and thus an imaginary multiple of I_{V}. This means that B consists of only complex multiples of I_{V}, because of the remarks in the previous paragraph.

Suppose from now on in this section that V is a real Hilbert space. If T is any bounded linear mapping from V into itself, then T^{∗_{V}} ∘ T is self-adjoint on V, and ⟨(T^{∗_{V}} ∘ T)(v), v⟩_{V} = ⟨T (v), T (v)⟩_{V} (21.9.8)

for every v ∈ V. If T ∈ B, then

T^{∗_{V}} ∘ T ∈ B, (21.9.9)

so that T^{∗_{V}} ∘ T = r I_{V} (21.9.10)

for some real number r, as before. It follows that

∥T (v)∥_{V}^{2} = r ∥v∥_{V}^{2} (21.9.11)

for every v ∈ V, where ∥ · ∥_{V} is the norm on V associated to the inner product. In particular, this means that

r = ∥T ∥_{op}^{2}, (21.9.12)

where ∥ · ∥_{op} is the operator norm on BL(V) corresponding to ∥ · ∥_{V}.

Similarly, T ∘ T^{∗_{V}} = ∥T^{∗_{V}} ∥_{op}^{2} I_{V} = ∥T ∥_{op}^{2} I_{V} (21.9.13)

when T ∈ B. If T ≠ 0, then it follows that T is invertible on V, with

T^{−1} = ∥T ∥_{op}^{−2} T^{∗_{V}} . (21.9.14)

Let T_{1}, T_{2} be bounded linear mappings from V into itself, and suppose that at least one of them is an element of B. Observe that

∥T_{1} ∘ T_{2}∥_{op} = ∥T_{1}∥_{op} ∥T_{2}∥_{op}, (21.9.15)

because of (21.9.11).

21.10 A helpful inner product

We would like to continue with the same notation and hypotheses as in the previous section. Thus V is a real Hilbert space, and E is a nonempty collection of bounded linear mappings from V into itself that is self-adjoint and topologically irreducible on V. The subalgebra B of BL(V) defined in (21.9.1) is also self-adjoint, and the only self-adjoint elements of B are multiples of the identity operator I_{V} on V. Let us ask that V ≠ {0} too, so that I_{V} ≠ 0.

If T_{1}, T_{2} are bounded linear mappings on V, then

(1/2) (T_{1} ∘ T_{2}^{∗_{V}} + T_{2} ∘ T_{1}^{∗_{V}}) (21.10.1)

is self-adjoint on V. If T_{1}, T_{2} ∈ B, then (21.10.1) is an element of B as well, and there is a unique real number ⟨T_{1}, T_{2}⟩_{B} (21.10.2) such that (1/2) (T_{1} ∘ T_{2}^{∗_{V}} + T_{2} ∘ T_{1}^{∗_{V}}) = ⟨T_{1}, T_{2}⟩_{B} I_{V} . (21.10.3)

It is easy to see that (21.10.2) is linear in T_{1} and T_{2}, and symmetric in T_{1} and T_{2}, because of the corresponding properties of (21.10.1). If T ∈ B, then

⟨T, T ⟩_{B} = ∥T ∥_{op}^{2}, (21.10.4)

by the remarks in the previous section. It follows that (21.10.2) is an inner product on B.

If one of T_{1}, T_{2} is self-adjoint and the other is anti-self-adjoint, then (21.10.1) is equal to 0. Of course, this implies that

⟨T_{1}, T_{2}⟩_{B} = 0. (21.10.5)

If T_{1} and T_{2} are both anti-self-adjoint, then (21.10.1) is equal to

−(1/2) (T_{1} ∘ T_{2} + T_{2} ∘ T_{1}). (21.10.6)

In this case, (21.10.5) holds if and only if (21.10.6) is equal to 0.

If T_{0} is a bounded linear mapping on V that is anti-self-adjoint, then

T_{0}^{2} = T_{0} ∘ T_{0} = −T_{0}^{∗_{V}} ∘ T_{0} (21.10.7)

is self-adjoint on V. If T_{0} ∈ B, then we get that

T_{0}^{2} = −∥T_{0}∥_{op}^{2} I_{V} , (21.10.8)

as in the previous section.

It may be that the elements of B are all self-adjoint, so that B consists only of multiples of I_{V}. Suppose for the moment that the space of anti-self-adjoint elements of B has dimension equal to one, as a vector space over the real numbers. This means that there is a nonzero anti-self-adjoint element T_{0} of B such that all other anti-self-adjoint elements of B are multiples of T_{0}. We can choose T_{0} so that ∥T_{0}∥_{op} = 1, (21.10.9)

which determines T_{0} up to multiplication by −1. It follows that

T_{0}^{2} = −I_{V} , (21.10.10)

and B basically corresponds to C.

Suppose now that the space of anti-self-adjoint elements of B has dimension at least two. In this case, we can use the inner product on B to get anti-self-adjoint elements T_{1}, T_{2} of B such that

∥T_{1}∥_{op} = ∥T_{2}∥_{op} = 1 (21.10.11)

and T_{1} is orthogonal to T_{2}, in the sense that (21.10.5) holds. Using (21.10.11), we get that T_{1}^{2} = T_{2}^{2} = −I_{V} , (21.10.12) as in (21.10.8). The orthogonality of T_{1} and T_{2} means that (21.10.6) is equal to 0, as before.

It is easy to see that T_{1} ∘ T_{2} is anti-self-adjoint, because (21.10.6) is equal to 0. Note that ∥T_{1} ∘ T_{2}∥_{op} = 1, (21.10.13)

as in (21.9.15). We also have that T_{1}∘T_{2} anti-commutes with T_{1} and T_{2}, because T_{1} and T_{2} anti-commute with each other. This implies that

⟨T_{1} ∘ T_{2}, T_{2}⟩_{B} = ⟨T_{1} ∘ T_{2}, T_{2}⟩_{B} = 0. (21.10.14)

It follows that the space of anti-self-adjoint elements of B has dimension at least three, as a vector space over the real numbers.

Suppose that T_{3} is an anti-self-adjoint element of B such that

⟨T_{1}, T_{3}⟩_{B} = ⟨T_{2}, T_{3}⟩_{B} = ⟨T_{1} ∘ T_{2}, T_{3}⟩_{B} = 0. (21.10.15)

This means that T_{3} anti-commutes with T_{1}, T_{2}, and T_{1} ∘ T_{2}. However, because T_{3} anti-commutes with T_{1} and T_{2}, T_{3} commutes with T_{1} ∘ T_{2}. This implies that

T_{1} ∘ T_{2} ∘ T_{3} = 0. (21.10.16)

It follows that T_{3} = 0, (21.10.17)

because T_{1} and T_{2} are invertible on V, as in the previous section.

This shows that the space of anti-self-adjoint elements of B has dimension equal to three under these conditions. More precisely, B basically corresponds to H.

21.11 Nondegeneracy on Hilbert spaces

Let k be a field, let V be a vector space over k, and let E be a nonempty collection of linear mappings from V into itself. Put

(21.11.1) N_{E} = ⋂_{T ∈E} ker T = {v ∈ V : T (v) = 0 for every T ∈ E},

which is a linear subspace of V. Let us say that E is nondegenerate if

N_{E} = {0}. (21.11.2)

Of course, the nondegeneracy of a representation on V, as in Section 20.9, is the same as the nondegeneracy in this sense of the corresponding collection of linear mappings on V.

Suppose that k = R or C, and let ∥ · ∥_{V} be a norm on V. Suppose also that every T ∈ E is a bounded linear mapping on V. This implies that the kernel of every T ∈ E is a closed linear subspace of V, with respect to the metric associated to ∥ · ∥_{V}, so that N_{E} is a closed linear subspace of V as well.

Suppose now that (V, ⟨·, ·⟩_{V}) is a Hilbert space, with corresponding norm ∥ · ∥_{V}. Consider the linear subspace V_{E} of V spanned by elements of the form

T (v), (21.11.3)

with T ∈ E and v ∈ V. Thus

(21.11.4) V_{E}^{⊥} = {w ∈ V : ⟨T (v), w⟩_{V} = 0 for every T ∈ E and v ∈ V }.

Equivalently,

V_{E}^{⊥} = {w ∈ V : ⟨v, T^{∗_{V}} (w)⟩_{V} = 0 for every T ∈ E and v ∈ V }
(21.11.5) = {w ∈ V : T^{∗_{V}} (w) = 0 for every T ∈ E} = N_{E^{∗_{V}}} ,

where E^{∗_{V}} is as in Section 21.7.

This implies that N_{E^{∗_{V}}}^{⊥} = \overline{V_{E}}, (21.11.6)

as in Section 2.15, where \overline{V_{E}} is the closure of V_{E} in V with respect to the metric associated to ∥ · ∥_{V}. We can replace E with E^{∗_{V}}, to get that

V_{E^{∗_{V}}}^{⊥} = N_{E}, (21.11.7)

and thus N_{E}^{⊥} = \overline{V_{E^{∗_{V}}}} . (21.11.8)

This shows that E is nondegenerate on V if and only if

\overline{V_{E^{∗_{V}}}} = V. (21.11.9)

Note that N_{E}, V_{E}, and \overline{V_{E}} are invariant under E. If E^{∗_{V}} = E, then V corresponds to the orthogonal direct sum of N_{E} and \overline{V_{E}}. In this case, the collection of the restrictions of the elements of E to \overline{V_{E}} is nondegenerate on \overline{V_{E}}, because N_{E} ∩ \overline{V_{E}} = \{0\}. (21.11.10)

This basically corresponds to Exercise (1) on p59 of [11], and to Corollary 21.10 on p314 of [151].

Let A be an associative algebra over the real or complex numbers, as appropriate, and let x ↦ x^{∗_{A}} be an opposite algebra isomorphism from A onto itself that is conjugate-linear in the complex case. Suppose that ρ^{V} is a ∗- representation of A on V, as in Section 21.5, which means in particular that ρ_{x}^{V} is a bounded linear mapping on V for each x ∈ A. Thus

E(ρ^{V}) = {ρ_{x}^{V} : x ∈ A} (21.11.11)

is a nonempty collection of bounded linear mappings on V, and

E(ρ^{V})^{∗_{V}} = E(ρ^{V}). (21.11.12)

As before, ρ^{V} is nondegenerate as a representation on V exactly when E(ρ^{V}) is nondegenerate on V.

Of course, V_{E(ρ^{V} )} is the same as the linear span in V of

ρ_{x}^{V} (v), (21.11.13)

with x ∈ A and v ∈ V. Using the earlier remarks, we get that ρ^{V} is nondegenerate on V if and only if \overline{V_{E(ρ^{V} )}} = V. (21.11.14)

In particular, this holds when

(21.11.15) ρ^{V} is topologically cyclic on V,

as in Section 20.13. This is related to Exercise (3) on p59 of [11].

21.12 Some topologically cyclic subspaces

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers, and let ∥ · ∥_{V} be the corresponding norm on V. Also let A be an associative algebra over the real or complex numbers, as appropriate, and let x ↦ x^{∗_{A}} be an opposite algebra isomorphism from A onto itself that is conjugate-linear in the complex case. Suppose that ρ^{V} is a ∗-representation of A on V, as in Section 21.5, so that ρ_{x}^{V} is a bounded linear mapping on V for each x ∈ A in particular. Put (21.12.1) N_{ρ^{V}} = ⋂_{x∈A} ker ρ_{x}^{V} = {v ∈ V : ρ_{x}^{V} (v) = 0 for every x ∈ A},

which is the same as N_{E(ρ^{V} )} in the notation of the previous section. This is a closed linear subspace of V, with respect to the metric associated to ∥ · ∥_{V}, as before.

If v_{0} ∈ V, then {ρ_{x}^{V} (v_{0}) : x ∈ A} (21.12.2)

is a linear subspace of V that is invariant under ρ^{V}, as in Section 20.13. This implies that V (ρ^{V} , v_{0}) = \overline{\{ρ_{x}^{V} (v_{0}) : x ∈ A\}} (21.12.3)

is a closed linear subspace of V that is invariant under ρ^{V}, as before. Suppose that v_{0} ∈ N_{ρ^{V}}^{⊥} . (21.12.4)

We would like to show that

v_{0} ∈ V (ρ^{V} , v_{0}). (21.12.5)

This corresponds to part of the proof of Theorem 21.13 on p315 of [151].

Note that N_{ρ^{V}}^{⊥} is invariant under ρ^{V}. This can be obtained from the remarks in the previous section, or from the fact that N_{ρ^{V}} is obviously invariant under ρ^{V}, as in Section 21.7. It follows that

V (ρ^{V} , v_{0}) ⊆ N_{ρ^{V}}^{⊥} , (21.12.6)

because of (21.12.4). Let V_{1}(ρ^{V} , v_{0}) be the linear span of v_{0} and V (ρ^{V} , v_{0}) in V. Clearly V_{1}(ρ^{V} , v_{0}) ⊆ N_{ρ^{V}}^{⊥} , (21.12.7)

by (21.12.4) and (21.12.6).

Suppose that ζ ∈ V_{1}(ρ^{V} , v_{0}) ∩ V (ρ^{V} , v_{0})^{⊥}. (21.12.8)

We would like to show that ζ = 0. (21.12.9)

Let y, z ∈ A be given, and observe that

⟨ζ, ρ_{y^{∗_{A}} z}^{V}(v_{0})⟩_{V} = 0, (21.12.10)

because ρ_{y^{∗_{A}} z}^{V}(v_{0}) ∈ V (ρ^{V} , v_{0}) and ζ ∈ V (ρ^{V} , v_{0})^{⊥}. This implies that

⟨ρ_{y}^{V} (ζ), ρ_{z}^{V} (v_{0})⟩_{V} = ⟨ζ, ρ_{y^{∗_{A}}}^{V} (ρ_{z}^{V} (v_{0}))⟩_{V} = 0, (21.12.11)

using the hypothesis that ρ^{V} be a ∗-representation of A on V in the first step. This means that ρ_{y}^{V} (ζ) ∈ V (ρ^{V} , v_{0})^{⊥}, (21.12.12)

by the definition (21.12.3) of V (ρ^{V} , v_{0}).

It is easy to see that

ρ_{y}^{V} (V_{1}(ρ^{V} , v_{0})) ⊆ V (ρ^{V} , v_{0}), (21.12.13)

by construction. In particular,

ρ_{y}^{V} (ζ) ∈ V (ρ^{V} , v_{0}). (21.12.14)

It follows that ρ_{y}(ζ) = 0, (21.12.15)

because of (21.12.12). This means that

ζ ∈ N_{ρ^{V}} . (21.12.16)

This implies (21.12.9), because ζ ∈ N_{ρ^{V}}^{⊥}, by (21.12.7) and (21.12.8).

Let us take ζ to be v_{0} minus its image under the orthogonal projection of V onto V (ρ^{V} , v_{0}). This satisfies (21.12.8), by construction. Thus (21.12.5) follows from (21.12.9).

Using (21.12.5), we get that the subrepresentation of ρ^{V} corresponding to V (ρ^{V} , v_{0}) is topologically cyclic, as in Section 20.13. This is also related to Exercise (3) on p59 of [11].

21.13 Direct sums and products

Let k be a field, let I be a nonempty set, and let V_{j} be a vector space over k for each j ∈ I. Remember that the Cartesian product

∏_{j∈I} V_{j} (21.13.1)

may be defined as the set of all functions on I whose value at each j ∈ I is an element of V_{j}. If I = {1, …, n} for some positive integer n, then the elements of (21.13.1) may be considered as n-tuples (v_{1}, …, v_{n}) with v_{j} ∈ V_{j} for each j. This was discussed in Section 5.12 when n = 2. Similarly, if I is the set Z_{+} of all positive integers, then the elements of (21.13.1) may be considered as infinite sequences {v_{j}}_{j=1}^{∞}, with v_{j} ∈ V_{j} for each j.

It is easy to see that (21.13.1) is a vector space over k, with respect to coordinatewise addition and scalar multiplication. This is called the direct product of the V_{j}’s, j ∈ I.

Let

⨁_{j∈I} V_{j} (21.13.2)

be the set of elements of (21.13.1) that are equal to 0 at all but finitely many j ∈ I. This is a linear subspace of (21.13.1), which is called the direct sum of the V_{j}’s, j ∈ I. Of course, if I has only finitely many elements, then the direct sum and direct product of the V_{j}’s are the same.

Let A be an associative algebra over k. If V_{j} is a left module over A for each j ∈ I, then (21.13.1) may be considered as a left module over A, where each element of A acts on elements of (21.13.1) coordinatewise. This is the direct product of the V_{j}’s, j ∈ I, as a left module over A. Note that (21.13.2) is a submodule of this module, which is the direct sum of the V_{j}’s. This is essentially the same as the definition in Section 20.8 when I has exactly two elements.

Equivalently, if ρ^{V_{j}} is a representation of A on V_{j} for each j, then we get a representation of A on (21.13.1), which is the direct product of the ρ^{V_{j}}’s. As before, (21.13.2) is invariant under the direct product representation, and the corresponding subrepresentation is called the direct sum of the ρ^{V_{j}}’s. Of course, there are analogous notions and statements when V_{j} is a right module over A for each j.

Let W be a vector space over k, and suppose for the moment that V_{j} = W for each j. In this case, the direct product (21.13.1) is the same as the space of all W -valued functions on I. Similarly, the direct sum (21.13.2) is the same as the space c_{00}(I, W) (21.13.3)

of all W -valued functions f on I that have finite support, in the sense that f(j) = 0 for all but finitely many j.

Suppose now that V_{j} = A for each j. Thus the direct sum and the direct product may be considered as both left and right modules over A. The direct sum is said to be free as a left or right module over A.

In particular, if n is a positive integer, then the space A^{n} of n-tuples of elements of A may be considered as a left and right module over A. This corresponds to taking I = {1, …, n}, as before.

21.14 Some norms on direct sums

Let I be a nonempty set again, and let us now take k = R or C. Also let V_{j} be a vector space over k for each j, and let

V = ∏_{j∈I} V_{j} (21.14.1)

be the direct product of the V_{j}’s, as in the previous section. If v ∈ V, then let v_{j} be the jth coordinate of v in V_{j} for each j ∈ I. The direct sum (21.13.2) of the V_{j}’s may be called an algebraic direct sum, to avoid ambiguity with some related constructions using norms. This consists of the v ∈ V such that v_{j} = 0 for all but finitely many j, as before.

Let ∥ · ∥_{V_{j}} be a norm on V_{j} for each j ∈ I. If v ∈ V, then put

∥v∥_{V,∞} = sup_{j∈I} ∥v_{j}∥_{V_{j}}, (21.14.2)

where the supremum on the right side is defined as a nonnegative extended real number when I has infinitely many elements. One can check that

{v ∈ V : ∥v∥_{V,∞} < +∞} (21.14.3)

is a linear subspace of V. Note that this subspace contains the algebraic direct sum (21.13.2). One may describe this linear subspace as the ℓ^{∞} direct sum of the V_{j}’s, with respect to this family of norms.

One can also verify that ∥v∥_{V,∞} defines a norm on (21.14.3). If I has exactly two elements, then (21.14.2) corresponds to a norm mentioned in Section 5.12.

Remember that the notion of vanishing at infinity for a real-valued function on I may be defined as in Section 1.13. One can check that

(21.14.4) {v ∈ V : ∥v_{j}∥_{V_{j}} vanishes at infinity on I}

is a linear subspace of (21.14.3) that contains the algebraic direct sum (21.13.2). More precisely, the algebraic direct sum is dense in (21.14.4), with respect to the metric associated to ∥v∥_{V,∞}. One can also verify that (21.14.4) is a closed set in (21.14.3), with respect to the metric associated to ∥v∥_{V,∞}. One may call (21.14.4) the c_{0} direct sum of the V_{j}’s, with respect to this family of norms.

If 1 ≤ p < ∞, then put

∥v∥_{V,p} = ( ∑_{j∈I} ∥v_{j}∥_{V_{j}}^{p})^{1/p} (21.14.5)

for each v ∈ V. More precisely, the sum on the right may be defined as a nonnegative extended real number as in Section 2.4 when I has infinitely many elements, and ∥v∥_{V,p} is interpreted as being +∞ when this sum is +∞. One can check that {v ∈ V : ∥v∥_{V,p} < +∞} (21.14.6)

is a linear subspace of V. Of course, this linear subspace contains the algebraic direct sum (21.13.2). One may call (21.14.6) the ℓ^{p} direct sum of the V_{j}’s, with respect to this family of norms.

One can verify that ∥v∥_{V,p} defines a norm on (21.14.6). This corresponds to a norm mentioned in Section 5.12 when I has exactly two elements. One can also check that the algebraic direct sum is dense in (21.14.6), with respect to the metric associated to ∥v∥_{V,p}. Observe that (21.14.6) is contained in (21.14.4), as in Section 2.5.

If 1 ≤ p_{1} ≤ p_{2} ≤ ∞, then

∥v∥_{V,p_{2}} ≤ ∥v∥_{V,p_{1}} (21.14.7)

for all v ∈ V, as in Section 2.5. This implies that the ℓ^{p_{1}} direct sum of the V_{j}’s is contained in the ℓ^{p_{2}} direct sum.

21.15 More on direct sums

Let us continue with the same notation and hypotheses as in the previous section. Suppose for the moment that V_{j} is complete with respect to the metric associated to ∥ · ∥_{V_{j}} for each j ∈ I. If 1 ≤ p ≤ ∞, then one can verify that the ℓ^{p} direct sum of the V_{j}’s is complete with respect to the metric associated to ∥ · ∥_{V,p}. This is analogous to the completeness of ℓ^{p} spaces, as in Section 2.6. It follows that the c_{0} direct sum of the V_{j}’s is complete with respect to the metric assocuated to ∥ · ∥_{V,∞}, because the c_{0} direct sum is a closed set in the ℓ^{∞} direct sum, as before.

Let W be a vector space over k, and suppose for the moment again that V_{j} = W for all j. Also let ∥ · ∥_{W} be a norm on W, and suppose that

∥ · ∥_{V_{j}} = ∥ · ∥_{W} (21.15.1)

for each j. If 1 ≤ p ≤ ∞, then the ℓ^{p} direct sum of the V_{j}’s corresponds to ℓ^{p}(I, W). Similarly, the c_{0} direct sum of the V_{j}’s corresponds to c_{0}(I, W).

If T_{j} is a linear mapping from V_{j} into iself for each j, then we get a linear mapping T from V into itself such that

(T (v))_{j} = T_{j}(v_{j}) (21.15.2)

for each v ∈ V and j ∈ I. Note that T maps the algebraic direct sum (21.13.2) into itself. Suppose that T_{j} is a bounded linear mapping from V_{j} into itself for each j, and that the corresponding operator norm

∥T_{j}∥_{op,V_{j}V_{j}} (21.15.3) is uniformly bounded. Under these conditions, T maps the ℓ^{p} direct sum of the V_{j}’s into itself for each p, 1 ≤ p ≤ ∞, and T maps the c_{0} direct sum of the V_{j}’s into itself. More precisely, the restriction of T to the ℓ^{p} or c_{0} direct sum of the V_{j}’s is a bounded linear mapping, with operator norm equal to

sup_{j∈I} ∥T_{j}∥_{op,V_{j}V_{j}}, (21.15.4)

with respect to ∥ · ∥_{V,p} or ∥ · ∥_{V,∞}, as appropriate.

Let A be an associative algebra over k, and suppose that V_{j} is a left module over A. Thus V may be considered as a left module over A too, as in Section 21.13. Suppose that the action of each x ∈ A is bounded on V_{j} for each j, with operator norm that is uniformly bounded in j. This implies that the action of x on V maps the ℓ^{p} direct sum of the V_{j}’s into itself for each p, 1 ≤ p ≤ ∞, and that the action of x on the ℓ^{p} direct sum is bounded with respect to ∥ · ∥_{V,p}, as in the preceding paragraph. Similarly, the action of x on V maps the c_{0} direct sum of the V_{j}’s into itself. This means that the ℓ^{p} and c_{0} direct sums of the V_{j}’s are submodules of V, as a left module over A. Of course, there are analogous statements when V_{j} is a right module over A for each j.

Algebras, norms, and representations

22.1 Direct sums of Hilbert spaces

Let I be a nonempty set, and let us take k = R or C, as in Section 21.14. Suppose now that (V_{j}, ⟨·, ·⟩_{V_{j}}) is an inner product space over k for each j ∈ I, with corresponding norm ∥ · ∥_{V_{j}}. Let

V = ∏_{j∈I} V_{j} (22.1.1)

be the direct product of the V_{j}’s, as before, and put

∥v∥_{V,2} = ( ∑_{j∈I} ∥v_{j}∥_{V_{j}}^{2})^{1/2} (22.1.2)

for each v ∈ V. Thus {v ∈ V : ∥v∥_{V,2} < +∞} (22.1.3)

is the ℓ^{2} direct sum of the V_{j}’s, as in Section 21.14. We would like to define a corresponding inner product on (22.1.3).

If u, v ∈ V, then |⟨u_{j}, v_{j}⟩_{V_{j}}| ≤ ∥u_{j}∥_{V_{j}} ∥v_{j}∥_{V_{j}} (22.1.4)

for each j, by the Cauchy–Schwarz inequality. If u and v are elements of (22.1.3), then ∥u_{j}∥_{V_{j}} ∥v_{j}∥_{V_{j}} (22.1.5)

is summable as a nonnegative real-valued function of j, by the Cauchy–Schwarz inequality for sums. This implies that

⟨u_{j}, v_{j}⟩_{V_{j}} (22.1.6) is summable as a real or complex-valued function of j on I, as appropriate. Thus

⟨u, v⟩_{V,2} = ∑_{j∈I}⟨u_{j}, v_{j}⟩_{V_{j}} (22.1.7)

is defined as a real or complex number, as appropriate, where the sum on the right converges in the generalized sense, as in Section 2.8. One can check that this defines an inner product on (22.1.3), for which the corresponding norm is equal to (22.1.2).

Suppose that V_{j} is a Hilbert space for each j, so that (22.1.3) is complete with respect to the metric associated to (22.1.2), as in Section 21.15. This means that (22.1.3) is a Hilbert space with respect to (22.1.7). We may also describe (22.1.3) as the orthogonal direct sum of the V_{j}’s with respect to the inner product (22.1.7) under these conditions.

Let T_{j} be a bounded linear mapping from V_{j} into itself for each j, and suppose that the corresponding operator norm ∥T_{j}∥_{op,V_{j}V_{j}} is bounded. This leads to a bounded linear mapping T_{V,2} from (22.1.3) into itself with respect to (22.1.2), as in Section 21.15. More precisely, this is the restriction to (22.1.3) of the linear mapping from the direct product of the V_{j}’s into itself that was mentioned in Section 21.15. Thus

(T_{V,2}(v))_{j} = T_{j}(v_{j}) (22.1.8)

for each v in (22.1.3) and j ∈ I, as in (21.15.2). If u is also an element of (22.1.3), then

⟨T_{V,2}(u), v⟩_{V,2} = ∑_{j∈I}⟨T_{j}(u_{j}), v_{j}⟩_{V_{j}} = ∑_{j∈I}⟨u_{j}, T_{j}^{∗_{V_{j}}} (v_{j})⟩_{V_{j}}. (22.1.9)

Let T_{V,2}^{∗} = T_{V,2}^{∗_{V,2}} be the adjoint of T_{V,2} on (22.1.3) with respect to (22.1.7). This is the same as the linear mapping from (22.1.3) into itself that corresponds to T_{j}^{∗_{V_{j}}} on V_{j} for each j.

Let A be an associative algebra over k, and let x ↦ x^{∗_{A}} be an opposite algebra isomorphism from A onto itself that is conjugate-linear in the complex case. Suppose that ρ^{V_{j}} is a ∗-representation of A on V_{j} for each j, as in Section 21.5. If x ∈ A, then ρ_{x}^{V_{j}} is supposed to be a bounded linear mapping on V_{j} for each j, and we also ask that the operator norm of ρ_{x}^{V_{j}} be uniformly bounded in j. Let ρ^{V} be the corresponding representation of A on the direct product of the V_{j}’s, as in Section 21.13, and remember that (22.1.3) is invariant under ρ^{V} under these conditions, as in Section 21.15. It is easy to see that the corresponding subrepresentation ρ^{V,2} of ρ^{V} on (22.1.3) is a ∗-representation of A on (22.1.3), using the remarks in the preceding paragraph.

22.2 Direct sums of linear subspaces

Let k be a field, and let W be a vector space over k. Also let I be a nonempty set, and suppose that for each j ∈ I, V_{j} is a linear subspace of W. Consider the linear subspace W_{0} of W spanned by the V_{j}’s, j ∈ I.

Remember that ⨁_{j∈I} V_{j} is the direct sum of the V_{j}’s, as in Section 21.13. There is an obvious linear mapping from ⨁_{j∈I} V_{j} into W, which sends an element v of ⨁_{j∈I} V_{j} to

∑_{j∈I} v_{j}. (22.2.1)

This sum defines an element of W, because v_{j} = 0 for all but finitely many j. Note that this mapping sends ⨁_{j∈I} V_{j} onto W_{0}.

Of course, this mapping is injective if and only if for every v ∈ ⨁_{j∈I} V_{j} with

∑_{j∈I} v_{j} = 0, (22.2.2)

we have that v = 0, which means that v_{j} = 0 for every j. In this case, if j_{1}, …, j_{n} are finitely many distinct elements of I, v_{j_{l}} ∈ V_{j_{l}} for each l = 1, …, n, and v_{j_{l}} ≠ 0 for each l = 1, …, n, then it is easy to see that

(22.2.3) v_{j_{1}}, …, v_{j_{n}} are linearly independent in W.

This condition also clearly implies the previous one. We may say that W_{0} corresponds to the direct sum of the V_{j}’s, j ∈ I, when this condition holds.

Let A be an associative algebra over k, and suppose that W is a left module over A. If V_{j} is a submodule of W for each j, then W_{0} is a submodule of W as well. Remember that ⨁_{j∈I} V_{j} may be considered as a left module over A too, as in Section 21.13. The usual mapping from ⨁_{j∈I} V_{j} onto W_{0} is a module homomorphism in this case. There are analogous statements when W is a right module over A, as usual.

Suppose now that k = R or C, and let ∥ · ∥_{W} be a norm on W. Suppose that W is a left module over A again, and that the action of each x ∈ A on W is bounded with respect to ∥ · ∥_{W}. If V_{j} is a submodule of W for each j ∈ I, then the closure \overline{V_{j}} of V_{j} in W with respect to the metric associated to ∥ · ∥_{W} is a submodule of W as well, as in Section 20.4. Similarly, the closure \overline{W_{0}} of W_{0} in W is a submodule of W under these conditions.

22.3 Orthogonal direct sums

Let us continue with the same notation and hypotheses as at the beginning of the previous section, with k = R or C. Suppose that ⟨·, ·⟩_{W} is an inner product on W, that the V_{j}’s are pairwise orthogonal in W. This means that

⟨v_{j_{1}}, v_{j_{2}}⟩_{W} = 0 (22.3.1)

when j_{1}, j_{2} ∈ I, j_{1} ≠ j_{2}, v_{j_{1}} ∈ V_{j_{1}}, and v_{j_{2}} ∈ V_{j_{2}}. This implies the condition mentioned in the previous section, so that the obvious mapping from ⨁_{j∈I} V_{j} onto W_{0} is injective.

Let ⟨·, ·⟩_{V_{j}} be the restriction of ⟨·, ·⟩_{W} to V_{j} for each j ∈ I. This leads to an inner product on ⨁_{j∈I} V_{j}, as in (22.1.7). This inner product corresponds exactly to the restriction of ⟨·, ·⟩_{W} to W_{0} under the usual mapping from ⨁_{j∈I} V_{j} onto W_{0}, because of the orthogonality condition (22.3.1).

Let us suppose from now on in this section that W is a Hilbert space with respect to ⟨·, ·⟩_{W}. Remember that ⨁_{j∈I} V_{j} is dense in the ℓ^{2} direct sum (22.1.3) of the V_{j}’s with respect to the metric associated to the norm ∥ · ∥_{V,2} defined in (22.1.2), as mentioned in Section 21.14. This implies that the usual mapping from ⨁_{j∈I} V_{j} onto W_{0} has a unique extension to a bounded linear mapping from the ℓ^{2} direct sum of the V_{j}’s into W, as in Section 2.2. More precisely, this mapping preserves the corresponding inner products, and thus their associated norms. This mapping sends the ℓ^{2} direct sum of the V_{j}’s into the closure \overline{W_{0}} of W_{0} in W, with respect to the norm ∥ · ∥_{W} determined by ⟨·, ·⟩_{W}.

Suppose that V_{j} is also a closed set in W for each j, with respect to the metric associated to ∥ · ∥_{W}. This implies that V_{j} is complete with respect to the metric associated to the norm ∥ · ∥_{V_{j}} determined by ⟨·, ·⟩_{V_{j}}, because W is complete, by hypothesis. Of course, ∥·∥_{V_{j}} is the same as the restriction of ∥·∥_{W} to V_{j} for each j. It follows that the ℓ^{2} direct sum of the V_{j}’s is complete with respect to the metric associated to ∥ · ∥_{V,2}, as mentioned in Section 21.15. One can use this to get that the image of the ℓ^{2} direct sum of the V_{j}’s in W under the mapping described in the preceding paragraph is a closed set, as in Section 1.6. This means that the image of the ℓ^{2} direct sum of the V_{j}’s in W under this mapping is equal to \overline{W_{0}}. We may say that \overline{W}_{0} corresponds to the orthogonal direct sum of the V_{j}’s in this case.

Let A be an associative algebra over k, and suppose that W is a left module over A, where the action of each x ∈ A on W is bounded with respect to ∥ · ∥_{W}. Suppose that V_{j} is a submodule of W for each j, so that W_{0} is a submodule of W, and thus \overline{W_{0}} is a submodule of W. If x ∈ A, then the operator norm of the action of x on V_{j} is less than or equal to the operator norm of the action of x on W, and is uniformly bounded in j in particular. This means that the ℓ^{2} direct sum of the V_{j}’s may be considered as a left module over A, as in Section 21.15, and the mapping from the ℓ^{2} direct sum of the V_{j}’s onto \overline{W_{0}} discussed in the previous two paragraphs is in fact a module isomorphism. There are analogous statements when W is a right module over A, as before.

22.4 Sums of topologically cyclic subspaces

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real or complex numbers, and let A be an associative algebra over the real or complex numbers, as appropriate, with an opposite algebra isomorphism x ↦ x^{∗_{A}} from A onto itself that is conjugatelinear in the complex case. Suppose that ρ^{V} is a ∗-representation of A on V, as in Section 21.5, so that ρ_{x}^{V} is a bounded linear mapping on V, with respect to the norm ∥ · ∥_{V} associated to the inner product, for each x ∈ A in particular. Put

N_{ρ^{V}} = ⋂_{x∈A} ker ρ_{x}^{V} , (22.4.1)

as in Section 21.12, which is a closed linear subspace of V, with respect to the metric associated to ∥ · ∥_{V}. Remember that the orthogonal complement N_{ρ^{V}}^{⊥} of N_{ρ^{V}} in V is invariant under ρ^{V}, as mentioned before. Let us suppose that

N_{ρ^{V}} ≠ V, (22.4.2)

so that N_{ρ^{V}}^{⊥} ≠ {0}, (22.4.3)

as in Section 2.15.

If v_{0} ∈ V, then V (ρ^{V} , v_{0}) = \overline{\{ρ_{x}^{V} (v_{0}) : x ∈ A\}} (22.4.4)

is a closed linear subspace of V that is invariant under ρ^{V}, as in Section 21.12. If v_{0} ∈ N_{ρ^{V}}^{⊥}, then V (ρ^{V} , v_{0}) ⊆ N_{ρ^{V}}^{⊥} (22.4.5)

and v_{0} ∈ V (ρ^{V} , v_{0}), (22.4.6)

as before. The latter implies that the subrepresentation of ρ^{V} corresponding to V (ρ^{V} , v_{0}) is topologically cyclic, as in Section 20.13. Of course, (22.4.3) implies that we can take v_{0} ≠ 0, so that

V (ρ^{V} , v_{0}) ≠ {0}. (22.4.7)

Note that if Z is any closed linear subspave of V that is invariant under ρ^{V} and v_{0} ∈ Z, then V (ρ^{V} , v_{0}) ⊆ Z. (22.4.8)

Let us say that a family of pairwise-orthogonal nonzero closed linear subspaces of N_{ρ^{V}}^{⊥} is “nice” if each of the subspaces is invariant under ρ^{V}, and the corresponding subrepresentations of ρ^{V} are topologically cyclic. Theorem 21.13 on p315 of [151] basically says that N_{ρ^{V}}^{⊥} can be expressed as the orthogonal direct sum of such a nice family. This is closely related to Exercise (3) on p59 of [11]. If one has a nice family of subspaces of N_{ρ^{V}}^{⊥}, then its orthogonal direct sum is a closed linear subspace of N_{ρ^{V}}^{⊥} that is invariant under ρ^{V}. The orthogonal complement of this closed linear subspace of N_{ρ^{V}}^{⊥} in N_{ρ^{V}}^{⊥} is also invariant under ρ^{V}, as in Section 21.7.

If one has a nice family of subspaces of N_{ρ^{V}}^{⊥} whose orthogonal direct sum is not equal to N_{ρ^{V}}^{⊥}, then one can add another nonzero closed linear subspace of N_{ρ^{V}}^{⊥} to the family to get a larger nice family, as in (22.4.8). If N_{ρ^{V}}^{⊥} has finite dimension, then one can do this finitely many times to get a nice family of finitely many subspaces of N_{ρ^{V}}^{⊥} whose orthogonal direct sum is N_{ρ_{V}}^{⊥}. If N_{ρ^{V}}^{⊥} is separable with respect to the metric associated to ∥ · ∥_{V}, then one can get a nice sequence of subspaces of N_{ρ^{V}}^{⊥} whose orthogonal direct sum is N_{ρ^{V}}^{⊥}. More precisely, if v_{1}, v_{2}, v3, … is any sequence of elements of N_{ρ^{V}}^{⊥}, then one can get a nice sequences of subspaces of N_{ρ^{V}}^{⊥} such that for each positive integer l, the orthogonal direct sum of the first l subspaces of N_{ρ^{V}}^{⊥} contains v_{1}, …, v_{l}. If the linear span of v_{1}, v_{2}, v_{3}, … is dense in N_{ρ^{V}}^{⊥}, then the orthogonal direct sum of this sequences of linear subspaces is equal to N_{ρ^{V}}^{⊥}.

Otherwise, the existence of a maximal nice family of linear subspaces of N_{ρ^{V}}^{⊥} can be obtained using the axiom of choice, by Zorn’s lemma or Hausdorff’s maximality principle. The orthogonal direct sum of the linear subspaces of N_{ρ^{V}}^{⊥} in a maximal nice family has to be equal to N_{ρ^{V}}^{⊥}, as before.

22.5 Boundedness of some homomorphisms

Let A, B be associative algebras over the complex numbers with multiplicative identity elements e_{A}, e_{B}, respectively, and let ϕ be a homomorphism from A into B such that ϕ(e_{A}) = e_{B}. If x ∈ A, then

σ_{B}(ϕ(x)) ⊆ σ_{A}(x), (22.5.1)

as in Section 19.3.

Suppose that A, B are Banach algebras with respect to norms ∥ · ∥_{A}, ∥ · ∥_{B}, respectively, with ∥e_{A}∥_{A} = ∥e_{B}∥_{B} = 1, and let r_{A}(·), r_{B}(·) be defined on A, B, respectively, as in Section 6.13. Observe that

r_{B}(ϕ(x)) ≤ r_{A}(x) (22.5.2)

for every x ∈ A, because of (22.5.1) and the fact that r_{A}(x), r_{B}(ϕ(x)) are equal to maxima of the absolute values of the elements of σ_{A}(x), σ_{B}(ϕ(x)), respectively, as mentioned in Section 6.14.

Let a ↦ a^{∗_{A}} and b ↦ b^{∗_{B}} be conjugate-linear involutions on A and B, respectively, and suppose that ϕ is a ∗-homomorphism from A into B, as in Section 12.7. If x ∈ A, then

ϕ(x)^{∗_{B}} ϕ(x) = ϕ(x^{∗_{A}} x), (22.5.3)

so that r_{B}(ϕ(x)^{∗_{B}} ϕ(x)) ≤ r_{A}(x^{∗_{A}} x), (22.5.4)

as in (22.5.2).

Suppose that B is a C^{∗} algebras, as in Section 12.6. If b is a normal element of B, then r_{B}(b) = ∥b∥_{B}, (22.5.5)

as in Section 14.12. In particular, this holds when b is self-adjoint, so that

r_{B}(ϕ(x)^{∗_{B}} ϕ(x)) = ∥ϕ(x)^{∗_{B}} ϕ(x)∥_{B} = ∥ϕ(x)∥_{B}^{2}. (22.5.6)

This implies that

∥ϕ(x)∥_{B}^{2} ≤ r_{A}(x^{∗_{A}} x) ≤ ∥x^{∗_{A}} x∥_{A} ≤ ∥x^{∗_{A}}∥_{A} ∥x∥_{A}, (22.5.7) using (22.5.4) in the first step, and the definition of r_{A}(·) in the second step. If A is a Banach ∗-algebra, so that the involution preserves the norm, then we get that ∥ϕ(x)∥_{B} ≤ ∥x∥_{A}. (22.5.8)

Suppose that instead of the involution on A preserving the norm, we ask that ∥x^{∗_{A}}∥_{A} ≤ C ∥x∥_{A} (22.5.9)

for some C ≥ 1 and all x ∈ A. In this case, we get that

∥ϕ(x)∥_{B} ≤ C^{1/2} ∥x∥_{A} (22.5.10)

for all x ∈ A, in place of (22.5.8).

22.6 Boundedness and identity elements

Let us continue with the same notation and hypotheses as in the previous section, with some modifications, as follows. We would first like to check that the same conclusions hold when A does not necessarily have a multiplicative identity element. One can use multiplication on A to define multiplication on A_{1} = A × C as in Section 6.15, so that A_{1} is an associative algebra over C with multiplicative identity element e_{A_{1}} = (0, 1). One can also use ∥ · ∥_{A} to define a norm ∥ · ∥_{A_{1}} on A_{1} as before, so that A_{1} is a Banach algebra.

Similarly, one can use the involution on A to get a conjugate-linear involution on A_{1}, as in Section 17.12. This involution preserves the norm on A_{1} when the involution on A is norm-preserving, as before. If the involution on A satisfies (22.5.9) for some C ≥ 1, then the involution on A_{1} satisfies the analogous condition, as in Section 17.12 again.

If ϕ is an algebra homomorphism from A into B, then we can get an algebra homomorphism ϕ_{1} from A_{1} into B that sends e_{A_{1}} to e_{B}, as in Section 20.10. If ϕ is a ∗-homomorphism, then it is easy to see that

(22.6.1) ϕ_{1} is a ∗-homomorphism,

by construction. The previous argument implies that

(22.6.2) ϕ_{1} satisfies the analogue of (22.5.8) on A_{1}

when the involution on A is norm-preserving. In particular, this implies that (22.5.8) holds on A. If the involution on A satisfies (22.5.9), then we get that

(22.6.3) ϕ_{1} satisfies the analogue of (22.5.10) on A_{1},

so that (22.5.10) holds on A, as before.

If A does have a multiplicative identity element e_{A}, then one can get the same conclusions without asking that ϕ(e_{A}) be equal to e_{B}. This could be obtained from the argument in the previous paragraphs, or by replacing B with

(22.6.4) the closure of ϕ(A), for which ϕ(e_{A}) is the multiplicative identity element.

If B does not have a multiplicative identity element, then one can adjoin a multiplicative identity element to it a get a C^{∗} algebra as in Section 18.5. If A is a C^{∗} algebra too, then (22.5.8) corresponds to part of Theorem 2.4 on p266 of [291].

22.7 Boundedness of ∗-representations

Let A be as in the previous two sections, and let (V, ⟨·, ·⟩_{V}) be a complex Hilbert space. Also let ρ^{V} be a ∗-representation of A on V, as in Section 21.5. Thus ρ^{V} corresponds to a ∗-homomorphism from A into the C^{∗} algebra BL(V) of bounded linear mappings from V into itself.

If the involution on A preserves the norm, then

∥ρ_{x}^{V} ∥_{op,V V} ≤ ∥x∥_{A} (22.7.1)

for every x ∈ A, as in (22.5.8). This is Theorem 2.5.5 on p58 of [11], and Theorem 21.22 on p320 of [151]. The argument in [151] will be discussed further in the next two sections.

Similarly, if the involution on A satisfies (22.5.9) for some C ≥ 1, then

∥ρ_{x}^{V} ∥_{op,V V} ≤ C^{1/2} ∥x∥_{A} (22.7.2)

for all x ∈ A, as in (22.5.10).

Note that one can reduce to the case where ρ^{V} is nondegenerate on V, using the remarks in Section 21.11, as in [11]. Remember that if ρ^{V} is nondegenerate on V, and A has a multiplicative identity element e_{A}, then ρ_{e_{A}}^{V} is the identity mapping on V, as in Section 20.9.

22.8 Some positive linear functionals

Let A be an associative algebra over the real or complex numbers with an involution x ↦ x^{∗_{A}} that is conjugate-linear in the complex case, and let (V, ⟨·, ·⟩_{V}) be an inner product space over the real or complex numbers, as appropriate. Also let ρ^{V} be a representation of A on V that is compatible with the involution on A and the inner product on V, in the sense that

⟨ρ_{x}^{V} (u), v⟩_{V} = ⟨u, ρ_{x^{∗_{A}}}^{V} (v)⟩_{V} (22.8.1)

for every u, v ∈ V, as in Section 21.5. If v_{0} ∈ V, then put

λ_{ρ^{V} ,v_{0}}(x) = ⟨ρ_{x}^{V} (v_{0}), v_{0}⟩_{V} (22.8.2)

for each x ∈ A, as in the proof of Theorem 21.22 on p320 of [151]. Of course, this defines a linear functional on A.

Observe that λ_{ρ^{V} ,v_{0}}(x^{∗_{A}}) = λ_{ρ^{V} ,v_{0}}(x) (22.8.3)

22.8. SOME POSITIVE LINEAR FUNCTIONALS 499 for every x ∈ A in the real case, and that λ_{ρ^{V} ,v_{0}}(x^{∗_{A}}) = \overline{λ_{ρ^{V} ,v_{0}}(x)} (22.8.4) for every x ∈ A in the complex case. It follows that λ_{ρ^{V} ,v_{0}}(x) ∈ R (22.8.5) when x ∈ A is self-adjoint in the complex case. In both cases, we have that λ_{ρ^{V} ,v_{0}}(x^{∗_{A}} x) = ⟨ρ_{x^{∗_{A}}}^{V} (ρ_{x}^{V} (v_{0})), v_{0}⟩_{V} = ⟨ρ_{x}^{V} (v_{0}), ρ_{x}^{V} (v_{0})⟩_{V} ≥ 0 (22.8.6) for every x ∈ A. This implies that λ_{ρ^{V} ,v_{0}} is positive in the strict sense with respect to the involution on A, as in Section 18.6.

If x ∈ A and ∥ · ∥_{V} is the norm on V associated to the inner product, then |λ_{ρ^{V} ,v_{0}}(x)| ≤ ∥ρ_{x}^{V} (v_{0})∥_{V} ∥v_{0}∥_{V} = λ_{ρ^{V} ,v_{0}}(x^{∗_{A}} x)^{1/2} ∥v_{0}∥_{V} , (22.8.7) using the Cauchy–Schwarz inequality in the first step. This implies that λ_{ρ^{V} ,v_{0}} is positive in the strong sense on A, as in Section 18.7. If A has a multiplicative identity element e_{A}, then e_{A}^{∗_{A}} = e_{A}, so that e_{A}^{∗_{A}} e_{A} = e_{A}, and λ_{ρ^{V} ,v_{0}}(e_{A}) = λ_{ρ^{V} ,v_{0}}(e_{A}^{∗_{A}} e_{A}) ≥ 0, as before. In this case, (22.8.7) implies that λ_{ρ^{V} ,v_{0}}(e_{A}) ≤ ∥v_{0}∥_{V}^{2} , (22.8.8) as in Section 18.7.

Suppose that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, and that the involution on A preserves the norm. If A has a multiplicative identity element e_{A}, then |λ_{ρ^{V} ,v_{0}}(x)| ≤ λ_{ρ^{V} ,v_{0}}(e_{A}) ∥x∥_{A} (22.8.9) for every x ∈ A, as in Section 18.11. This implies that |λ_{ρ^{V} ,v_{0}}(x)| ≤ ∥x∥_{A} ∥v_{0}∥_{V}^{2} (22.8.10) for every x ∈ A, because of (22.8.8). It follows that ∥ρ_{x}^{V} (v_{0})∥_{V}^{2} = λ_{ρ^{V} ,v_{0}}(x^{∗_{A}} x) ≤ ∥x^{∗_{A}} x∥_{A} ∥v_{0}∥_{V}^{2} ≤ ∥x∥_{A}^{2} ∥v_{0}∥_{V}^{2} (22.8.11) for every x ∈ A, using (22.8.6) in the first step. Of course, this is the same as saying that ∥ρ_{x}^{V} (v_{0})∥_{V} ≤ ∥x∥_{A} ∥v_{0}∥_{V} (22.8.12) for every x ∈ A.

Suppose that the involution on A does not necessarily preserve the norm, but is continuous, so that (22.5.9) holds for some C ≥ 1 and all x ∈ A. If A has a multiplicative identity element e_{A}, and w ∈ A is self-adjoint, then |λ_{ρ^{V} ,v_{0}}(w)| ≤ λ_{ρ^{V} ,v_{0}}(e_{A}) ∥w∥_{A}, (22.8.13) as in Section 18.11. This implies that

|λ_{ρ^{V} ,v_{0}}(w)| ≤ ∥w∥_{A} ∥v_{0}∥_{V}^{2} , (22.8.14)

because of (22.8.8). If x is any element of A, then we can take w = x^{∗_{A}} x, to get that

∥ρ_{x}^{V} (v_{0})∥_{V}^{2} = λ_{ρ^{V} ,v_{0}}(x^{∗_{A}} x) ≤ ∥x^{∗_{A}} x∥_{A} ∥v_{0}∥_{V}^{2} ≤ C ∥x∥_{A}^{2} ∥v_{0}∥_{V}^{2} , (22.8.15)

using (22.8.6) in the first step, as before. This means that

∥ρ_{x}^{V} (v_{0})∥_{V} ≤ C^{1/2} ∥x∥_{A} ∥v_{0}∥_{V} . (22.8.16)

22.9 Positivity and identity elements

Let us continue with the same notation and hypotheses as in the previous section. If A does not necessarily have a multiplicative identity element, then we can consider the algebra A_{1} defined in Section 6.15, which is a Banach algebra with respect to the corresponding norm ∥ · ∥_{A_{1}}, as before. We can use the involution on A to get an involution on A_{1}, which is conjugate-linear in the complex case, as in Section 17.12. Suppose first that the involution on A preserves the norm, so that the corresponding involution on A_{1} preserves the norm there, as before.

We can also use λ_{ρ^{V} ,v_{0}} to get a positive linear functional λ_{1,ρ^{V} ,v_{0}} on A_{1}, as in Section 18.8. More precisely, the value of λ_{1,ρ^{V} ,v_{0}} at the multiplicative identity element of A_{1} can be taken to be ∥v_{0}∥_{V}^{2} , because of (22.8.7), as before. This leads to an analogue of (22.8.9) for λ_{1,ρ^{V} ,v_{0}} on A_{1}, as in Section 18.11, which means that λ_{1,ρ^{V} ,v_{0}} satisfies the analogue of (22.8.10) on A_{1}. It follows that (22.8.10) holds on A, so that (22.8.12) holds, as before.

Alternatively, we can use ρ^{V} to get a representation of A_{1} on V, as in Section 20.10. One can check that this representation satisfies the analogous compatibility with the inner product on V. We can use this representation to get a positive linear functional on A_{1} associated to v_{0} in the same way as before. The previous argument implies that this representation of A_{1} satisfies the analogue of (22.8.12). It follows that (22.8.12) holds, as in Theorem 21.22 on p320 of [151].

Suppose now that the involution on A does not necessarily preserve the norm, but it continuous, so that (22.5.9) holds for some C ≥ 1 again. This implies that the involution on A_{1} satisfies the analogous condition, as in Section 17.12. One can use the same types of arguments as before to get that (22.8.16) holds under these conditions too.

22.10 A result about unitary equivalence

Let A be an associative algebra over the real or complex numbers with an involution x ↦ x^{∗_{A}} that is conjugate-linear in the complex case, and let (V, ⟨·, ·⟩_{V}), (W, ⟨·, ·⟩_{W}) be Hilbert spaces over the real or complex numbers, as appropriate. Also let ρ^{V}, ρ^{W} be representations of A on V, W, respectively, that are compatible with the involution on A and the inner products on V, W, in the sense of (22.8.1) and its analogue for ρ^{W}. This implies that ρ_{x}^{V}, ρ_{x}^{W} are bounded linear mappings on V, W, respectively, for each x ∈ A, as in Section 21.6. Thus ρ^{V}, ρ^{W} are ∗-representations of A on V, W, respectively, as in Section 21.5.

Suppose that ρ^{V}, ρ^{W} are topologically cyclic, with topologically cyclic vectors v_{0} ∈ V and w_{0} ∈ W, respectively, as in Section 20.13. If

⟨ρ_{x}^{V} (v_{0}), v_{0}⟩_{V} = ⟨ρ_{x}^{W} (w_{0}), w_{0}⟩_{W} (22.10.1)

for every x ∈ A, then ρ^{V}, ρ^{W} are unitarily equivalent, as in Section 20.7. This corresponds to Theorem 21.23 on p320 of [151]. This also corresponds to the uniqueness parts of Theorem 4.7.3 on p123 of [11], and Theorem 4.3 on p273 of [291].

More precisely, it suffices to ask that

⟨ρ_{z^{∗_{A}} y}^{V}(v_{0}), v_{0}⟩_{V} = ⟨ρ_{z^{∗_{A}} y}^{W}(w_{0}), w_{0}⟩_{W} (22.10.2)

for every y, z ∈ A. Of course, (22.10.2) implies (22.10.1) when A has a multiplicative identity element. Another argument without this additional hypothesis will be discussed in the next section.

To see this, observe that

⟨ρ_{y}^{V} (v_{0}), ρ_{z}^{V} (v_{0})⟩_{V} = ⟨ρ_{z^{∗_{A}} y}^{V}(v_{0}), v_{0}⟩_{V}
(22.10.3)  = ⟨ρ_{z^{∗_{A}} y}^{W}(w_{0}), w_{0}⟩_{W} = ⟨ρ_{y}^{W} (w_{0}), ρ_{z}^{W} (w_{0})⟩_{W}

for every y, z ∈ A. This implies that

∥ρ_{y}^{V} (v_{0})∥_{V} = ∥ρ_{y}^{W} (w_{0})∥_{W} (22.10.4)

for every y ∈ A, where ∥ · ∥_{V}, ∥ · ∥_{W} are the norms on V, W, associated to the inner products. In particular, this means that

(22.10.5) ρ_{y}^{V} (v_{0}) = 0 in V if and only if ρ_{y}^{W} (w_{0}) = 0 in W.

We would like to define a linear mapping U from

{ρ_{y}^{V} (v_{0}) : y ∈ A} (22.10.6)

to {ρ_{y}^{W} (w_{0}) : y ∈ A} (22.10.7)

with U(ρ_{y}^{V} (v_{0})) = ρ_{y}^{W} (w_{0}) (22.10.8)

for every y ∈ A. This mapping is well defined, because of (22.10.5). In fact, U is a unitary mapping from (22.10.6) onto (22.10.7), with respect to the restrictions of the inner products on V, W to these linear subspaces, because of (22.10.3).

Remember that (22.10.6) and (22.10.7) are dense in V, W, respectively, with respect to the metrics associated to their norms, by hypothesis. It follows that U has a unique extension to a unitary mapping from V onto W, as in Section 2.2.

Observe that

ρ_{x}^{W} (U(ρ_{y}^{V} (v_{0}))) = ρ_{x}^{W} (ρ_{y}^{W} (w_{0})) = ρ_{x y}^{W}(w_{0})
(22.10.9)  = U(ρ_{x y}^{V}(v_{0})) = U(ρ_{x}^{V} (ρ_{y}^{V} (v_{0})))

for every x, y ∈ A. This implies that

ρ_{x}^{W} ∘ U = U ∘ ρ_{x}^{V} (22.10.10)

on (22.10.6) for every x ∈ A. It follows that (22.10.10) holds on V for every x ∈ A, because (22.10.6) is dense in V, and because ρ_{x}^{V}, ρ_{x}^{W} are bounded linear mappings on V, W, respectively. Thus U is a unitary equivalence between these two representations.

It would also be nice to have

U(v_{0}) = w_{0}. (22.10.11)

If A has a multiplicative identity element e_{A}, then ρ_{e_{A}}^{V}(ρ_{y}^{V} (v_{0})) = ρ_{y}^{V} (v_{0}) and ρ_{e_{A}}^{W}(ρ_{y}^{W} (w_{0})) = ρ_{y}^{W} (w_{0}) for every y ∈ A. This implies that ρ_{e_{A}}^{V}, ρ_{e_{A}}^{W} are the identity mappings on V, W, respectively, because ρ_{e_{A}}^{V}, ρ_{e_{A}}^{W} are bounded linear mappings on V, W, respectively, and (22.10.6), (22.10.7) are dense in V, W, respectively, by hypothesis. In this case, (22.10.11) follows by taking y = e_{A} in (22.10.8).

We shall look at (22.10.11) further in the next section.

22.11 More on this unitary mapping

Let us continue with the same notation and hypotheses as in the previous section, without asking that A have a multiplicative identity element. If (22.10.11) holds, then it is easy to see that (22.10.1) holds, using (22.10.8) and the fact that U is a unitary mapping. This corresponds to a remark on p327 of [151].

Conversely, suppose for the moment that (22.10.1) holds. If y ∈ A, then

⟨U(ρ_{y}^{V} (v_{0})), w_{0}⟩_{W} = ⟨ρ_{y}^{W} (w_{0}), w_{0}⟩_{W} = ⟨ρ_{y}^{V} (v_{0}), v_{0}⟩_{V} , (22.11.1)

using (22.10.8) in the first step, and (22.10.1) in the second step. This implies that ⟨U(v), w_{0}⟩_{W} = ⟨v, v_{0}⟩_{V} (22.11.2)

for every v ∈ V, because (22.10.6) is dense in V, and U is a bounded linear mapping from V into W. Equivalently, this means that

⟨v, U^{−1}(w_{0})⟩_{V} = ⟨v, v_{0}⟩_{V} (22.11.3)

22.12. SOME MORE POSITIVE LINEAR FUNCTIONALS 503 for every v ∈ V, because U is a unitary mapping from V onto W. It follows that U^{−1}(w_{0}) = v_{0}, (22.11.4) which is the same as saying that (22.10.11) holds.

The question of getting (22.10.1) or equivalently (22.10.11) from (22.10.2) is essentially part of the proof of Theorem 21.32 on p325 of [151]. More precisely, if x ∈ A, then ⟨U(v_{0}), ρ_{x}^{W} (w_{0})⟩_{W} = ⟨ρ_{x^{∗_{A}}}^{W} (U(v_{0})), w_{0}⟩_{W} = ⟨U(ρ_{x^{∗_{A}}}^{V} (v_{0})), w_{0}⟩_{W} , (22.11.5) using (22.10.10) in the second step. This implies that ⟨U(v_{0}), ρ_{x}^{W} (w_{0})⟩_{W} = ⟨ρ_{x^{∗_{A}}}^{W} (w_{0}), w_{0}⟩_{W} , (22.11.6) because of (22.10.8). Of course, this means that ⟨U(v_{0}), ρ_{x}^{W} (w_{0})⟩_{W} = ⟨w_{0}, ρ_{x}^{W} (w_{0})⟩_{W} . (22.11.7) It is easy to obtain (22.10.11) from this, because (22.10.7) is dense in W, as on p327 of [151].

22.12 Some more positive linear functionals

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗_{A}} that is conjugate-linear in the complex case. Also let λ be a linear functional on A that is positive in the strict sense, as in Section 18.6. If x ∈ A, then λ_{x}(y) = λ(x^{∗_{A}} y x) (22.12.1) defines another linear functional on A.

If y ∈ A, then λ_{x}(y^{∗_{A}}) = λ((x^{∗_{A}} y^{∗_{A}}) x) = λ((y x)^{∗_{A}} x) = \overline{λ(x^{∗_{A}} (y x))} = \overline{λ_{x}(y)} (22.12.2) in the complex case, and similarly λ_{x}(y^{∗_{A}}) = λ_{x}(y) (22.12.3) in the real case. In particular, λ_{x}(y) ∈ R (22.12.4) when y is self-adjoint in the complex case. We also have that λ_{x}(y^{∗_{A}} y) = λ(x^{∗_{A}} y^{∗_{A}} y x) = λ((y x)^{∗_{A}} (y x)) ≥ 0 (22.12.5) in both the real and complex cases.

504 CHAPTER 22. ALGEBRAS, NORMS, AND REPRESENTATIONS

Observe that |λ_{x}(y)| = |λ(x^{∗_{A}} (y x))| ≤ λ((y x)^{∗_{A}} (y x))^{1/2} λ(x^{∗_{A}} x)^{1/2}, (22.12.6) where the second step is as in Section 18.6. This means that |λ_{x}(y)| ≤ λ(x^{∗_{A}} x)^{1/2} λ_{x}(y^{∗_{A}} y)^{1/2}. (22.12.7) Thus λ_{x} is positive in the strong sense on A, as in Section 18.7. If A has a multiplicative identity element e_{A}, then λ_{x}(e_{A}) = λ(x^{∗_{A}} x). (22.12.8) In this case, (22.12.7) can also be obtained from the positivity of λ_{x}, as before.

Suppose that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, and that ∥y^{∗_{A}}∥_{A} ≤ C ∥y∥_{A} (22.12.9) for some C ≥ 1 and every y ∈ A, so that y ↦ y^{∗_{A}} is continuous on A. Suppose also for the moment that A has a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1. If w ∈ A is self-adjoint, then |λ_{x}(w)| ≤ λ_{x}(e_{A}) ∥w∥_{A}, (22.12.10) as in Section 18.11. This means that |λ_{x}(w)| ≤ λ(x^{∗_{A}} x) ∥w∥_{A}, (22.12.11) because of (22.12.8). It follows that |λ_{x}(y)| ≤ C^{1/2} λ_{x}(e_{A}) ∥y∥_{A} (22.12.12) for every y ∈ A, as before, so that |λ_{x}(y)| ≤ C^{1/2} λ(x^{∗_{A}} x) ∥y∥_{A}. (22.12.13)

If A does not necessarily have a multiplicative identity element, then we can consider the algebra A_{1} defined in Section 6.15, with the involution defined in Section 17.12. We can also use λ_{x} to get a positive linear functional λ_{1,x} on A_{1}, as in Section 18.8. We can take the value of λ_{1,x} at the multiplicative identity element in A_{1} to be λ(x^{∗_{A}} x), (22.12.14) because of (22.12.7), as before. This leads to analogues of (22.12.10) and (22.12.12) for λ_{1,x} on A_{1}, as in Section 18.11, which imply the analogues of (22.12.11) and (22.12.13) for λ_{1,x} on A_{1}. It follows that λ_{x} satisfies (22.12.11) and (22.12.13) on A under these conditions.

22.13 Kernels of bilinear mappings

Let k be a field, let V, W, and Z be vector spaces over k, and let B be a bilinear mapping from V × W into Z, as in Section 19.1. The kernel of B on the left is defined by

(22.13.1) ker_{L} B = {v ∈ V : B(v, w) = 0 for every w ∈ W },

and is a linear subspace of V. Similarly, the kernel of B on the right is defined by (22.13.2) ker_{R} B = {w ∈ W : B(v, w) = 0 for every v ∈ V },

and is a linear subspace of W. These notions are considered more braodly for commutative groups in place of vector spaces on p52 of [206].

Put B_{1,w}(v) = B(v, w) and B_{2,v}(w) = B(v, w) for all v ∈ V and w ∈ W, as in Sections 5.13, 5.15, and 19.11. Thus B_{1,w} is a linear mapping from V into Z for each w ∈ W, and w ↦ B_{1,w} is a linear mapping from W into the space L(V, Z) of linear mappings from V into Z. Note that (22.13.2) is the same as the kernel of w ↦ B_{1,w}. Similarly, B_{2,v} is a linear mapping from W into Z for each v ∈ V, v ↦ B_{2,v} is a linear mapping from V into L(W, Z), and (22.13.1) is the same as the kernel of v ↦ B_{2,v}.

We say that B is nondegenerate on the left if

ker_{L} B = {0}, (22.13.3)

and that B is nondegenerate on the right if

ker_{R} B = {0}, (22.13.4)

as on p337 of [206]. The former is equivalent to the injectivity of v ↦ B_{2,v}, and the latter is equivalent to the injectivity of w ↦ B_{1,w}.

Let q_{V} be the natural quotient mapping from V onto V/ ker_{L} B, and let q_{W} be the natural quotient mapping from W onto W/ ker_{R} B. We would like to define a bilinear mapping b from (V/ ker_{L} B) × (W/ ker_{R} B) into Z by

b(q_{V} (v), q_{W} (w)) = B(v, w). (22.13.5)

It is easy to see that b is well defined in this way, and that b is nondegenerate on both the left and the right, as on p338 of [206].

Suppose now that Z = k, as on p90 of [206]. Remember that V^{alg}, W^{alg} are the algebraic dual spaces of all linear functionals on V, W, respectively, as in Sections 3.1 and 19.9. Using b, we get an injective linear mapping from W/ ker_{R} B into (V/ ker_{L} B)^{alg}, and an injective linear mapping from V/ ker_{L} B into (W/ ker_{R} B)^{alg}, as before.

Remember that the algebraic dual of a finite-dimensional vector space has the same finite dimension. If either V/ ker_{L} B or W/ ker_{R} B has finite dimension, then it follows that the other has finite dimension too, with

(22.13.6) dim(V/ ker_{L} B) ≤ dim(W/ ker_{B} B)^{alg} = dim(W/ ker_{R} B) or (22.13.7) dim(W/ ker_{R} B) ≤ dim(V/ ker_{L} B)^{alg} = dim(V/ ker_{L} B),

as appropriate, by the remark in the preceding paragraph. In this case, we get that V/ ker_{L} B and W/ ker_{R} B both have finite dimension, with

(22.13.8) dim(V/ ker_{L} B) = dim(W/ ker_{R} B).

This means that the injective linear mappings from V/ ker_{L} B and W/ ker_{R} B into (W/ ker_{R} B)^{alg} and (V/ ker_{L} B)^{alg}, respectively, mentioned before are surjective. This corresponds to Theorem 6 on p91 of [206], and its proof.

Suppose that V and W are both finite dimensional, with the same dimension. Under these conditions, (22.13.8) implies that

(22.13.9) dim ker_{L} B = dim ker_{R} B.

In particular, this implies that B is nondegenerate on the left if and only if it is nondegenerate on the right.

22.14 Kernels of sesquilinear functionals

Let V and W be vector spaces over the complex numbers, and let B be a complex-valued function on V ×W that is sesquilinear. This means that B(v, w) is complex-linear in v and conjugate-linear in w, as in Section 5.15. The kernels of B on the left and on the right may be defined as in (22.13.1) and (22.13.2) again, and are linear subspaces of V and W, respectively. Similarly, the nondegeneracy of B on the left and on the right may be defined as in (22.13.3) and (22.13.4), respectively.

Of course, B may be considered as a bilinear mapping from V ×W into Z = C as vector spaces over the real numbers in this case. If w ∈ W, then B_{1,w}(v) = B(v, w) is a complex linear functional on V, and w ↦ B_{1,w} is a conjugate-linear mapping from W into the algebraic dual V^{alg} of V. In particular, w ↦ B_{1,w} is a real-linear mapping, whose kernel is equal to ker_{R} B. Thus this mapping is injective exactly when B is nondegenerate on the right.

Similarly, if v ∈ V, then B_{2,v}(w) = B(v, w) is a conjugate-linear functional on W, which is the same as the complex-conjugate of a complex-linear functional on W. The space of conjugate-linear functionals on W is a vector space over the complex numbers with respect to pointwise addition and scalar multiplication of complex-valued functions on W, and v ↦ B_{2,v} is a complex-linear mapping from V into the space of conjugate-linear functionals on W. The kernel of this mapping is equal to ker_{L} B, so that this mapping is injective exactly when B is nondegenerate on the left, as before.

One can define b as a sesquilinear mapping from (V/ ker_{L} B) × (W/ ker_{R} B) into C as in (22.13.5). One can also check that b is automatically nondegenerate on the left and right, as before. Using b, we get an injective conjugate-linear mapping from W/ ker_{R} B into (V/ ker_{L} B)^{alg}. Similarly, we get an injective complex-linear mapping from V/ ker_{L} B into the space of conjugate-linear functionals on W/ ker_{R} B.

The space of conjugate-linear functionals on a finite-dimensional vector space over the complex numbers has the same dimension as the algebraic dual space of complex-linear functionals, which is the same as the dimension of the vector space. If V/ ker_{L} B or W/ ker_{R} B has finite dimension, then one can use the previous remarks to get that the other has finite dimension as well, with

(22.14.1) dim(V/ ker_{L} B) ≤ dim(W/ ker_{R} B)

or (22.14.2) dim(W/ ker_{R} B) ≤ dim(V/ ker_{L} V),

as appropriate, as in the previous section. Thus V/ ker_{L} B and W/ ker_{R} B both have finite dimension, with

(22.14.3) dim(V/ ker_{L} B) = dim(W/ ker_{R} B),

as before.

In this case, the injective conjugate-linear mapping from W/ ker_{R} B into (V/ ker_{L} B)^{alg} mentioned before is surjective. Similarly, the injective complexlinear mapping from V/ ker_{L} B into the space of conjugate-linear functionals on W/ ker_{R} B mentioned earlier is surjective.

If V and W are both finite-dimensional, with the same dimension, then (22.14.3) implies that (22.14.4) dim ker_{L} B = dim ker_{R} B,

as in the previous section. This implies that B is nondegenerate on the left if and only if it is nondegenerate on the right, as before.

22.15 Bilinear forms and algebra involutions

Let k be a field, and let A be an associative algebra over k with an involution x ↦ x^{∗_{A}}, which may be conjugate-linear when k = C. Also let V be a vector space over k, and let B be a bilinear form on V, or a sesquilinear form on V when k = C and x ↦ x^{∗_{A}} is conjugate-linear on V. Suppose that ρ^{V} is a representation of A on V such that B is compatible with ρ^{V} and the involution on A, in the sense that

B(ρ_{x}^{V} (u), v) = B(u, ρ_{x^{∗_{A}}}^{V} (v)) (22.15.1)

for every x ∈ A and u, v ∈ V, as in Section 21.5.

Under these conditions, one can check that the left and right kernels of B are invariant under ρ^{V}. We may be particularly concerned with the cases where B is symmetric or antisymmetric, or when k = C, the involution on A is conjugatelinear, and B is a Hermitian-symmetric sesquilinear form on V, as in Section 8.5. In each of these cases, it is easy to see that

ker_{L} B = ker_{R} B, (22.15.2) and we may refer to this simply as the kernel of B, and denote it ker B. In these cases, we may say that B is nondegenerate when

ker B = {0}, (22.15.3)

which means that B is nondegenerate on the left and on the right, as in the previous two sections.

Suppose that we are in one of the three cases mentioned in the preceding paragraph, and let q_{V} be the natural quotient mapping from V onto V/ ker B. As in the previous two sections, we get a bilinear or sesquilinear form b on V/ ker B as appropriate, such that

b(q_{V} (u), q_{V} (v)) = B(u, v) (22.15.4)

for every u, v ∈ V. We also have that b is symmetric, antisymmetric, or Hermitian symmetric on V/ ker B when B has the corresponding property on V. Note that b is automatically nondegenerate on V/ ker B, as before.

Let ρ = ρ^{V/ ker B} be the quotient representation of A on V/ ker B corresponding to ρ^{V} as in Section 20.6, so that

ρ_{x} ∘ q_{V} = q_{V} ∘ ρ_{x}^{V} (22.15.5)

for every x ∈ A. If x ∈ A and u, v ∈ V, then

b(ρ_{x}(q_{V} (u)), q_{V} (v)) = b(q_{V} (ρ_{x}^{V} (u)), q_{V} (v)) = B(ρ_{x}^{V} (u), v) = B(u, ρ_{x^{∗_{A}}}^{V} (v))
(22.15.6)  = b(q_{V} (u), q_{V} (ρ_{x^{∗_{A}}}^{V} (v))) = b(u, ρ_{x^{∗_{A}}} (q_{V} (v))),

using (22.15.1) in the third step. This means that b is compatible with ρ and the involution on A.

Suppose now that either k = R and B is symmetric on V, or that k = C, the involution on A is conjugate-linear, and B is a Hermitian-symmetric sesquilinear form on V. In the complex case, remember that

B(v, v) ∈ R (22.15.7)

for every v ∈ V, as in Section 8.5. In either case, suppose that B is nonnegative on V, in the sense that B(v, v) ≥ 0 (22.15.8)

for every v ∈ V, as in Section 8.10. This implies that the analogue

|B(u, v)| ≤ B(u, u)^{1/2} B(v, v)^{1/2} (22.15.9)

of the Cauchy–Schwarz inequality holds for all u, v ∈ V, as before.

It follows that ker B = {v ∈ V : B(v, v) = 0} (22.15.10)

under these conditions. More precisely, the left side is automatically contained in the right side, and the opposite inclusion holds by (22.15.9).

22.15. BILINEAR FORMS AND ALGEBRA INVOLUTIONS 509

In the complex case, we have that b(q_{V} (v), q_{V} (v)) ∈ R (22.15.11) for every v ∈ V, by (22.15.7). In both cases, b(q_{V} (v), q_{V} (v)) ≥ 0 (22.15.12) for every v ∈ V, by (22.15.8). In fact, b is an inner product on V/ ker B in both cases, because b is nondegenerate on V/ ker B.

Linear functionals and representations

23.1 Some more isomorphisms

Let k be a field, and let A be an associative algebra over k with an involution x ↦ x^{∗_{A}} again, which may be conjugate-linear when k = C. Also let V, W be vector spaces over k, and let b_{V}, b_{W} be bilinear forms on V, W, respectively, or sesquilinear forms on V, W, respectively, when k = C and x ↦ x^{∗_{A}} is conjugate-linear on A. Suppose that ρ^{V}, ρ^{W} are representations of A on V, W, respectively, and that (23.1.1) b_{V} , b_{W} are compatible with ρ^{V} , ρ^{W} , respectively,

and the involution on A,

as in Sections 21.5 and 22.15. In this section, we ask that (23.1.2) ρ^{V} , ρ^{W} be algebraically cyclic too,

with cyclic vectors v_{0} ∈ V, w_{0} ∈ W, respectively,

as in Section 20.13.

Consider the condition that b_{V} (ρ_{x}^{V} (v_{0}), v_{0}) = b_{W} (ρ_{x}^{W} (w_{0}), w_{0}) (23.1.3) for every x ∈ A. Using the compatibility of b_{V}, b_{W} with ρ^{V}, ρ^{W} and the involution on A, we get that (23.1.3) is the same as saying that b_{V} (v_{0}, ρ_{x^{∗_{A}}}^{V} (v_{0})) = b_{W} (w_{0}, ρ_{x^{∗_{A}}}^{W} (w_{0})) (23.1.4) for every x ∈ A. Of course, this is equivalent to the condition that b_{V} (v_{0}, ρ_{y}^{V} (v_{0})) = b_{W} (w_{0}, ρ_{y}^{W} (w_{0})) (23.1.5)

23.1. SOME MORE ISOMORPHISMS 511

for every y ∈ A.

Note that (23.1.3) implies that b_{V} (ρ_{z^{∗_{A}} y}^{V}(v_{0}), v_{0}) = b_{W} (ρ_{z^{∗_{A}} y}^{W}(w_{0}), w_{0}) (23.1.6) for every y, z ∈ A. If A has a multiplicative identity element, then (23.1.6) implies (23.1.3). Using (23.1.6), we get that

b_{V} (ρ_{y}^{V} (v_{0}), ρ_{z}^{V} (v_{0})) = b_{V} (ρ_{z^{∗_{A}} y}^{V}(v_{0}), v_{0})
(23.1.7)  = b_{W} (ρ_{z^{∗_{A}} y}^{W}(w_{0}), w_{0}) = b_{W} (ρ_{y}^{W} (w_{0}), ρ_{z}^{W} (w_{0}))

for every y, z ∈ A. In particular, this implies that (23.1.8) ρ_{y}^{V} (v_{0}) ∈ ker_{L} b_{V} if and only if ρ_{y}^{W} (w_{0}) ∈ ker_{L} b_{W} , because of (23.1.2), where the left kernels of b_{V}, b_{W} are as in Sections 22.13 and 22.14. Similarly, (23.1.7) implies that (23.1.9) ρ_{z}^{V} (v_{0}) ∈ ker_{R} b_{V} if and only if ρ_{z}^{W} (w_{0}) ∈ ker_{R} b_{W} , where the right kernels of b_{V}, b_{W} are as before.

Let us ask that (23.1.10) b_{V} , b_{W} be nondegenerate on V, W, respectively,

both on the left or both on the right,

as in Sections 22.13 and 22.14. If (23.1.6) holds, then we would like to show that there is a one-to-one linear mapping U from V onto W that is an isomorphism between ρ^{V} and ρ^{W}, with b_{W} (U(v), U(v′)) = b_{V} (v, v′) (23.1.11) for all v, v′ ∈ V. This is analogous to the result mentioned in Section 22.10. If b_{V}, b_{W} are nondegenerate on V, W, respectively, on the left, then (23.1.8) implies that (23.1.12) ρ_{y}^{V} (v_{0}) = 0 in V if and only if ρ_{y}^{W} (w_{0}) = 0 in W. Similarly, this holds when b_{V}, b_{W} are nondegenerate on V, W, respectively, on the right, because of (23.1.9).

We would like to define U so that U(ρ_{y}^{V} (v_{0})) = ρ_{y}^{W} (w_{0}) (23.1.13) for every y ∈ A. One can use (23.1.12) to get that U is well defined, and injective. More precisely, U maps V onto W, because of (23.1.2).

Using this definition of U, (23.1.11) follows from (23.1.7). If x, y ∈ A, then it is easy to see that ρ_{x}^{W} (U(ρ_{y}^{V} (v_{0}))) = U(ρ_{x}^{V} (ρ_{y}(v_{0}))), (23.1.14)

512 CHAPTER 23. LINEAR FUNCTIONALS AND REPRESENTATIONS

as in Section 22.10. This means that ρ_{x}^{W} ∘ U = U ∘ ρ_{x}^{V} (23.1.15) on V for every x ∈ A, because of (23.1.2).

As in Section 22.10, it would be nice to have that U(v_{0}) = w_{0}. (23.1.16) If A has a multiplicative identity element e_{A}, then it is easy to see that ρ_{e_{A}}^{V}, ρ_{e_{A}}^{W} are the identity mappings on V, W, respectively, as in Section 22.10, and because of (23.1.2). This permits one to obtain (23.1.16) from (23.1.13) with y = e_{A}, as before. We shall discuss (23.1.16) further in the next section.

23.2 More on this isomorphism

We continue with the same notation and hypotheses as in the previous section, without asking that A have a multiplicative identity element. If (23.1.16) holds, then it is easy to see that (23.1.3) holds, using (23.1.11) and (23.1.13). This is analogous to a remark in Section 22.11.

Conversely, if (23.1.3) holds, then b_{W} (U(ρ_{y}^{V} (v_{0})), w_{0}) = b_{W} (ρ_{y}^{W} (w_{0}), w_{0}) = b_{V} (ρ_{y}^{V} (v_{0}), v_{0}) (23.2.1) for every y ∈ A, using (23.1.13) in the first step. This means that b_{W} (U(v), w_{0}) = b_{V} (v, v_{0}) (23.2.2) for every v ∈ V, because ρ^{V} is algebraically cyclic, with cyclic vector v_{0}. This is the same as saying that b_{V} (v, U^{−1}(w_{0})) = b_{V} (v, v_{0}) (23.2.3) for every v ∈ V, because of (23.1.11). If b_{V} is nondegenerate on V on the right, then it follows that U^{−1}(w_{0}) = v_{0}, (23.2.4) or equivalently that (23.1.16) holds.

Alternatively, b_{W} (w_{0}, U(ρ_{y}^{V} (v_{0}))) = b_{W} (w_{0}, ρ_{y}^{W} (w_{0})) = b_{V} (v_{0}, ρ_{y}^{V} (v_{0})) (23.2.5) for every y ∈ A, using (23.1.13) in the first step again, and (23.1.5) in the second step. This implies that b_{W} (w_{0}, U(v)) = b_{V} (v_{0}, v) (23.2.6) for every v ∈ V, because ρ^{V} is algebraically cyclic, with cyclic vector v_{0}, as before. Equivalently, this means that b_{V} (U^{−1}(w_{0}), v) = b_{V} (v_{0}, v) (23.2.7) for every v ∈ V, because of (23.1.11). If b_{V} is nondegenerate on the left, then we get that (23.2.4) holds, so that (23.1.16) holds.

We can also get (23.1.16) from (23.1.6), the algebraic cyclicity of ρ^{W}, and the nondegeneracy of b_{W}, as in Section 22.11. If x ∈ A, then

b_{W} (U(v_{0}), ρ_{x}^{W} (w_{0})) = b_{W} (ρ_{x^{∗_{A}}}^{W} (U(v_{0}), w_{0})) = b_{W} (U(ρ_{x^{∗_{A}}}^{V} (v_{0})), w_{0}), (23.2.8)

usng (23.1.15) in the second step. This implies that

b_{W} (U(v_{0}), ρ_{x}^{W} (w_{0})) = b_{W} (ρ_{x^{∗_{A}}}^{W} (w_{0}), w_{0}) = b_{W} (w_{0}, ρ_{x}^{W} (w_{0})), (23.2.9)

using (23.1.13) in the first step. This implies (23.1.16) when b_{W} is nondegenerate on the left, because ρ^{W} is algebraically cyclic with cyclic vector w_{0}.

Similarly,

b_{W} (ρ_{x}^{W} (w_{0}), U(w_{0})) = b_{W} (w_{0}, ρ_{x^{∗_{A}}}^{W} (U(w_{0}))) = b_{W} (w_{0}, U(ρ_{x^{∗_{A}}}^{V} (v_{0})))
(23.2.10)  = b_{W} (w_{0}, ρ_{x^{∗_{A}}}^{W} (w_{0})) = b_{W} (ρ_{x}^{W} (w_{0}), w_{0}),

which implies (23.1.16) when b_{W} is nondegenerate on the right, because ρ^{W} is algebraically cyclic with cyclic vector w_{0}.

23.3 Linear functionals from bilinear forms

Let k be a field, and let A be an associative algebra over k with an involution x ↦ x^{∗_{A}}, which may be conjugate-linear when k = C, as usual. Also let V be a vector space over k, and let b_{V} be a bilinear form on V, or a sesquilinear form on V when k = C and x ↦ x^{∗_{A}} is conjugate-linear on A. Suppose that ρ^{V} is a representation of A on V, and that b_{V} is compatible with ρ^{V} and the involution on A, as in Sections 21.5 and 22.15.

If v_{0} ∈ V, then put

λ_{ρ^{V} ,v_{0}}(x) = b_{V} (ρ_{x}^{V} (v_{0}), v_{0}) (23.3.1)

for each x ∈ A, as in Section 22.8. This defines a linear functional on A, with

λ_{ρ^{V} ,v_{0}}(x) = b_{V} (v_{0}, ρ_{x^{∗_{A}}}^{V} (v_{0})) (23.3.2)

for every x ∈ A. If b_{V} is a symmetric bilinear form on V, then

λ_{ρ^{V} ,v_{0}}(x^{∗_{A}}) = λ_{ρ^{V} ,v_{0}}(x) (23.3.3)

for every x ∈ A. If b_{V} is an antisymmetric bilinear form on V, then

λ_{ρ^{V} ,v_{0}}(x^{∗_{A}}) = −λ_{ρ^{V} ,v_{0}}(x) (23.3.4)

for every x ∈ A. If k = C, x ↦ x^{∗_{A}} is conjugate-linear on A, and b_{V} is a Hermitian-symmetric sesquilinear form on V, then

514 CHAPTER 23. LINEAR FUNCTIONALS AND REPRESENTATIONS

λ_{ρ^{V} ,v_{0}}(x^{∗_{A}}) = \overline{λ_{ρ^{V} ,v_{0}}(x)} (23.3.5) for every x ∈ A.

Remember that ker_{L} b_{V} and ker_{R} b_{V} are the left and right kernels of b_{V}, as in Sections 22.13 and 22.14. Note that

λ_{ρ^{V} ,v_{0}}(y^{∗_{A}} x) = b_{V} (ρ_{y^{∗_{A}}}^{V} (ρ_{x}^{V} (v_{0})), v_{0}) = b_{V} (ρ_{x}^{V} (v_{0}), ρ_{y}^{V} (v_{0})) (23.3.6)

for all x, y ∈ A. If ρ_{x}^{V} (v_{0}) ∈ ker_{L} b_{V} , (23.3.7)

then it follows that λ_{ρ^{V} ,v_{0}}(y^{∗_{A}} x) = 0 for all y ∈ A. (23.3.8)

The converse holds when ρ^{V} is algebraically cyclic, with cyclic vector v_{0}, as in Section 20.13. Of course, (23.3.8) is the same as saying that

λ_{ρ^{V} ,v_{0}}(z x) = 0 for all z ∈ A. (23.3.9)

Similarly, if ρ_{y}^{V} (v_{0}) ∈ ker_{R} b_{V} , (23.3.10)

then λ_{ρ^{V} ,v_{0}}(y^{∗_{A}} x) = 0 for all x ∈ A. (23.3.11)

The converse holds when ρ^{V} is algebraically cyclic, with cyclic vector v_{0}, as before. We also have that (23.3.11) is the same as saying that

λ_{ρ^{V} ,v_{0}}((z y)^{∗_{A}}) = λ_{ρ^{V} ,v_{0}}(y^{∗_{A}} z^{∗_{A}}) = 0 for all z ∈ A. (23.3.12)

23.4 Bilinear forms from linear functionals

Let k be a field, and let A be an associative algebra over k with an involution x ↦ x^{∗_{A}}, which may be conjugate-linear when k = C. Also let λ be a linear functional on A, and put

B_{λ}(x, y) = λ(y^{∗_{A}} x) (23.4.1)

for every x, y ∈ A. This is related to the parts of the proofs of Theorem 4.7.3 on p123 of [11], Theorem 21.24 on p321 of [151], and Theorem 12.40 on p320 of [286], and to a remark on p273 of [291]. Note that (23.4.1) is a bilinear form on A, except when k = C and the involution on A is conjugate-linear, in which case (23.4.1) is a sesquilinear form on A.

If a ∈ A, then

B_{λ}(a x, y) = λ(y^{∗_{A}} a x) = λ((a^{∗_{A}} y)^{∗_{A}} x) = B_{λ}(x, a^{∗_{A}} y) (23.4.2)

for every x, y ∈ A. Thus B_{λ} is compatible with the involution on A, and the representation of A on itself corresponding to left multiplication on A, as in Section 20.2.

Let us say that λ is symmetric in the strict sense with respect to the involution on A if λ(x^{∗_{A}} y) = λ(y^{∗_{A}} x) (23.4.3)

for all x, y ∈ A, so that (23.4.1) is a symmetric bilinear form on A. If

λ(x^{∗_{A}} y) = −λ(y^{∗_{A}} x) (23.4.4)

for all x, y ∈ A, so that (23.4.1) is an antisymmetric bilinear form on A, then we may say that λ is antisymmetric in the strict sense with respect to the involution on A. If k = C, the involution on A is conjugate-linear, and

λ(x^{∗_{A}} y) = \overline{λ(y^{∗_{A}} x)}, (23.4.5)

then (23.4.1) is a Hermitian-symmetric sesquilinear form on A, and we may say that λ is Hermitian symmetric in the strict sense with respect to the involution on A.

Let us say that λ is symmetric in the strong sense with respect to the involution on A if λ(x^{∗_{A}}) = λ(x) (23.4.6)

for all x ∈ A, and that λ is antisymmetric in the strong sense if

λ(x^{∗_{A}}) = −λ(x) (23.4.7)

for all x ∈ A. If k = C, the involution on A is conjugate-linear, and

λ(x^{∗_{A}}) = \overline{λ(x)} (23.4.8)

for all x ∈ A, then we may say that λ is Hermitian symmetric in the strong sense with respect to the involution on A. It is easy to see that these conditions imply the corresponding strict versions. If A has a multiplicative identity element, then the converse holds too. In this case, we may simply say that λ is symmetric, antisymmetric, or Hermitian symmetric, as appropriate.

The left kernel of B_{λ} is given by

(23.4.9)  ker_{L} B_{λ} = {x ∈ A : λ(y^{∗_{A}} x) = 0 for every y ∈ A}
= {x ∈ A : λ(z x) = 0 for every z ∈ A}.

This is clearly a left ideal in A. Similarly,

(23.4.10) ker_{R} B_{λ} = {y ∈ A : λ(y^{∗_{A}} x) = 0 for every x ∈ A}
= {y ∈ A : λ((z y)^{∗_{A}}) = 0 for every z ∈ A}

is a left ideal in A as well. Suppose from now on in this section that λ is symmetric, antisymmetric, or Hermitian symmetric in the strict sense with respect to the involution on A, so that (23.4.9) and (23.4.10) are the same, as in Section 22.15. In this case, we may refer to (23.4.9), (23.4.10) as the kernel of B_{λ}, and denote it ker B_{λ}, as before.

Let q_{λ} be the natural quotient mapping from A onto A/ ker B_{λ}. Remember that b_{λ}(q_{λ}(x), q_{λ}(y)) = B_{λ}(x, y) (23.4.11)

defines a bilinear or sesquilinear form on A/ ker B_{λ}, as appropriate, as in Section 22.15. More precisely, b_{λ} is symmetric, antisymmetric, or Hermitian symmetric on A/ ker B_{λ}, as appropriate, and nondegenerate, as before.

We also get a representation of A on A/ ker B_{λ}, corresponding to left multiplication on A, as in Section 20.6. If a, x ∈ A, then

a · q_{λ}(x) = q_{λ}(a x) (23.4.12)

is the action of a on q_{λ}(x) in A/ ker B_{λ}, as a left module over A. Remember that b_{λ} is compatible with this representation and the involution on A, as in Section 22.15.

Suppose that A has a multiplicative identity element e_{A}, and note that the action of e_{A} on A/ ker B_{λ} is the identity mapping on A/ ker B_{λ}. We also have that the representation of A on A/ ker B_{λ} just mentioned is algebraically cyclic, with cyclic vector q_{λ}(e_{A}), as in Section 20.13. If a ∈ A, then

b_{λ}(a · q_{λ}(e_{A}), q_{λ}(e_{A})) = b_{λ}(q_{λ}(a), q_{λ}(e_{A})) = B_{λ}(a, e_{A}) = λ(a). (23.4.13)

This means that λ is the same as the linear functional on A associated to b_{λ}, this representation of A on A/ ker B_{λ}, and v_{0} = q_{λ}(e_{A}) as in the previous section. This is related to parts of Theorem 4.7.3 on p123 of [11], Theorem 21.24 on p321 of [151], and Theorem 4.3 on p273 of [291].

23.5 Using positive linear functionals

Let us continue with the same notation and hypotheses as in the previous section, now with k = R or C, and with the involution on A being conjugate-linear when k = C. Suppose also now that λ is positive in the strict sense on A, as in Section 18.6. In particular, this means that λ is symmetric or Hermitian symmetric with respect to the involution on A, as appropriate, as in the previous section. We also get that

ker B_{λ} = {x ∈ A : B_{λ}(x, x) = 0} = {x ∈ A : λ(x^{∗_{A}} x) = 0}, (23.5.1)

as in Section 22.15.

Suppose from now on in this section that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, and that the involution on A is continuous on A, so that

∥z^{∗_{A}}∥_{A} ≤ C ∥z∥_{A} (23.5.2)

for some C ≥ 1 and every z ∈ A. If x, w ∈ A and w is self-adjoint, then

|λ(x^{∗_{A}} w x)| ≤ λ(x^{∗_{A}} x) ∥w∥_{A}, (23.5.3) as in Section 22.12. If a ∈ A, then we can take w = a^{∗_{A}} a, to get that

λ(x^{∗_{A}} a^{∗_{A}} a x) ≤ λ(x^{∗_{A}} x) ∥a^{∗_{A}} a∥_{A} ≤ C λ(x^{∗_{A}} x) ∥a∥_{A}^{2} (23.5.4)

for every x ∈ A. This means that

B_{λ}(a x, a x) ≤ B_{λ}(x, x) ∥a^{∗_{A}} a∥_{A} ≤ C ∥a∥_{A}^{2} B_{λ}(x, x) (23.5.5)

for every a, x ∈ A.

Remember that q_{λ} is the natural quotient mapping from A onto A/ ker B_{λ}, and that b_{λ} is the bilinear or sesquilinear form on A/ ker B_{λ}, as appropriate, associated to B_{λ} as in (23.4.11). More precisely, b_{λ} is an inner product on A/ ker B_{λ}, as in Section 22.15. If a, x ∈ A, then

b_{λ}(a · q_{λ}(x), a · q_{λ}(x)) = b_{λ}(q_{λ}(a x), q_{λ}(a x)) = B_{λ}(a x, a x)
(23.5.6)  ≤ b_{λ}(q_{λ}(x), q_{λ}(x)) ∥a^{∗_{A}} a∥_{A}
≤ C ∥a∥_{A}^{2} b_{λ}(q_{λ}(x), q_{λ}(x)),

as in (23.5.5).

Let (V_{λ}, ⟨·, ·⟩_{V_{λ}}) be a completion of A/ ker B_{λ} with respect to b_{λ} and the corresponding norm and metric, so that V_{λ} is a Hilbert space over k. If a ∈ A, then the action of a on A/ ker B_{λ} is a bounded linear mapping with respect to the norm associated to b_{λ}, as in (23.5.6). This means that the action of a on A/ ker B_{λ} has a unique extension to a bounded linear mapping ρ_{a}^{V_{λ}} on V_{λ}, as in Section 2.2. More precisely,

∥ρ_{a}^{V_{λ}}∥_{op,V_{λ} V_{λ}} ≤ ∥a^{∗_{A}} a∥_{A}^{1/2} ≤ C^{1/2} ∥a∥_{A}, (23.5.7)

because of (23.5.6).

This defines a representation ρ^{V_{λ}} of A on V_{λ}. In fact, it is easy to see that ρ^{V_{λ}} is a ∗-representation of A on V_{λ}, as in Section 21.5, because b_{λ} is compatible with the representation of A on A/ ker B_{λ} and the involution on A, as mentioned in the previous section. This corresponds to parts of Theorem 4.7.3 on p123 of [11], Theorem 21.24 on p321 of [151], and Theorem 4.3 on p273 of [291]. This also corresponds to part of the proof of Theorem 12.40 on p320 of [286].

Suppose for the moment that A has a multiplicative identity element e_{A}, and remember that the action of e_{A} on A/ ker B_{λ} is the identity mapping, which implies that ρ_{e_{A}}^{V_{λ}} is the identity mapping on V_{λ}. In this case, the representation of A on A/ ker B_{λ} is algebraically cyclic, with cyclic vector q_{λ}(e_{A}), as mentioned in the previous section. This implies that ρ^{V_{λ}} is topologically cyclic, with topologically cyclic vector q_{λ}(e_{A}), as an element of V_{λ}. This corresponds to parts of Theorem 4.7.3 on p123 of [11], Theorem 21.24 on p321 of [151], and Theorem 4.3 on p273 of [151].

If a ∈ A, then

⟨ρ_{a}^{V_{λ}}(q_{λ}(e_{A})), q_{λ}(e_{A})⟩_{V_{λ}} = b_{λ}(a · q_{λ}(e_{A}), q_{λ}(e_{A})) = λ(a), (23.5.8)

using (23.4.13) in the second step. This means that λ is the same as the linear functional on A associated to ρ^{V_{λ}}, ⟨·, ·⟩_{V_{λ}}, and v_{0} = q_{λ}(e_{A}) as in Section 23.3, as in [11, 151, 291].

23.6 Some analogous arguments using A_{1}

Let us continue with the same notation and hypotheses as in the previous section, without asking that A necessarily have a multiplicative identity element. Suppose now that λ is positive in the strong sense on A, as in Section 18.7. Let A_{1} be A × R or A × C, as appropriate, considered as a Banach algebra, as in Section 6.15. We can define an involution on A_{1} as in Section 17.12, and which satisfies the analogue of (23.5.2). Let λ_{1} be the positive linear functional on A obtained from λ as in Section 18.8.

This leads to a bilinear or sesquilinear form B_{λ_{1}} on A_{1}, as appropriate, as in Section 23.4. Let q_{λ_{1}} be the natural quotient mapping from A_{1} onto A_{1}/ ker B_{λ_{1}}, and let b_{λ_{1}} be the corresponding bilinear or sesquilinear form on A_{1}/ ker B_{λ_{1}}, as before. In fact, b_{λ_{1}} is an inner product on A_{1}/ ker B_{λ_{1}}, as in the previous section. We also have that b_{λ_{1}} is compatible with the representation of A_{1} on A_{1}/ ker B_{λ_{1}} corresponding to left multiplication on A_{1} and the involution on A_{1}, as before.

Let (V_{λ_{1}}, ⟨·, ·⟩_{V_{λ_{1}}} ) be a completion of A_{1}/ ker B_{λ_{1}} with respect to b_{λ_{1}}, as in the previous section. We also get a ∗-representation ρ^{V_{λ_{1}}} of A_{1} on V_{λ_{1}}, as before. Remember that e_{A_{1}} = (0, 1) is the multiplicative identity element in A_{1}, as in Section 6.15. Thus ρ^{V_{λ_{1}}} is topologically cyclic, as a representation of A_{1} on V_{λ_{1}}, with topologically cyclic vector q_{λ_{1}}(e_{A_{1}}), as an element of V_{λ_{1}}, as mentioned in the previous section. More precisely, the representation of A_{1} on A_{1}/ ker B_{λ_{1}} corresponding to multiplication on the left is algebraically cyclic, with cyclic vector q_{λ_{1}}(e_{A_{1}}), as before.

If a ∈ A, then (a, 0) ∈ A_{1}, and ρ_{(a,0)}^{V_{λ_{1}}} is a bounded linear mapping on V_{λ_{1}}. This defines a ∗-representation of A on V_{λ_{1}}. Put

(23.6.1) Z_{λ_{1}} = {v ∈ V_{λ_{1}} : ρ_{(a,0)}^{V_{λ_{1}}}(v) = 0 for every a ∈ A},

which is a closed linear subspace of V_{λ_{1}}. Consider

W_{λ_{1}} = Z_{λ_{1}}^{⊥}, (23.6.2)

which is to say that W_{λ_{1}} is the orthogonal complement of Z_{λ_{1}} in V_{λ_{1}}, as in Section 2.15. This is a closed linear subspace of V_{λ_{1}} as well, and every element of V_{λ_{1}} can be expressed in a unique way as a sum of elements of W_{λ_{1}} and Z_{λ_{1}}, as before.

Observe that ρ_{(a,0)}^{V_{λ_{1}}}(W_{λ_{1}}) ⊆ W_{λ_{1}} (23.6.3)

for every a ∈ A, as in Section 21.7, because ρ_{(a^{∗_{A}} ,0)}^{V_{λ_{1}}} maps Z_{λ_{1}} to {0}, by construction. Let w_{0}, z_{0} be the unique elements of W_{λ_{1}}, Z_{λ_{1}}, respectively, such that q_{λ_{1}}(e_{A_{1}}) = w_{0} + z_{0}, (23.6.4)

as in the preceding paragraph. If a ∈ A, then

⟨ρ_{(a,0)}^{V_{λ_{1}}}(q_{λ_{1}}(e_{A_{1}})), q_{λ_{1}}(e_{A_{1}})⟩_{V_{λ_{1}}} = λ_{1}((a, 0)) = λ(a), (23.6.5) where the first step is as in (23.5.8), and the second step follows from the definition of λ_{1} on A_{1}, as in Section 18.8. Of course, ρ_{(a,0)}^{V_{λ_{1}}}(z_{0}) = 0, because z_{0} is an element of (23.6.1), so that

ρ_{(a,0)}^{V_{λ_{1}}}(q_{λ_{1}}(e_{A_{1}})) = ρ_{(a,0)}^{V_{λ_{1}}}(w_{0}), (23.6.6)

by (23.6.4). We also have that

⟨ρ_{(a,0)}^{V_{λ_{1}}}(w_{0}), z_{0}⟩_{V_{λ_{1}}} = 0, (23.6.7)

because of (23.6.2) and (23.6.3).

It follows that ⟨ρ_{(a,0)}^{V_{λ_{1}}}(w_{0}), w_{0}⟩_{V_{λ_{1}}} = λ(a) (23.6.8)

for every a ∈ A, because of (23.6.6). Let

(23.6.9) ρ_{a}^{W_{λ_{1}}} be the restriction of ρ_{(a,0)}^{V_{λ_{1}}} to W_{λ_{1}}.

This defines a representation ρ^{W_{λ_{1}}} of A on W_{λ_{1}}, because of (23.6.3). More precisely, it is easy to see that ρ^{W_{λ_{1}}} is a ∗-representation of A on W_{λ_{1}}, because ρ^{V_{λ_{1}}} is a ∗-representation of A_{1} on V^{λ_{1}}. Of course, this uses the restriction of ⟨·, ·⟩_{V_{λ_{1}}} to W_{λ_{1}} as the inner product on W_{λ_{1}}, so that W_{λ_{1}} may be considered as a Hilbert space.

This shows that λ is the same as the linear functional on A associated to ρ^{W_{λ_{1}}} , the restriction of ⟨·, ·⟩_{V_{λ_{1}}} to W_{λ_{1}}, and w_{0} as in Section 23.3. This corresponds to part of Theorem 21.24 on p321 of [151], and its proof. Another part of the theorem is that

(23.6.10) ρ^{W_{λ_{1}}} is topologically cyclic, with topologically cyclic vector w_{0},

and this will be discussed in the next section.

23.7 Positivity and topological cyclicity

We continue with the same notation and hypotheses as before. Note that ρ_{e_{A_{1}}}^{V_{λ_{1}}} is the identity mapping on V_{λ_{1}}, because the action of e_{A_{1}} on A_{1}/ ker B_{λ_{1}} is the identity mapping, by construction. If a ∈ A and t ∈ R or C, as appropriate, then (a, t) ∈ A_{1}, and

ρ_{(a,t)}^{V_{λ_{1}}}(q_{λ_{1}}(e_{A_{1}})) = ρ_{(a,0)}^{V_{λ_{1}}}(q_{λ_{1}}(e_{A_{1}})) + t q_{λ_{1}}(e_{A_{1}})
(23.7.1)  = ρ_{(a,0)}^{V_{λ_{1}}}(w_{0}) + t q_{λ_{1}}(e_{A_{1}}) = ρ_{a}^{W_{λ_{1}}} (w_{0}) + t q_{λ_{1}}(e_{A_{1}}),

using the previous remark in the first step, (23.6.6) in the second step, and the definition of ρ_{a}^{W_{λ_{1}}} in the third step. The set of vectors in V_{λ_{1}} of this form is the same as A_{1}/ ker B_{λ_{1}}, which is dense in V_{λ_{1}}, by construction.

520 CHAPTER 23. LINEAR FUNCTIONALS AND REPRESENTATIONS

Combining (23.7.1) with (23.6.4), we get that ρ_{(a,t)}^{V_{λ_{1}}}(q_{λ_{1}}(e_{A_{1}})) = ρ_{a}^{W_{λ_{1}}} (w_{0}) + t w_{0} + t z_{0}. (23.7.2) It follows that the set of vectors of the form ρ_{a}^{W_{λ_{1}}} (w_{0}) + t w_{0}, (23.7.3) with a ∈ A and t ∈ R or C, as appropriate, is dense in W_{λ_{1}}. Of course, (23.7.3) is the same as the image of (23.7.2) under the orthogonal projection from V_{λ_{1}} onto W_{λ_{1}}, whose kernel is Z_{λ_{1}}.

In order to get (23.6.10), we would like to show that the set of vectors of the form ρ_{a}^{W_{λ_{1}}} (w_{0}), a ∈ A, is dense in W_{λ_{1}}. Suppose that η ∈ W_{λ_{1}} satisfies ⟨η, ρ_{a}^{W_{λ_{1}}} (w_{0})⟩_{V_{λ_{1}}} = 0 (23.7.4) for every a ∈ A. It suffices to show that η = 0 under these conditions.

Let y, z ∈ A and t ∈ R or C, as appropriate, be given, and consider a = y^{∗_{A}} z + t y^{∗_{A}}, (23.7.5) which is an element of A. In this case, ⟨η, ρ_{a}^{W_{λ_{1}}} (w_{0})⟩_{V_{λ_{1}}} = ⟨ρ_{y}^{W_{λ_{1}}} (η), ρ_{z}^{W_{λ_{1}}} (w_{0}) + t w_{0}⟩_{V_{λ_{1}}} . (23.7.6) Thus (23.7.4) implies that ⟨ρ_{y}^{W_{λ_{1}}} (η), ρ_{z}^{W_{λ_{1}}} (w_{0}) + t w_{0}⟩_{V_{λ_{1}}} = 0. (23.7.7) It follows that ρ_{y}^{W_{λ_{1}}} (η) = 0, (23.7.8) because ρ_{y}^{W_{λ_{1}}} (η) ∈ W_{λ_{1}}, and the set of elements of W_{λ_{1}} of the form ρ_{z}^{W_{λ_{1}}} (w_{0}) + t w_{0} (23.7.9) is dense in W_{λ_{1}}, as before.

Equivalently, ρ_{(y,0)}^{V_{λ_{1}}}(η) = 0 (23.7.10) for every y ∈ A, by (23.6.9). This means that η ∈ Z_{λ_{1}}, (23.7.11) as in (23.6.1). This implies that η = 0, because η ∈ W_{λ_{1}}, and W_{λ_{1}} is the orthogonal complement of Z_{λ_{1}} in V_{λ_{1}}, as in (23.6.2). Thus (23.6.10) holds, as in [151].

Of course, W_{λ_{1}} = V_{λ_{1}} (23.7.12)

23.8. SOME REMARKS ABOUT V_{λ}, W_{λ} 521 if and only if Z_{λ_{1}} = {0}. In this case, z_{0} = 0, so that q_{λ_{1}}(e_{A_{1}}) = w_{0}, (23.7.13) by (23.6.4). Conversely, if z_{0} = 0, then (23.7.12) holds, because the set of vectors of the form (23.7.2) is dense in V_{λ_{1}}, as before.

Remark 4.7.4 on p125 of [11] also deals with Banach algebras with involution that may not have a multiplicative identity element. In Theorem 4.3 on p273 of [291], A is a complex C^{∗} algebra that may not have a multiplicative identity element.

23.8 Some remarks about V_{λ}, W_{λ}

Let us return to the discussion in Section 23.5, and suppose for the moment that A has a multiplicative identity element e_{A}. If z ∈ A is self-adjoint, then |λ(z)| ≤ λ(e_{A}) ∥z∥_{A}, (23.8.1) as in Section 18.11. This implies that λ(x^{∗_{A}} x) ≤ λ(e_{A}) ∥x^{∗_{A}} x∥_{A} ≤ λ(e_{A}) ∥x^{∗_{A}}∥_{A} ∥x∥_{A} ≤ C λ(e_{A}) ∥x∥_{A}^{2} (23.8.2) for every x ∈ A. It follows that b_{λ}(q_{λ}(x), q_{λ}(x)) = B_{λ}(x, x) = λ(x^{∗_{A}} x) ≤ C λ(e_{A}) ∥x∥_{A}^{2} (23.8.3) for every x ∈ A. This means that (23.8.4) q_{λ} is bounded as a linear mapping from A onto A/ ker B_{λ}, with respect to the norm associated to b_{λ} on A/ ker B_{λ}, so that (23.8.5) q_{λ} is bounded as a linear mapping from A into V_{λ}, with respect to the norm associated to the inner product.

In particular, if (23.8.6) A is separable with respect to the metric associated to ∥ · ∥_{A}, then (23.8.7) A/ ker B_{λ} is separable with respect to the metric associated to the norm corresponding to b_{λ}. In this case, (23.8.8) V_{λ} is separable with respect to the metric associated to the norm corresponding to the inner product defined earlier.

Now let us return to the discussion in Section 23.6, so that A does not necessarily have a multiplicative identity element, and we consider the algebra A_{1} associated to A as before. Note that

(23.8.9) q_{λ_{1}} is bounded as a linear mapping from A_{1} into V_{λ_{1}},

with respect to the appropriate norms, as in (23.8.5). Remember that q_{λ_{1}}(A_{1}) = A_{1}/ ker B_{λ_{1}} is dense in V_{λ_{1}}, by construction. Of course, the orthogonal projection from V_{λ_{1}} onto W_{λ_{1}} is a bounded linear mapping too. Composing these two mappings, we get a bounded linear mapping from A_{1} onto a dense linear subspace of W_{λ_{1}}.

If (23.8.6) holds, then it is easy to see that

(23.8.10) A_{1} is separable with respect to the metric associated to ∥ · ∥_{A_{1}},

where ∥ · ∥_{A_{1}} is as in Section 6.15. This implies that

(23.8.11) V_{λ_{1}} is separable

with respect to the metric associated to the norm corresponding to the inner product mentioned earlier, as in (23.8.8). It follows that

(23.8.12) W_{λ_{1}} is separable

under these conditions, with respect to the metric associated to the norm corresponding to the restriction of the inner product on V_{λ_{1}} to W_{λ_{1}}.

Separability of algebras over the real or complex numbers with submultiplicative norms will be discussed further in Section 24.1.

23.9 Some remarks about convex sets

Let (V, ⟨·, ·⟩_{V}) be a Hilbert space over the real numbers, with the associated norm ∥ · ∥_{V}, and let E be a nonempty closed convex set in V. Of course, a Hilbert space over the complex numbers may be considered as a Hilbert space over the real numbers, with the same norm, using the real part of the complex inner product.

If y ∈ V, then there is a z ∈ E such that

∥y − z∥_{V} = dist(y, E), (23.9.1)

as in Section 2.15. Suppose that w ∈ E, so that

(1 − t) z + t w ∈ E (23.9.2)

for every t ∈ R with 0 ≤ t ≤ 1. Equivalently,

z + t (w − z) ∈ E (23.9.3)

23.9. SOME REMARKS ABOUT CONVEX SETS 523

for every t ∈ [0, 1]. This implies that ∥y − z∥_{V} ≤ ∥y − z + t (z − w)∥_{V} (23.9.4) for every t ∈ [0, 1]. This means that

(23.9.5) ∥y − z∥_{V}^{2} ≤ ∥y − z + t (z − w)∥_{V}^{2}
= ∥y − z∥_{V}^{2} + 2 t ⟨y − z, z − w⟩_{V} + t^{2} ∥z − w∥_{V}^{2}

for every t ∈ [0, 1].

It follows that ⟨y − z, z − w⟩_{V} ≥ 0 (23.9.6) for every w ∈ E, as in calculus. Suppose now that y ∉ E, so that y ≠ z. (23.9.7) Put v(r) = z + r (z − y) (23.9.8) for each r ∈ R. Note that ∥v(r) − y∥_{V} = |r + 1| ∥y − z∥_{V} (23.9.9) for every r. If w ∈ E, then

∥v(r) − w∥_{V}^{2} = ∥r (z − y) + z − w∥_{V}^{2}
(23.9.10)  = r^{2} ∥z − y∥_{V}^{2} + 2 r ⟨z − y, z − w⟩_{V} + ∥z − w∥_{V}^{2}
≤ r^{2} ∥z − y∥_{V}^{2} + ∥z − w∥_{V}^{2} ,

using (23.9.6) in the third step.

Suppose that E is also bounded in V, so that ∥z − w∥_{V} ≤ C (23.9.11) for some C ≥ 0 and every w ∈ E. Combining this with (23.9.10), we get that ∥v(r) − w∥_{V}^{2} ≤ r^{2} ∥z − y∥_{V}^{2} + C^{2} (23.9.12) for every w ∈ E. We also have that

(23.9.13) ∥v(r) − y∥_{V}^{2} = (r + 1)^{2} ∥y − z∥_{V}^{2}
= r^{2} ∥y − z∥_{V}^{2} + 2 r ∥y − z∥_{V}^{2} + ∥y − z∥_{V}^{2}

for every r ∈ R, by (23.9.9). If r > 0 is sufficiently large, then (23.9.13) is strictly larger than the right side of (23.9.12), because y ≠ z. This shows that E is contained in the closed ball in V centered at v(r) with radius (r^{2} ∥y − z∥_{V}^{2} + C^{2})^{1/2} (23.9.14) for each r, and that y is not in this ball when r > 0 is sufficiently large.

This corresponds to Remark 1.10.2 on p28 of [11]. More precisely, that remark is stated for the case where V is the complex plane, which will be used in the next section.

23.10 Some bounded linear functionals

Let X be a nonempty compact Hausdorff topological space, and let A be the algebra C(X, R) of continuous real-valued functions on X, or the algebra C(X, C) of continuous complex-valued functions on X, equipped with the supremum norm. As usual, 1_{X} denotes the constant function on X equal to 1 at every point, which is the multiplicative identity element of A. Suppose that λ is a bounded linear functional on A such that

∥λ∥_{A′} = λ(1_{X}) = 1, (23.10.1)

where the left side is the dual norm of λ with respect to the supremum norm on A, as in Section 3.1. If f ∈ A, then Lemma 1.10.3 on p28 of [11] states that

λ(f) ∈ \overline{conv f(X)}, (23.10.2)

where conv f(X) is the convex hull of f(X) in R or C, as appropriate. More precisely, this is stated in [11] in the complex case, and the real case is a bit simpler.

To see this, let w ∈ R or C be given, as appropriate, and suppose that

|f(x) − w| ≤ R (23.10.3)

for some R ≥ 0 and all x ∈ X. This is the same as saying that the supremum norm of f − w 1_{X} is less than or equal to R. This implies that

|λ(f − w 1_{X})| ≤ R, (23.10.4)

because of (23.10.1). Equivalently, this means that

|λ(f) − w| ≤ R, (23.10.5)

using (23.10.1) again. One can get (23.10.2) from this and the remarks in the previous section, as in [11].

In the complex case, (23.10.2) implies in particular that

λ(f) ∈ R (23.10.6)

when f is a continuous real-valued function on X. In both the real and complex cases, if f is a nonnegative real-valued continuous function on X, then (23.10.2) implies that λ(f) ≥ 0. (23.10.7)

This means that λ is a nonnegative linear functional on A in the sense of Section 18.9. Remember that we can define an algebra involution on A using the identity mapping on A in the real case, and the mapping that sends a complex-valued continuous function on X to its complex conjugate in the complex case. It follows that λ is a positive linear functional on A with respect to this involution as in Section 18.7 in both cases, as before.

23.11. POSITIVITY ON C^{∗} ALGEBRAS 525

Alternatively, it is well known that any bounded linear functional λ on C(X, R) or C(X, C) may be expressed as

λ(f) = ∫_{X} f dν (23.10.8)

for a unique real or complex Borel measure on X, as appropriate, with suitable regularity properties. This is a version of another type of Riesz representation theorem. More precisely, the dual norm of λ is equal to |ν|(X), where |ν| is the total variation measure on X associated to ν. Using this representation, (23.10.1) is the same as saying that

|ν|(X) = ν(X) = 1. (23.10.9)

One can check that ν is a nonnegative real measure on X under these conditions.

23.11 Positivity on C^{∗} algebras

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra, as in Section 12.6, with a multiplicative identity element e_{A}. Suppose that λ is a bounded linear functional on A, with

∥λ∥_{A′} = λ(e_{A}) = 1, (23.11.1)

where ∥ · ∥_{A′} is the dual norm associated to ∥ · ∥_{A} as in Section 3.1. Under these conditions, Corollary 1 on p128 of [11] states that

(23.11.2) λ is a positive linear functional on A,

with respect to the involution x ↦ x^{∗_{A}} on A, as a C^{∗} algebra.

If A is commutative, then A is equivalent to the algebra of continuous complex-valued functions on a nonempty compact Hausdorff topological space equipped with the supremum norm, as in Sections 12.10 and 12.11. In this case, (23.11.2) follows from the remarks in the previous section.

Suppose for the moment that z ∈ A is normal, as in Section 7.5, so that z commutes with z^{∗_{A}}. Let B = B(z) be the closure of the subalgebra of A generated by e_{A}, z, and z^{∗_{A}}. This is a C^{∗} algebra with respect to the restriction of the norm and involution on A to B, and B is commutative, because z is normal.

Thus B is equivalent to the algebra of continuous complex-valued functions on a nonempty compact Hausdorff topological space with the supremum norm, as before. More precisely, one can take this compact Hausdorff space to be the spectrum σ(z) of z as an element of B, as in Section 13.9. This is the same as the spectrum σ_{A}(z) of z as an element of A, as in Section 13.10.

Note that the restriction of λ to B has dual norm equal to one, with respect to the restriction of ∥ · ∥_{A} to B. This permits us to use the remarks in the previous section, for the corresponding linear functional on the appropriate algebra of continuous complex-valued functions. It follows that

(23.11.3) λ(z) ∈ \overline{conv σ_{B}(z)} = \overline{conv σ_{A}(z)}, because of (23.10.2), as in [11]. In particular,

λ(z) ∈ R (23.11.4)

when z is self-adjoint. If σ_{A}(z) is contained in the set of nonnegative real numbers, then λ(z) ≥ 0. (23.11.5)

If a ∈ A, then a^{∗_{A}} a is self-adjoint, and thus normal. We can take z = a^{∗_{A}} a in the previous paragraphs, to get that

λ(a^{∗_{A}} a) ∈ R. (23.11.6)

Remember that σ_{A}(a^{∗_{A}} a) is contained in the set of nonnegative real numbers, as in Sections 14.14 and 14.15. This implies that

λ(a^{∗_{A}} a) ≥ 0, (23.11.7)

as in (23.11.5).

A version of (23.11.2) when A does not have a multiplicative identity element is given in 4.1 on p271 of [291]. The analogue of the argument in [291] for the case where A has a multiplicative identity element will be discussed in the next section. If A does not have a multiplicative identity element, then Corollary 1 on p271 deals with extending a bounded positive linear functional on A to the C^{∗} algebra obtained by adjoining a multiplicative identity element to A as in Section 18.5. This is a variant of the remarks in Section 18.6 for C^{∗} algebras.

23.12 Another argument to get positivity

Let A be a C^{∗} algebra with multiplicative identity element e_{A} again, and let λ be a bounded linear functional on A that satisfies (23.11.1). We would like to mention another way to get (23.11.2), which is a simplification of part of the proof of 4.1 on p271 of [291], as in the previous section.

If x ∈ A is self-adjoint and t ∈ R, then

(23.12.1)  ∥x − i t e_{A}∥_{A}^{2} = ∥(x − i t e_{A})^{∗_{A}} (x − i t e_{A})∥_{A}
= ∥(x + i t e_{A}) (x − i t e_{A})∥_{A},

using the C^{∗} identity in the first step. This implies that

∥x − i t e_{A}∥_{A}^{2} = ∥x^{2} + t^{2} e_{A}∥_{A} ≤ ∥x∥_{A}^{2} + t^{2}. (23.12.2)

This means that |λ(x − i t e_{A})|^{2} ≤ ∥x∥_{A}^{2} + t^{2}, (23.12.3)

because of (23.11.1). We also have that

λ(x − i t e_{A}) = λ(x) − i t, (23.12.4)

23.12. ANOTHER ARGUMENT TO GET POSITIVITY 527 because of (23.11.1). Thus

|λ(x − i t e_{A})|^{2} = (Re λ(x))^{2} + (Im λ(x) − t)^{2}
(23.12.5)  = (Re λ(x))^{2} + (Im λ(x))^{2} − 2 t Im λ(x) + t^{2}.

Combining (23.12.3) and (23.12.5), we obtain that (Re λ(x))^{2} + (Im λ(x))^{2} − 2 t Im λ(x) ≤ ∥x∥_{A}^{2} (23.12.6) for every t ∈ R. It follows that Im λ(x) = 0. (23.12.7)

Remember that the spectrum σ_{A}(x) of x with respect to A is contained in the real line, because x is self-adjoint and A is a C^{∗} algebra, as in Section 12.8. More precisely, σ_{A} ⊆ [−∥x∥_{A}, ∥x∥_{A}], (23.12.8) because the absolute value of every element of σ_{A}(x) is less than or equal to ∥x∥_{A}, as in Section 6.8. Suppose that σ_{A}(x) is contained in the set of nonnegative real numbers, so that σ_{A}(x) ⊆ [0, ∥x∥_{A}]. (23.12.9) This implies that σ_{A}(∥x∥_{A} e_{A} − x) ⊆ [0, ∥x∥_{A}]. (23.12.10) It follows that ∥∥x∥_{A} e_{A} − x∥_{A} ≤ ∥x∥_{A}, (23.12.11) because ∥x∥_{A} e_{A} − x is self-adjoint, and thus normal, and A is a C^{∗} algebra, as mentioned in Section 14.12.

Note that λ(∥x∥_{A} e_{A} − x) = ∥x∥_{A} − λ(x). (23.12.12) We also have that |λ(∥x∥_{A} e_{A} − x)| ≤ ∥x∥_{A}, (23.12.13) because of (23.11.1) and (23.12.11). This implies that λ(x) ≥ 0. (23.12.14) If a ∈ A, then a^{∗_{A}} a is self-adjoint, and its spectrum is contained in the set of nonnegative real numbers, as in Sections 14.14 and 14.15. It follows that (23.11.7) holds, as in (23.12.14).

23.13 Getting some positive linear functionals

Let A be a C^{∗} algebra with multiplicative identity element e_{A}, and suppose that z ∈ A is normal. Under these conditions, there is a bounded linear functional λ on A that satisfies (23.11.1) and

|λ(z)| = ∥z∥_{A}. (23.13.1)

Of course, λ is a positive linear functional on A, as in the previous two sections. This corresponds to the lemma on p272 of [291].

If a ∈ A, then a^{∗_{A}} a is self-adjoint, and thus normal. If we take z = a^{∗_{A}} a, then we get that λ(a^{∗_{A}} a) = ∥a^{∗_{A}} a∥_{A} = ∥a∥_{A}^{2}. (23.13.2)

More precisely, this uses the fact that λ(a^{∗_{A}} a) is a nonnegative real number, because λ is a positive linear functional on A. This corresponds to Corollary 2 on p128 of [11].

If A is commutative, then one can find a homomorphism λ from A onto C that satisfies (23.13.1), because ∥z∥_{A} is the same as the maximum of the absolute values of the elements of the spectrum of z, as mentioned in Section 14.12. This uses the fact that the elements of the spectrum of z correspond to the values of homomorphisms from A onto C at z, as in Section 6.12. Alternatively, one can use the fact that A is isometrically isomorphic to the algebra of continuous complex-valued functions on a nonempty compact Hausdorff topological space, as in Sections 12.10 and 12.11.

Otherwise, let B = B(z) be the closure of the subalgebra of A generated by e_{A}, z, and z^{∗_{A}}, as in Section 23.11. This is a commutative C^{∗} algebra, because z is normal, as before. The previous argument implies that there is a bounded linear functional λ on B whose dual norm is equal to λ(e_{A}) = 1, and which satisfies (23.13.1). The Hahn–Banach theorem implies that there is an extension of λ to a bounded linear functional on A with the same dual norm, as in Section 3.12.

Theorem 12.39 on p319 of [286] gives another approach to getting positive linear functionals on A that satisfy λ(e_{A}) = 1 and (23.13.2), and this will be discussed in the next section.

23.14 Some remarks about positivity

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra with involution x ↦ x^{∗_{A}} and multiplicative identity element e_{A}, as before. The set of self-adjoint elements of A may be denoted A_{r}, as on p319 of [286], or A_{sa}, as on p262, 267 of [291]. This is a real-linear subspace of A, as in Section 7.5.

Remember that a self-adjoint element of A is said to be nonnegative if its spectrum with respect to A is contained in the set of nonnegative real numbers, as in Section 14.10. The set of nonnegative self-adjoint elements of A may be denoted P = P (A), as on p319 of [286], or A_{+}, as on p267 of [291]. If a is a nonnegative self-adjoint element of A, and t is a nonnegative real number, then it is easy to see that t a is a nonnegative self-adjoint element of A too. If b is another nonnegative self-adjoint element of A, then a + b is nonnegative and self-adjoint as well, as in Section 14.12. Thus the set of nonnegative self-adjoint elements of A is a convex cone in the space of all self-adjoint elements of A, as in Section 3.7.

If a ∈ A, then a^{∗_{A}} a is self-adjoint, and it is also nonnegative, as in Sections 14.14 and 14.15. This is simpler when a is self-adjoint, as in Section 14.10.

Suppose that b ∈ A is self-adjoint, and let B = B(b) the closure of the subalgebra of A generated by e_{A} and b. Note that B is invariant under the involution on A, so that the spectrum of any element of B with respect to B is the same as its spectrum with respect to A, as in Section 13.10. Because B is a commutative C^{∗} algebra, it is isometrically ∗-isomorphic to the algebra of continuous complex-valued functions on a nonempty compact Hausdorff topological space X, as in Section 12.10 and 12.11. More precisely, one can take X to be the spectrum of b, as in Section 13.9. One can use this to express b as the square of a nonnegative self-adjoint element of B.

Remember that a linear functional λ on A is said to be Hermitian symmetric with respect to the involution on A if λ(x^{∗_{A}}) = \overline{λ(x)} for every x ∈ A, as in Section 23.4. This implies that λ(x) ∈ R when x is self-adjoint, so that the restriction of λ to the self-adjoint elements of A defines a linear functional on A_{r}, as a vector space over the real numbers. It is easy to see that λ is uniquely determined by its restriction to A_{r}, and that any linear functional on A_{r}, as a vector space over the real numbers, can be extended to a Hermitian symmetric linear functional on A.

A Hermitian symmetric linear functional on A is positive if λ(a^{∗_{A}} a) ≥ 0 for every a ∈ A, as in Section 18.7. This is the same as saying that

λ(b) ≥ 0 (23.14.1)

for every b ∈ A that is nonnegative and self-adjoint, by the previous remarks. In the proof of Theorem 12.39 on p319 of [286], only the fact that (23.14.1) implies positivity is needed. In order to get a positive linear functional on A, it is enough to obtain a linear functional on A_{r}, as a vector space over the real numbers, that satisfies (23.14.1).

If a ∈ A, then we would like to get a positive linear functional λ on A that satisfies λ(e_{A}) = 1 and (23.13.2), as in the previous section. It suffices to obtain a linear functional on A_{r}, as a vector space over the real numbers, with the analogous properties, as in the preceding paragraph. The argument in [286] starts by defining such a linear functional on the linear span of e_{A} and a^{∗_{A}} a. This linear functional can be extended to larger linear subspaces of A_{r} while maintaining the positivity condition (23.14.1), as in the Hahn–Banach theorem.

23.15 Some related ∗-representations

Let (A, ∥·∥_{A}) be a C^{∗} algebra with involution x ↦ x^{∗_{A}} and multiplicative identity element e_{A} again, and let λ be a positive linear functional on A. As in Section 23.5, there are a complex Hilbert space (V_{λ}, ⟨·, ·⟩_{V_{λ}}) and a ∗-representation ρ^{V_{λ}} of A on V_{λ} with the following properties. First,

ρ_{e_{A}}^{V_{λ}} = I_{V_{λ}}, (23.15.1)

the identity mapping on V_{λ}. Second,

∥ρ_{a}^{V_{λ}}∥_{op,V_{λ}V_{λ}} ≤ ∥a∥_{A} (23.15.2)

for every a ∈ A, because the involution on A preserves the norm, as in Sections 7.7 and 12.6. Third,

⟨ρ_{a}^{V_{λ}}(q_{λ}(e_{A})), q_{λ}(e_{A})⟩_{V_{λ}} = λ(a) (23.15.3)

for every a ∈ A, where q_{λ}(e_{A}) is an element of V_{λ} that was defined in the construction.

It follows from (23.15.3) that

∥ρ_{a}^{V_{λ}}(q_{λ}(e_{A}))∥_{A}^{2} = ⟨ρ_{a^{∗_{A}} a}^{V_{λ}}(q_{λ}(e_{A})), q_{λ}(e_{A})⟩_{V_{λ}} = λ(a^{∗_{A}} a) (23.15.4)

for every a ∈ A, where ∥·∥_{V_{λ}} is the norm on V_{λ} associated to the inner product, as usual. This implies that

∥q_{λ}(e_{A})∥_{V_{λ}}^{2} = λ(e_{A}), (23.15.5)

because of (23.15.1). In particular,

∥q_{λ}(e_{A})∥_{V_{λ}} = 1 (23.15.6)

when λ(e_{A}) = 1. (23.15.7)

If a ∈ A satisfies λ(a^{∗_{A}} a) = ∥a∥_{A}^{2}, (23.15.8)

then ∥ρ_{a}^{V_{λ}}(q_{λ}(e_{A}))∥_{A} = ∥a∥_{A}, (23.15.9)

by (23.15.4). This means that

∥ρ_{a}^{V_{λ}}∥_{op,V_{λ}V_{λ}} = ∥a∥_{A} (23.15.10)

when (23.15.7) and (23.15.8) hold.

Of course, for each a ∈ A, there is a positive linear functional λ = λ_{a} on A that satisfies (23.15.7) and (23.15.8), as in the previous two sections. This leads to a Hilbert space V_{λ} = V_{λ_{a}} and a representation ρ^{V_{λ}} = ρ^{V_{λa}} of A as in the previous paragraphs, for which (23.15.10) holds. This corresponds to Theorem 23.15. SOME RELATED ∗-REPRESENTATIONS 531

12.40 on p320 of [286], and it is also mentioned on p128 of [11]. This corresponds to part of the proof of the corollary on p274 of [291] as well.

One can use this to get that A is isometrically ∗-isomorphic to a closed subalgebra of the algebra of bounded linear mappings on a complex Hilbert space that is invariant with respect to taking adjoints, as in Theorem 4.8.4 on p129 of [11], Theorem 12.41 on p322 of [286], and the corollary on p274 of [291].

Involutions and representations

24.1 Some subsets of algebras

Let k be a field, and let A be an algebra over k. Also let E_{1} be a nonempty subset of A. If n ≥ 2 is an integer, then let E_{n} be the set of elements of A that can be obtained by taking the product of n elements of E_{1}, which need not be distinct. It is easy to see that the linear span of

⋃_{n=1}^{∞} E_{n} (24.1.1)

in A is a subalgebra of A. This is the smallest subalgebra that contains E_{1}, which may also be described as the subalgebra of A generated by E_{1}. If A has a multiplicative identity element e_{A}, then one can take E_{0} = {e_{A}}. In this case, the linear span of

⋃_{n=0}^{∞} E_{n} (24.1.2)

is the smallest subalgebra of A that contains E_{1} and e_{A}.

If E_{1} has only finitely many elements, then it is easy to see that E_{n} has only finitely many elements for each n. This implies that (24.1.1), (24.1.2) have only finitely or countably many elements, as appropriate. If E_{1} is countably infinite, then one can check that E_{n} has only finitely or countably many elements for each n. This can be obtained from the fact that the set of n-tuples of elements is countably infinite when n ≥ 1. It follows that (24.1.1), (24.1.2) are countably infinite, as appropriate.

Suppose now that k = R or C, and that ∥ · ∥_{A} is a norm on A. Suppose also that the subalgebra of A generated by E_{1} is dense in A, with respect to the metric associated to ∥ · ∥_{A}, or that A has a multiplicative identity element e_{A}, and that the subalgebra of A generated by E_{1} and e_{A} is dense in A. In both cases, we get that (24.1.3) A is separable,

with respect to the metric associated to ∥ · ∥_{A}, as in Section 1.14.

Suppose that x ↦ x^{∗_{A}} is an involution on A, which may be conjugate-linear when k = C. If E is a subset of A, then put

E^{∗_{A}} = {x^{∗_{A}} : x ∈ E}. (24.1.4)

The subalgebra of A generated by

E_{1} = E ∪ E^{∗_{A}} (24.1.5)

is the smallest subalgebra of A that contains E and is invariant under the involution. If A has a multiplicative identity element, then the subalgebra of A generated by E_{1} and e_{A} is the smallest subalgebra of A that contains E and e_{A} and is invariant under the involution.

Of course, if E has only finitely or countably many elements, then (24.1.5) has the same property. Thus (24.1.3) holds under the conditions mentioned previously. This basically corresponds to Exercise (3) on p129 of [11].

24.2 Some separability conditions

Let k be a field, let I be a nonempty set, and let V_{j} be a vector space over k for each j ∈ I. If V_{j} has finite dimension for each j ∈ I, and if I has only finitely many elements, then the direct sum of the V_{j}’s, j ∈ I, has finite dimension too, which is the sum of the dimensions of the V_{j}’s. If V_{j} is the linear span of finitely or countably many elements for each j, and if I has only finitely or countably many elements, then one can check that the direct sum of the V_{j}’s is the linear span of finitely or countably many elements as well.

Suppose now that k = R or C, and let ∥ · ∥_{V_{j}} be a norm on V_{j} for each j. Suppose also that (24.2.1) V_{j} is separable

for each j, with respect to the metric associated to ∥ · ∥_{V_{j}}. Equivalently, this means that there is a subset of V_{j} with only finitely or countably many elements whose linear span is dense in V_{j}, as in Section 1.14.

If I has only finitely or countably many elements, then one can check that

(24.2.2) the c_{0} direct sum of V_{j}, j ∈ I, is separable,

with respect to the metric associated to the appropriate norm, as in Section 21.14. Of course, it suffices to verify that there is a subset of the c_{0} direct sum with only finitely or countably many elements whose linear span is dense in the c_{0} direct sum, as before. Remember that the algebraic direct sum of the V_{j}’s is dense in the c_{0} direct sum of the V_{j}’s, as in Section 21.14. Similarly, if 1 ≤ p < ∞, then

(24.2.3) the ℓ^{p} direct sum of V_{j}, j ∈ I, is separable,

with respect to the metric assocuated to the appropriate norm under these conditions.

Let (A, ∥ · ∥_{A}) be a C^{∗} algebra with a multiplicative identity element e_{A}, and suppose that A is separable, with respect to the metric associated to ∥ · ∥_{A}. If λ is a positive linear functional on A, then there are a complex Hilbert space (V_{λ}, ⟨·, ·⟩_{V_{λ}}) and a ∗-representation ρ^{V_{λ}} of A on V_{λ} as in Section 23.5. Note that V_{λ} is separable with respect to the metric associated to the norm corresponding to the inner product, because A is separable, as in Section 23.8.

One can show that A is isometrically ∗-isomorphic to a closed subalgebra of the algebra of bounded linear mappings on a separable complex Hilbert space that is invariant with respect to taking adjoints, as in Exercise (4) on p129 of [11]. Of course, this is related to the remarks in Section 23.15.

24.3 Comparing positive linear functionals

Let A be an associative algebra over the real or complex numbers with an involution x ↦ x^{∗_{A}} that is conjugate-linear in the complex case. Also let λ, μ be positive linear functionals on A in the strict sense, as in Section 18.6. In particular, this means that λ, μ are symmetric in the strict sense with respect to the involution on A in the real case, and they are Hermitian symmetric in the strict sense in the complex case, as in Section 23.4. If c_{1} is a real number, then it follows that c_{1} λ − μ (24.3.1)

is symmetric in the strict sense in the real case, and Hermitian symmetric in the strict sense in the complex case. Similarly, if λ, μ are symmetric in the strong sense in the real case, or if they are Hermitian symmetric in the strong sense in the complex case, as in Section 23.4, then (24.3.1) has the same property.

If there is a positive real number c_{1} such that (24.3.1) is positive in the strict sense on A, then this may be expressed by

μ ⪯ λ, (24.3.2)

as in Definition 21.31 on p325 of [151]. Equivalently, this means that

μ(x^{∗_{A}} x) ≤ c_{1} λ(x^{∗_{A}} x) (24.3.3)

for every x ∈ A. Of course, this holds automatically when μ = 0, and when μ = λ, with c_{1} = 1. Let ν be another positive linear functional on A in the strict sense. If (24.3.2) holds, and if

ν ⪯ μ, (24.3.4) then it is easy to see that ν ⪯ λ, (24.3.5)

as mentioned in [151].

24.4 Linear functionals from linear mappings

Let us continue with the same notation and hypotheses as in the previous section. Suppose that (W_{λ}, ⟨·, ·⟩_{W_{λ}}) is a Hilbert space over the real or complex numbers, as appropriate, and that ρ^{W_{λ}} is a ∗-representation of A on W_{λ}. Remember that this implies that

(24.4.1) ρ_{a}^{W_{λ}} is bounded

with respect to the norm ∥ · ∥_{W_{λ}} on W_{λ} associated to the inner product for each a ∈ A, as in Section 21.6. Suppose also that w_{0} ∈ W_{λ} has the property that

⟨ρ_{a}^{W_{λ}}(w_{0}), w_{0}⟩_{W_{λ}} = λ(a) (24.4.2)

for every a ∈ A. Note that

λ(x^{∗_{A}} x) = ⟨ρ_{x}^{W_{λ}}(w_{0}), ρ_{x}^{W_{λ}}(w_{0})⟩_{W_{λ}} = ∥ρ_{x}^{W_{λ}}(w_{0})∥_{W_{λ}}^{2} (24.4.3)

for each x ∈ A, as in Section 22.8.

Suppose for the moment that (A, ∥ · ∥_{A}) is a Banach algebra, with

∥z^{∗_{A}}∥_{A} ≤ C ∥z∥_{A} (24.4.4)

for some C ≥ 1 and every z ∈ A. If A has a multiplicative identity element e_{A}, then one can use V_{λ}, ρ^{V_{λ}}, and q_{λ}(e_{A}) ∈ V_{λ} as in Section 23.5. If λ is positive in the strong sense on A, then one can get W_{λ}, ρ^{W_{λ}}, and w_{0} as in Section 23.6. Remember that (24.4.2) implies that λ is positive in the strong sense on A, as in Section 22.8.

Let R be a bounded linear mapping from W_{λ} into itself that commutes with ρ^{W_{λ}}, in the sense that R ∘ ρ_{a}^{W_{λ}} = ρ_{a}^{W_{λ}} ∘ R (24.4.5)

for every a ∈ A. Consider the linear functional μ_{R} defined on A by

μ_{R}(a) = ⟨R(ρ_{a}^{W_{λ}}(w_{0})), w_{0}⟩_{W_{λ}} (24.4.6)

for every a ∈ A. Suppose from now on in this section that

(24.4.7) R is self-adjoint on W_{λ}.

This implies that μ_{R} is symmetric in the strong sense on A in the real case, and that μ_{R} is Hermitian symmetric in the strong sense in the complex case.

Remember that ⟨R(v), w⟩_{W_{λ}} (24.4.8) is a symmetric bilinear form on W_{λ} in the real case, and a Hermitian symmetric sesquilinear form on V in the complex case, as in Section 8.9. Suppose from now on in this section that

(24.4.9) R is also nonnegative

with respect to the inner product on W_{λ}, as in Section 8.10. This means that (24.4.8) is nonnegative as a symmetric bilinear form or Hermitian symmetric sesquilinear form on W, as appropriate. It follows that

|⟨R(v), w⟩_{W_{λ}}| ≤ ⟨R(v), v⟩_{W_{λ}}^{1 2} ⟨R(w), w⟩_{W_{λ}}^{1/2} (24.4.10)

for every v, w ∈ W_{λ}, as before.

If x ∈ A, then it is easy to see that

μ_{R}(x^{∗_{A}} x) = ⟨R(ρ_{x}^{W_{λ}}(w_{0})), ρ_{x}^{W_{λ}}(w_{0})⟩_{W_{λ}}. (24.4.11)

This implies that μ_{R} is positive in the strict sense on A, because R is nonnegative, by hypothesis. Using (24.4.10), we get that

(24.4.12) |μ_{R}(x)| ≤ ⟨R(ρ_{x}^{W_{λ}}(w_{0})), ρ_{x}^{W_{λ}}(w_{0})⟩_{W_{λ}}^{1/2} ⟨R(w_{0}), w_{0}⟩_{W_{λ}}^{1/2}
= μ_{R}(x^{∗_{A}} x)^{1/2} ⟨R(w_{0}), w_{0}⟩_{W_{λ}}^{1/2}.

It follows that (24.4.13) μ_{R} is positive in the strong sense on A,

as in Section 18.7.

Observe that

(24.4.14)  μ_{R}(x^{∗_{A}} x) ≤ ∥R(ρ_{x}^{W_{λ}}(w_{0}))∥_{W_{λ}} ∥ρ_{x}^{W_{λ}}(w_{0})∥_{W_{λ}}
≤ ∥R∥_{op,W_{λ} W_{λ}} ∥ρ_{x}^{W_{λ}}(w_{0})∥_{W_{λ}}^{2},

where ∥R∥_{op,W_{λ} W_{λ}} is the operator norm of R with respect to ∥ · ∥_{W_{λ}}, as usual. Combining this with (24.4.3), we get that

μ_{R}(x^{∗_{A}} x) ≤ ∥R∥_{op,W_{λ}W_{λ}} λ(x^{∗_{A}} x). (24.4.15)

This shows that μ_{R} ⪯ λ (24.4.16)

under these conditions. This corresponds to part of Theorem 21.32 on p325 of [151].

24.5 Linear mappings from linear functionals

Let us return to the same notation and hypotheses as in the first paragraph in the previous section. Let us also ask now that

(24.5.1) ρ^{W_{λ}} be topologically cyclic, with topologically cyclic vector w_{0}.

Remember that this holds when ρ^{W_{λ}} is obtained as in Section 23.5 and A has a multiplicative identity element e_{A}, and when λ is positive in the strong sense and ρ^{W_{λ}} is obtained as in Section 23.6.

Suppose that μ is a positive linear functional on A in the strong sense that satisfies (24.3.2). It would be nice if there were a bounded linear mapping R from W_{λ} into itself that commutes with ρ^{W_{λ}}, that is self-adjoint and nonnegative, and so that μ is the same as (24.4.6). This corresponds to another part of Theorem 21.32 on p325 of [151], under suitable conditions.

Put W = {ρ_{a}^{W_{λ}}(w_{0}) : a ∈ A}, (24.5.2)

which is a linear subspace of W_{λ}. More precisely, W is dense in W_{λ}, by hypothesis. We would like to define Φ as a real or complex-valued function on W × W, as appropriate, by Φ(ρ_{x}^{W_{λ}}(w_{0}), ρ_{y}^{W_{λ}}(w_{0})) = μ(y^{∗_{A}} x) (24.5.3)

for all x, y ∈ A, as in (2) on p326 of [151].

Suppose that x, x_{1}, y, y_{1} ∈ A satisfy ρ_{x}^{W_{λ}}(w_{0}) = ρ_{x_{1}}^{W_{λ}}(w_{0}) and ρ_{y}^{W_{λ}}(w_{0}) = ρ_{y_{1}}^{W_{λ}}(w_{0}), so that ρ_{x−x_{1}}^{W_{λ}}(w_{0}) = ρ_{y−y_{1}}^{W_{λ}}(w_{0}) = 0. (24.5.4)

We would like to verify that

μ(y^{∗_{A}} x) = μ(y_{1}^{∗_{A}} x_{1}). (24.5.5)

Observe that

λ((x − x_{1})^{∗_{A}} (x − x_{1})) = λ((y − y_{1})^{∗_{A}} (y − y_{1})) = 0, (24.5.6)

by (24.4.3) and (24.5.4). This implies that

μ((x − x_{1})^{∗_{A}} (x − x_{1})) = μ((y − y_{1})^{∗_{A}} (y − y_{1})) = 0, (24.5.7)

because of (24.3.3).

Remember that

|μ(b^{∗_{A}} a)| ≤ μ(a^{∗_{A}} a)^{1/2} μ(b^{∗_{A}} b)^{1/2} (24.5.8)

for every a, b ∈ A, as in Section 18.6. In particular, μ(b^{∗_{A}} a) = 0 when μ(a^{∗_{A}} a) = 0 or μ(b^{∗_{A}} b) = 0. It follows that

μ((y − y_{1})^{∗_{A}} x) = μ(y_{1}^{∗_{A}} (x − x_{1})) = 0, (24.5.9)

because of (24.5.7). It is easy to obtain (24.5.5) from (24.5.9).

This shows that Φ is well-defined on W × W by (24.5.3). Note that Φ is a symmetric bilinear form on W in the real case, and a Hermitian symmetric sesquilinear form on W in the complex case. If x, y ∈ A and c_{1} > 0 is as in (24.3.3), then

|Φ(ρ_{x}^{W_{λ}}(w_{0}), ρ_{y}^{W_{λ}}(w_{0}))| = |μ(y^{∗_{A}} x)| ≤ μ(x^{∗_{A}} x)^{1/2} μ(y^{∗_{A}} y)^{1/2}
(24.5.10)  ≤ c_{1} λ(x^{∗_{A}} x)^{1/2} λ(y^{∗_{A}} y)^{1/2}
= c_{1} ∥ρ_{x}^{W_{λ}}(w_{0})∥_{W_{λ}} ∥ρ_{y}^{W_{λ}}(w_{0})∥_{W_{λ}},

using (24.5.8) in the second step, and (24.4.3) in the fourth step. Thus Φ is bounded as a bilinear or sesquilinear form on W, as appropriate, as in Sections 5.13 and 5.15. It follows that Φ has a unique extension to a bounded bilinear or sesquilinear form on W_{λ}, as appropriate, as in Section 5.13, because W is dense in W_{λ}.

If we use Φ to denote this extension as well, then there is a unique bounded linear mapping R from W_{λ} into itself such that

Φ(v, w) = ⟨R(v), w⟩_{W_{λ}} (24.5.11)

for every v, w ∈ W_{λ}, as in Section 5.15. Of course, Φ is symmetric or Hermitian symmetric on W_{λ}, as appropriate, because of the corresponding property on W, and because W is dense in W_{λ}. This implies that R is self-adjoint on W_{λ}. Similarly, Φ is nonnegative on W by construction, so that Φ is nonnegative on W_{λ}, because W is dense in W_{λ}. This means that R is nonnegative as a bounded self-adjoint linear mapping on W_{λ}.

If x, y, z ∈ A, then

⟨R(ρ_{x}^{W_{λ}}(ρ_{y}^{W_{λ}}(w_{0}))), ρ_{z}^{W_{λ}}(w_{0})⟩_{W_{λ}} = μ(z^{∗_{A}} x y), (24.5.12)

by (24.5.3) and (24.5.11). Similarly,

⟨ρ_{x}^{W_{λ}}(R(ρ_{y}^{W_{λ}}(w_{0}))), ρ_{z}^{W_{λ}}(w_{0})⟩_{W_{λ}} = ⟨R(ρ_{y}^{W_{λ}}(w_{0})), ρ_{x^{∗_{A}} z}^{W_{λ}}(w_{0})⟩_{W_{λ}}
(24.5.13)  = μ((x^{∗_{A}} z)^{∗_{A}} y) = μ(z^{∗_{A}} x y).

One can use this to get that R commutes with ρ^{W_{λ}}, as in (24.4.5), because W is dense in W_{λ}.

It follows that

⟨R(ρ_{y^{∗_{A}} x}^{W_{λ}}(w_{0})), w_{0}⟩_{W_{λ}} = ⟨R(ρ_{x}^{W_{λ}}(w_{0})), ρ_{y}^{W_{λ}}(w_{0})⟩_{W_{λ}} = μ(y^{∗_{A}} x) (24.5.14)

for every x, y ∈ A. We would like to have

⟨R(ρ_{a}^{W_{λ}}(w_{0})), w_{0}⟩_{W_{λ}} = μ(a) (24.5.15)

for every a ∈ A. Of course, if A has a multiplicative identity element, then (24.5.15) follows from (24.5.14). Otherwise, an additional argument is needed, and this will be discussed in the next section.

24.6 Showing that μ = μ_{R}

Let us continue with the same notation and hypotheses as in the previous section. Using the bounded linear mapping R on W_{λ} as in (24.5.11), we can define a linear functional μ_{R} on A as in (24.4.6). This is the same as the left side of (24.5.15), so that (24.5.15) is the same as saying that

μ = μ_{R} (24.6.1)

24.6. SHOWING THAT μ = μ_{R} 539

on A. Similarly, (24.5.14) is the same as saying that

μ(y^{∗_{A}} x) = μ_{R}(y^{∗_{A}} x) (24.6.2)

for every x, y ∈ A. Note that μ_{R} is positive in the strong sense on A, as in Section 24.4.

In this section, we also ask that (A, ∥ · ∥_{A}) be a Banach algebra, and that the involution on A be continuous, so that (24.4.4) holds for some C ≥ 1. This implies that there are a Hilbert space (W_{μ}, ⟨·, ·⟩_{W_{μ}}) over the real or complex numbers, as appropriate, a ∗-representation ρ^{W_{μ}} of A on W_{μ}, and w_{0,μ} ∈ W_{μ} with the following properties, as in Sections 23.6 and 23.7. First,

(24.6.3) ρ_{a}^{W_{μ}} is bounded

with respect to the norm ∥ · ∥_{W_{μ}} on W_{μ} associated to the inner product for each a ∈ A. Second, ⟨ρ_{a}^{W_{μ}}(w_{0,μ}), w_{0,μ}⟩_{W_{μ}} = μ(a) (24.6.4)

for every a ∈ A. Third,

(24.6.5) ρ^{W_{μ}} is topologically cyclic, with topologically cyclic vector w_{0,μ}.

Similarly, there are a Hilbert space (W_{μ_{R}}, ⟨·, ·⟩_{W_{μ_{R}}}) over the real or complex numbers, as appropriate, a ∗-representation ρ^{W_{μ_{R}}} of A on W_{μ_{R}}, and w_{0,μ_{R}} in W_{μ_{R}} with the analogous properties. Namely,

(24.6.6) ρ_{a}^{W_{μ_{R}}} is bounded

with respect to the norm ∥ · ∥_{W_{μ_{R}}} on W_{μ_{R}} associated to the inner product for each a ∈ A, and ⟨ρ_{a}^{W_{μ_{R}}} (w_{0,μ_{R}}), w_{0,μ_{R}}⟩_{W_{μ_{R}}} = μ_{R}(a) (24.6.7)

for every a ∈ A. We also have that

(24.6.8) ρ^{W_{μ_{R}}} is topologically cyclic with topologically cyclic vector w_{0,μ_{R}},

as before.

Using (24.6.2), (24.6.4), and (24.6.7), we get that

⟨ρ_{z^{∗_{A}} y}^{W_{μ}}(w_{0,μ}), w_{0,μ}⟩_{W_{μ}} = ⟨ρ_{z^{∗_{A}} y}^{W_{μ_{R}}}(w_{0,μ_{R}}), w_{0,μ_{R}}⟩_{W_{μ_{R}}} (24.6.9)

for every y, a ∈ A. This implies that

⟨ρ_{x}^{W_{μ}}(w_{0,μ}), w_{0,μ}⟩_{W_{μ}} = ⟨ρ_{x}^{W_{μ_{R}}} (w_{0,μ_{R}}), w_{0,μ_{R}}⟩_{W_{μ_{R}}} (24.6.10)

for every x ∈ A, as in Sections 22.10 and 22.11. This is the same as (24.6.1), because of (24.6.4) and (24.6.7).

This corresponds to part of Theorem 21.32 on p325 of [151], as mentioned earlier. More precisely, this argument basically corresponds to part of the discussion on p327 of [151]. Part of that discussion in [151] is used here to get (24.6.10), as before.

24.7 More on positive linear functionals

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗_{A}} that is conjugate-linear in the complex case. Note that in the real case, the collections of linear functionals on A that are symmetric or antisymmetric in the strict or strong senses, as in Section 23.4, are linear subspaces of the algebraic dual of A. Similarly, in the complex case, the collections of linear functionals on A that are Hermitian symmetric in the strict or strong sense are real-linear subspaces of the algebraic dual of A.

Let λ be a positive linear functional on A in the strong sense, as in Section 18.7. Thus λ is symmetric or Hermitian symmetric, as appropriate, in the strong sense on A, and |λ(x)|^{2} ≤ c λ(x^{∗_{A}} x) (24.7.1)

for some c ≥ 0 and all x ∈ A. Put

M(λ) = sup \{\frac{|λ(x)|^{2}}{λ(x^{∗_{A}} x)} : x ∈ A, λ(x^{∗_{A}} x) > 0\}, (24.7.2)

as on p319 in [151], in the statement of Corollary 21.20. More precisely, if λ(x^{∗_{A}} x) = 0 for every x ∈ A, then λ ≡ 0 on A, because of (24.7.1), and we interpret (24.7.2) as being equal to 0. Equivalently, one can check that

(24.7.3) M(λ) = min{c ≥ 0 : (24.7.1) holds}.

If r is a nonnegative real number, then r λ is a positive linear functional on A in the strong sense, and it is easy to see that

M(r λ) = r M(λ). (24.7.4)

Note that M(λ) = 0 only when λ ≡ 0 on A. If A has a multiplicative identity element e_{A}, then M(λ) = λ(e_{A}), (24.7.5)

as in Section 18.7.

Without asking that A have a multiplicative identity element, we can take A_{1} = A × R or A × C, as appropriate, with multiplication as defined in Section 6.15. We can also define the involution

(x, t) ↦ (x, t)^{∗_{A_{1}}} (24.7.6)

on A_{1} as in Section 17.12. Remember that e_{A_{1}} = (0, 1) is the multiplicative identity element in A_{1}, as in Section 6.15.

If c is any real number, then

λ_{1,c}((x, t)) = λ(x) + t c (24.7.7)

defines a linear functional on A_{1} that is symmetric or Hermitian symmetric, as appropriate, in the strong sense, as in Section 18.8. We have also seen that λ_{1,c} is positive on A_{1} if and only if c ≥ 0 and (24.7.1) holds. This means that

(24.7.8) M(λ) = min{c ≥ 0 : λ_{1,c} is a positive linear functional on A_{1}}, 24.8. MORE ON M(λ) 541

as in (24.7.3).

If λ and μ are positive linear functionals on A in the strict sense, then

(24.7.9) λ + μ is a positive linear functional on A in the strict sense.

If λ and μ are positive linear functionals on A in the strong sense, then one can verify that

(24.7.10) λ + μ is a positive linear functional on A in the strong sense.

In fact, M(λ + μ) ≤ M(λ) + M(μ) (24.7.11)

in this case, as in Corollary 21.20 on p318 of [151].

Indeed, if λ_{1,c} and μ_{1,c} are as in (24.7.7), then λ_{1,M(λ)} and μ_{1,M(μ)} are positive linear functionals on A_{1}, by hypothesis, as before. Observe that

(λ + μ)_{1,M(λ)+M(μ)} = λ_{1,M(λ)} + μ_{1,M(μ)}, (24.7.12)

where (λ + μ)_{1,c} is defined as in (24.7.7) as well. This implies that the left side of (24.7.12) is a positive linear functional on A_{1}, as before. We can get (24.7.11) from this, as in (24.7.8).

24.8 More on M(λ)

Let us continue with the same notation and hypotheses as in the previous section. Suppose for the moment that A is a Banach algebra with respect to a norm ∥ · ∥_{A}, and that the involution on A is continuous, so that

∥y^{∗_{A}}∥_{A} ≤ C ∥y∥_{A} (24.8.1)

for some C ≥ 1 and all y ∈ A. Remember that A_{1} is a Banach algebra with respect to the corresponding norm ∥ · ∥_{A_{1}} defined in Section 6.15, and that the involution on A_{1} satisfies the analogue of (22.12.9) with respect to this norm, as in Section 17.12. Let λ be a positive linear functional in the strong sense on A, so that λ_{1,M(λ)} is a positive linear functional on A_{1}.

Under these conditions,

|λ_{1,M(λ)}((x, t))| ≤ C^{1/2} λ_{1,M(λ)}(e_{A_{1}}) ∥(x, t)∥_{A_{1}} (24.8.2)

for every (x, t) ∈ A_{1}, as in Section 18.11. This means that

|λ_{1,M(λ)}((x, t))| ≤ C^{1/2} M(λ) ∥(x, t)∥_{A_{1}} (24.8.3)

for every (x, t) ∈ A_{1}, because λ_{1,M(λ)}(e_{A_{1}}) = λ_{1,M(λ)}((0, 1)) = M(λ). In particular, we can take t = 0, to get that

|λ(x)| ≤ C^{1/2} M(λ) ∥x∥_{A} (24.8.4)

542 CHAPTER 24. INVOLUTIONS AND REPRESENTATIONS for every x ∈ A. This corresponds to part of Corollary 21.20 on p318 of [151].

Now let (V, ⟨·, ·⟩_{V}) be an inner product space over the real or complex numbers, as appropriate, and let ρ^{V} be a representation of A on V that is compatible with the inner product, in the sense that ⟨ρ_{x}^{V} (u), v⟩_{V} = ⟨u, ρ_{x^{∗_{A}}}^{V} (v)⟩_{V} (24.8.5) for every u, v ∈ V, as in Section 21.5. If v_{0} ∈ V, then λ_{ρ^{V} ,v_{0}}(x) = ⟨ρ_{x}^{V} (v_{0}), v_{0}⟩_{V} (24.8.6) is a positive linear functional on A in the strong sense, as in Section 22.8. More precisely, the same argument shows that M(λ_{ρ^{V} ,v_{0}}) ≤ ∥v_{0}∥_{V}^{2} , (24.8.7) where ∥ · ∥_{V} is the norm associated to the inner product. This corresponds to the first part of the proof of Theorem 21.25 on p323 of [151].

If ρ^{V} is topologically cyclic, with topologically cyclic vector v_{0}, then M(λ_{ρ^{V} ,v_{0}}) = ∥v_{0}∥_{V}^{2} , (24.8.8) as in Theorem 21.25 on p323 of [151]. This can be seen by considering x ∈ A such that ρ_{x}^{V} (v_{0}) approximates v_{0}. This also uses the fact that λ_{ρ^{V} ,v_{0}}(x^{∗_{A}} x) = ∥ρ_{x}^{V} (v_{0})∥_{V}^{2} (24.8.9) for every x ∈ A, as in Section 22.8.

If A has a multiplicative identity element e_{A}, then M(λ_{ρ^{V} ,v_{0}}) = λ_{ρ^{V} ,v_{0}}(e_{A}) = ⟨ρ_{e_{A}}^{V}(v_{0}), v_{0}⟩_{V} , (24.8.10) as in (24.7.5). Observe that

⟨ρ_{e_{A}}^{V}(v_{0}), v_{0} − ρ_{e_{A}}^{V}(v_{0})⟩_{V} = ⟨ρ_{e_{A}}^{V}(v_{0}), v_{0}⟩_{V} − ⟨ρ_{e_{A}}^{V}(v_{0}), ρ_{e_{A}}^{V}(v_{0})⟩_{V}
(24.8.11)  = ⟨ρ_{e_{A}}^{V}(v_{0}), v_{0}⟩_{V} − ⟨ρ_{e_{A}^{∗_{A}} e_{A}}^{V}(v_{0}), v_{0}⟩_{V} = 0,

using (24.8.5) in the second step. This implies that

(24.8.12)  ∥v_{0}∥_{V}^{2} = ⟨ρ_{e_{A}}^{V}(v_{0}), v_{0}⟩_{V} + ⟨v_{0} − ρ_{e_{A}}^{V}(v_{0}), v_{0}⟩_{V}
= M(λ_{ρ^{V} ,v_{0}}) + ∥v_{0} − ρ_{e_{A}}^{V}(v_{0})∥_{V}^{2} ,

using also (24.8.10) in the second step. Of course, if ρ_{e_{A}}^{V}(v_{0}) = v_{0}, then (24.8.8) holds. In particular, this happens when ρ_{e_{A}}^{V} is the identity mapping on V.

24.9 Some related convex sets

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗_{A}} that is conjugate-linear in the complex case, as usual. Consider the space A^{sym} of linear functionals that are symmetric or Hermitian symmetric in the strong sense, as appropriate, with respect to the involution, as in Section 23.4. This is a linear subspace of the algebraic dual A^{alg} of A as a vector space in the real case, and a real-linear subspace of A^{alg} in the complex case, as mentioned in Section 24.7.

Let A_{sa} be the space of self-adjoint elements of A, as in Section 23.14, and let A_{asa} be the space of anti-self-adjoint elements of A. These are linear subspaces of A in the real case, and real-linear subspaces of A in the complex case, as in Section 7.5. In both cases, A corresponds to the direct sum of A_{sa} and A_{asa} as a vector space over the real numbers, as in Section 7.5 again. Remember that self-adjoint elements of A are also said to be Hermitian in the complex case, as in Section 7.5, and note that A_{sa} may be denoted A_{h}, as in the statement of Theorem 21.33 on p327 of [151].

In the real case, a linear functional λ on A is symmetric in the strong sense with respect to the involution if and only of λ ≡ 0 on A_{asa}, as mentioned in Section 18.7. In the complex case, a linear functional λ on A is Hermitian symmetric in the strong sense with respect to the involution if and only if λ is real-valued on A_{sa}, as before. Of course,

A_{asa} = {i a : a ∈ A_{sa}} (24.9.1)

in the complex case, as mentioned in Section 7.5. In both the real and complex cases, if λ ∈ A^{sym}, then λ is uniquely determined by its restriction to A_{sa}.

Let μ be a linear functional on A_{sa}, as a vector space over the real numbers, in either the real or complex case. In both cases, μ can be extended to an element of A^{sym}, by defining the extension in a suitable way on A_{asa}. In the real case, one can simply define the extension to be 0 on A_{asa}. In the complex case, one can define the extension on A_{asa} by

i a ↦ i μ(a) (24.9.2)

for every a ∈ A_{sa}.

In both the real and complex cases, the map that sends λ ∈ A^{sym} to its restriction to A_{asa} defines a linear mapping from A^{sym} to the algebraic dual (A_{sa})^{alg} of A_{sa}, as a vector space over the real numbers. More precisely, this is an isomorphism from A^{sym} onto (A_{sa})^{alg}, as vector spaces over the real numbers, as in the previous two paragraphs.

Let P = P (A) be the set of positive linear functionals λ in the strong sense on A, as in Section 18.7, such that

M(λ) ≤ 1, (24.9.3)

where M(λ) is as in Section 24.7. Equivalently, P consists of the λ ∈ A^{sym} that are positive in the strict sense and satisfy

|λ(x)| ≤ λ(x^{∗_{A}} x)^{1/2} (24.9.4) for every x ∈ A, as in the statement of Theorem 21.33 on p327 of [151].

One can check that P is convex (24.9.5)

as a subset of A^{sym}, using the remarks in Section 24.7. This corresponds to part of Theorem 21.33 on p327 of [151].

Suppose now that A has a multiplicative identity element e_{A}. In this case, P is the same as the set of positive linear functionals λ on A such that

λ(e_{A}) ≤ 1, (24.9.6)

as in Section 24.7. Note that the set of positive linear functionals λ on A such that λ(e_{A}) = 1 (24.9.7)

is a convex set in A^{sym} too.

24.10 Extreme points

Let V be a vector space over the real or complex numbers, let E be a subset of V, and let u be an element of E. We say that u is an extreme point of E if for every v, w ∈ E such that u = (1 − t) v + t w (24.10.1)

for some t ∈ R with 0 < t < 1, we have that

u = v = w. (24.10.2)

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗_{A}} that is conjugate linear in the complex case again, and let P be as in the previous section. One can check that

(24.10.3) 0 is an extreme point of P,

as mentioned at the beginning of the proof of Theorem 21.34 on p328 of [151].

If λ ∈ P and 0 < M(λ) < 1, then one can express λ as a nontrivial convex combination of 0 and M(λ)^{−1} λ ∈ P, so that λ is not an extreme point of P. This means that if λ ∈ P is an extreme point of P, and if λ ≠ 0, then

M(λ) = 1, (24.10.4)

as in Theorem 21.34 on p328 of [151].

Suppose now that λ ∈ P satisfies (24.10.4). Suppose also that (W_{λ}, ⟨·, ·⟩_{W_{λ}}) is a Hilbert space over the real or complex numbers, as appropriate, ρ^{W_{λ}} is a ∗-representation of A on W_{λ}, w_{0} ∈ W_{λ}, and

⟨ρ_{a}^{W_{λ}}(w_{0}), w_{0}⟩_{W_{λ}} = λ(a) (24.10.5)

for every a ∈ A. If (24.10.6) ρ^{W_{λ}} is topologically irreducible, as in Section 20.14, then we would like to show that

(24.10.7) λ is an extreme point of P,

at least under suitable conditions, as in Theorem 21.34 on p328 of [151]. This is related to Remark 4.8.5 on p129 of [11]. Remember that ρ_{a}^{W_{λ}} is bounded on W_{λ} with respect to the norm ∥ · ∥_{W_{λ}} associated to the inner product for each a ∈ A, as in Section 21.6.

If λ is not an extreme point of P, then there are λ_{1}, λ_{2} ∈ P such that

λ = t λ_{1} + (1 − t) λ_{2} (24.10.8)

for some t ∈ (0, 1) and λ_{1} ≠ λ_{2}. Note that

M(λ) ≤ t M(λ_{1}) + (1 − t) M(λ_{2}), (24.10.9)

as in Section 24.7. This implies that

M(λ_{1}) = M(λ_{2}) = 1, (24.10.10)

because of (24.10.4), and because M(λ_{1}), M(λ_{2}) ≤ 1, by hypothesis. Using (24.10.8), we get that

t^{−1} λ − λ_{1} = t^{−1} (1 − t) λ_{2}. (24.10.11)

This means that λ_{1} ⪯ λ, (24.10.12) in the notation of 24.3.

Of course, w_{0} ≠ 0, because of (24.10.4) and (24.10.5). It follows from this and (24.10.6) that ρ^{W_{λ}} is topologically cyclic, with topologically cyclic vector w_{0}, as in Section 20.14.

Thus the conditions mentioned at the beginning of Section 24.5 are satisfied, with μ = λ_{1}. As before, it would be nice if there were a bounded linear mapping R_{1} from W_{λ} into itself that commutes with ρ^{W_{λ}}, that is self-adjoint and nonnegative, and so that

⟨R_{1}(ρ_{a}^{W_{λ}}(w_{0})), w_{0}⟩_{W_{λ}} = λ_{1}(a) (24.10.13)

for every a ∈ A. We saw in Section 24.5 that this holds when A has a multiplicative identity element. We also saw that this holds when A is a Banach algebra with respect to a norm ∥ · ∥_{A}, and the involution on A is continuous, so that (24.8.1) holds for some C ≥ 1, as in Section 24.6.

If R_{1} is as in the preceding paragraph, then

R_{1} = r_{1} I_{W_{λ}} (24.10.14)

for some real number r_{1}, where I_{W_{λ}} is the identity mapping on W_{λ}, as in Section 21.8, because of (24.10.6). More precisely, r_{1} ≥ 0 because R_{1} is nonnegative as a bounded self-adjoint linear mapping from W_{λ} into itself. It follows that

λ_{1} = r_{1} λ, (24.10.15)

because of (24.10.5) and (24.10.13). In fact, r_{1} = 1, because of (24.10.4) and (24.10.10). This means that λ_{1} = λ, so that λ_{2} = λ as well, and (24.10.7) holds.

24.11 Irreducibility from extreme points

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗_{A}} that is conjugate-linear in the complex case, and let P be as in Section 24.9 again. Suppose that λ ∈ P satisfies (24.10.4), W_{λ} is a Hilbert space over the real or complex numbers, as appropriate, and that ρ^{W_{λ}} is a ∗-representation of A. Suppose also that ρ^{W_{λ}} is topologically cyclic, with topologically cyclic vector w_{0} ∈ W_{λ}, as in Section 20.13, and that (24.10.5) holds for every a ∈ A.

If λ is an extreme point of P, then we would like to show that ρ^{W_{λ}} is topologically irreducible, as in Theorem 21.34 on p328 of [151]. This is related to Remark 4.8.5 on p129 of [11], as before.

If ρ^{W_{λ}} is not topologically irreducible, then there is a nonzero proper closed linear subspace Z_{1} of W_{λ} that is invariant under ρ^{W_{λ}}. Let

Z_{2} = (Z_{1})^{⊥} (24.11.1)

be the orthogonal complement of Z_{1} in W_{λ}, so that Z_{2} is invariant under ρ^{W_{λ}} as well, as in Section 21.7. Of course,

w_{0} = z_{1} + z_{2} (24.11.2)

for some z_{1} ∈ Z_{1} and z_{2} ∈ Z_{2}, as in Section 2.15. It is easy to see that

z_{1}, z_{2} ≠ 0, (24.11.3)

because ρ^{W_{λ}} is topologically cyclic with topologically cyclic vector w_{0}, by hypothesis.

Put λ_{j}(a) = ⟨ρ_{a}^{W_{λ}}(z_{j}), z_{j}⟩_{W_{λ}} (24.11.4)

for j = 1, 2 and each a ∈ A. Note that λ_{1}, λ_{2} are positive linear functionals on A in the strong sense, as in Section 22.8. We also have that

λ = λ_{1} + λ_{2} (24.11.5)

on A, because of (24.10.5) and (24.11.2), and because Z_{1}, Z_{2} are invariant under ρ^{W_{λ}}.

More precisely, M(λ_{j}) ≤ ∥z_{j}∥_{W_{λ}}^{2} (24.11.6)

for j = 1, 2, as in Section 24.8, where ∥·∥_{W_{λ}} is the norm on W_{λ} associated to the inner product, as usual. Because ρ^{W_{λ}} is topologically cyclci with topologically cyclic vector w_{0}, M(λ) = ∥w_{0}∥_{W_{λ}}^{2}, (24.11.7)

as in Section 24.8 again. It follows that

M(λ_{1}) + M(λ_{2}) ≤ ∥z_{1}∥_{W_{λ}}^{2} + ∥z_{2}∥_{W_{λ}}^{2} = ∥w_{0}∥_{W_{λ}}^{2} = M(λ). (24.11.8)

24.12. SHOWING THAT λ IS NOT AN EXTREME POINT 547

This means that M(λ) = M(λ_{1}) + M(λ_{2}), (24.11.9)

because the left side is less than or equal to the right side, as in Section 24.7. In fact, M(λ_{j}) = ∥z_{j}∥_{W_{λ}}^{2} (24.11.10)

for j = 1, 2, because of (24.11.9).

In particular, M(λ_{j}) ≤ M(λ) = 1 (24.11.11)

for j = 1, 2, so that λ_{1}, λ_{2} ∈ P. (24.11.12)

Observe that λ_{1}, λ_{2} ≠ 0, (24.11.13)

by (24.11.3) and (24.11.10).

Alternatively, because ρ^{W_{λ}} is topologically cyclic, with topologically cyclic vector w_{0}, there is an x ∈ A such that

∥ρ_{x}^{W_{λ}}(w_{0}) − z_{1}∥_{W_{λ}} < ∥z_{1}∥_{W_{λ}}. (24.11.14)

Note that ⟨ρ_{x}^{W_{λ}}(w_{0}), z_{1}⟩_{W_{λ}} = ⟨ρ_{x}^{W_{λ}}(z_{1}), z_{1}⟩_{W_{λ}} (24.11.15)

because of (24.11.2), and the fact that ρ_{x}^{W_{λ}}(z_{2}) ∈ Z_{2}. Using this and the definition (24.11.4) of λ_{1}(x), we obtain that

(24.11.16) λ_{1}(x) − ∥z_{1}∥_{W_{λ}}^{2} = ⟨ρ_{x}^{W_{λ}}(z_{1}), z_{1}⟩_{W_{λ}} − ⟨z_{1}, z_{1}⟩_{W_{λ}}
= ⟨ρ_{x}^{W_{λ}}(w_{0}) − z_{1}, z_{1}⟩_{W_{λ}}.

It follows that

|λ_{1}(x) − ∥z_{1}∥_{W_{λ}}^{2} | ≤ ∥ρ_{x}^{W_{λ}}(w_{0}) − z_{1}∥_{W_{λ}} ∥z_{1}∥_{W_{λ}} < ∥z_{1}∥_{W_{λ}}^{2}, (24.11.17)

using the Cauchy–Schwarz inequality in the first step, and (24.11.14) in the second step. This implies that λ_{1}(x) ≠ 0, and the argument for λ_{2} is the same.

We shall continue the proof in the next section, showing that λ is not an extreme point of P under these conditions.

24.12 Showing that λ is not an extreme point

Let us continue with the same notation and hypotheses as in the previous section. Note that M(λ_{1}) + M(λ_{2}) = 1, (24.12.1)

by (24.10.4) and (24.11.9). Put

μ_{j} = M(λ_{j})^{−1} λ_{j} (24.12.2) for j = 1, 2, so that M(μ_{j}) = 1 and thus μ_{j} ∈ P. Using (24.11.5), we get that

λ = M(λ_{1}) μ_{1} + M(λ_{2}) μ_{2}. (24.12.3)

We would like to show that μ_{1} ≠ μ_{2}, to obtain that λ is not an extreme point of P.

Suppose for the sake of a contradiction that there is a real number r such that λ_{2} = r λ_{1} (24.12.4)

on A. Of course, r ≠ 0, by (24.11.13), and one can check that r > 0, because λ_{1}, λ_{2} are positive linear functionals on A in the strong sense. Let δ be a positive real number such that

δ < \frac{r ∥z_{1}∥_{W_{λ}}^{2}}{∥z_{2}∥_{W_{λ}} + r ∥z_{1}∥_{W_{λ}}} , (24.12.5)

as in (1) on p329 of [151]. Let a be an element of A such that

∥ρ_{a}^{W_{λ}}(w_{0}) − z_{1}∥_{W_{λ}} < δ, (24.12.6)

using the hypothesis that ρ^{W_{λ}} be topologically cyclic with topologically cyclic vector w_{0} again.

Observe that

⟨ρ_{a}^{W_{λ}}(z_{1}), z_{1}⟩_{W_{λ}} − ∥z_{1}∥_{W_{λ}}^{2} = ⟨ρ_{a}^{W_{λ}}(w_{0}), z_{1}⟩_{W_{λ}} − ⟨z_{1}, z_{1}⟩_{W_{λ}}
(24.12.7)  = ⟨ρ_{a}^{W_{λ}}(w_{0}) − z_{1}, z_{1}⟩_{W_{λ}},

where the first step is as in (24.11.15). This implies that

|⟨ρ_{a}^{W_{λ}}(z_{1}), z_{1}⟩_{W_{λ}} − ∥z_{1}∥_{W_{λ}}^{2} | ≤ ∥ρ_{a}^{W_{λ}}(w_{0}) − z_{1}∥_{W_{λ}} ∥z_{1}∥_{W_{λ}}
(24.12.8)  < δ ∥z_{1}∥_{W_{λ}},

using the Cauchy–Schwarz inequality in the first step, and (24.12.6) in the second step. It follows that

|⟨ρ_{a}^{W_{λ}}(z_{1}), z_{1}⟩_{W_{λ}} | > ∥z_{1}∥_{W_{λ}}^{2} − δ ∥z_{1}∥_{W_{λ}}, (24.12.9)

as in (2) on p329 of [151].

Similarly,

(24.12.10) ⟨ρ_{a}^{W_{λ}}(z_{2}), z_{2}⟩_{W_{λ}} = ⟨ρ_{a}^{W_{λ}}(w_{0}), z_{2}⟩_{W_{λ}} − ⟨z_{1}, z_{2}⟩_{W_{λ}}
= ⟨ρ_{a}^{W_{λ}}(w_{0}) − z_{1}, z_{2}⟩_{W_{λ}},

using (24.11.2) and the fact that ρ_{a}^{W_{λ}}(z_{1}) ∈ Z_{1} in the first step. This means that |⟨ρ_{a}^{W_{λ}}(z_{2}), z_{2}⟩_{W_{λ}} | ≤ ∥ρ_{a}^{W_{λ}}(w_{0}) − z_{1}∥_{W_{λ}} ∥z_{2}∥_{W_{λ}} < δ ∥z_{2}∥_{W_{λ}}, (24.12.11)

using the Cauchy–Schwarz inequality in the first step, and (24.12.6) in the second step, as in (3) on p329 of [151]. It is easy to see that

δ ∥z_{2}∥_{W_{λ}} < r (∥z_{1}∥_{W_{λ}}^{2} − δ ∥z_{1}∥_{W_{λ}}), (24.12.12) using (24.12.5), as in (4) on p329 of [151]. It follows from this, (24.12.9), and (24.12.11) that

|⟨ρ_{a}^{W_{λ}}(z_{2}), z_{2}⟩_{W_{λ}} | < r |⟨ρ_{a}^{W_{λ}}(z_{1}), z_{1})⟩_{W_{λ}} |, (24.12.13)

which contradicts (24.12.4).

24.13 Multiplicativity and extreme points

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗_{A}} that is conjugate-linear in the complex case, and let P be as in Section 24.9. Also let λ be a linear functional on A that is symmetric or Hermitian symmetric in the strong sense with respect to the involution on A, as in Section 23.4. Suppose that

λ(x^{∗_{A}} x) = λ(x^{∗_{A}}) λ(x) (24.13.1)

for all x ∈ A. This means that

λ(x^{∗_{A}} x) = |λ(x)|^{2} (24.13.2)

for every x ∈ A, because λ ∈ A^{sym}. This implies that λ is positive in the strong sense on A with respect to the involution, as in Section 18.7.

More precisely, either λ ≡ 0 on A, or

M(λ) = 1, (24.13.3)

where M(λ) is as in Section 24.7. Of course, λ ∈ P in both cases. We would like to show that λ is an extreme point of P. This basically corresponds to the fact that (b) implies (c) in Theorem 11.33 on p286 of [286]. Although there are some additional hypotheses in that theorem, they are not really needed for this part, as follows.

It is convenient to look at this in terms of the comparisons considered in Section 24.3. Let μ be another positive linear functional on A in the strong sense, and suppose that μ ⪯ λ, so that

μ(x^{∗_{A}} x) ≤ c_{1} λ(x^{∗_{A}} x) (24.13.4)

for some c_{1} ≥ 0 and all x ∈ A. If x ∈ A and λ(x) = 0, then λ(x^{∗} x) = 0, by (24.13.2), which implies that μ(x^{∗_{A}} x) = 0. It follows that μ(x) = 0, because μ is positive in the strong sense. This means that

ker λ ⊆ ker μ. (24.13.5)

Using (24.13.5), we get that

μ = r λ (24.13.6) for some real or complex number r, as appropriate, by standard arguments in linear algebra. More precisely, if λ ≡ 0 on A, then μ ≡ 0 on A, and we can take r = 0. Otherwise, if λ(x) ≠ 0 for some x ∈ A, then λ(x^{∗_{A}} x) > 0, and we can take

r = \frac{μ(x^{∗_{A}} x)}{λ(x^{∗_{A}} x)}, (24.13.7)

which is a nonnegative real number.

If λ ≡ 0 on A, then λ is an extreme point of P, as mentioned in Section 24.10, and so we may as well suppose that (24.13.3) holds. If λ_{1}, λ_{2} ∈ P and λ = t λ_{1} + (1 − t) λ_{2} for some t ∈ (0, 1), then it is easy to see that

λ_{1}, λ_{2} ⪯ λ, (24.13.8)

as in Section 24.10. This implies that λ_{1}, λ_{2} are nonnegative multiples of λ, as in the preceding paragraph. We also have that M(λ_{1}) = M(λ_{2}) = 1, as in Section 24.10. One can use this to get that λ_{1} = λ_{2} = λ.

24.14 Commutativity and extreme points

Let (A, ∥ · ∥_{A}) be a commutative Banach algebra over the real or complex numbers with a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1 and an involution y ↦ y^{∗_{A}} that is conjugate-linear in the complex case. Let P be as in Section 24.9, which consists of the positive linear functionals λ on A such that λ(e_{A}) ≤ 1 in this case. Suppose that λ is an extreme point of P. If A is complex, or if the involution on A is continuous, then we would like to show that

λ(x^{∗_{A}} x y) = λ(x^{∗_{A}} x) λ(y) (24.14.1)

for every x, y ∈ A, as in (2) on p287 of [286].

Of course, (24.14.1) is trivial when λ ≡ 0 on A. Otherwise, M(λ) = 1, as in Section 24.10, which means that

λ(e_{A}) = 1 (24.14.2)

in this case, as in Section 24.7.

If x ∈ A, then λ_{x}(y) = λ(x^{∗_{A}} y x) (24.14.3)

defines a positive linear functional on A, as in Section 22.12. Note that

λ_{x}(e_{A}) = λ(x^{∗_{A}} x) ≥ 0. (24.14.4)

If λ_{x}(e_{A}) = 0, then λ_{x} ≡ 0 on A, as in Section 22.12, and (24.14.1) holds for all y ∈ A. Thus we may as well suppose that λ_{x}(e_{A}) > 0.

Suppose that ∥x^{∗_{A}} x∥_{A} < 1, (24.14.5)

24.14. COMMUTATIVITY AND EXTREME POINTS 551

so that z = b_{1/2}(−x^{∗_{A}} x) ∈ A may be defined as in Section 17.2. Thus z^{2} = e_{A} − x^{∗_{A}} x, (24.14.6) as before. If the involution on A is continuous, then z^{∗} = z, (24.14.7) because x^{∗_{A}} x is self-adjoint. This also holds when A is complex, without asking that the involution on A be continuous, as in Sections 14.4 and 17.10.

If y ∈ A, then

(24.14.8) λ(y^{∗_{A}} y) − λ_{x}(y^{∗_{A}} y) = λ((e_{A} − x^{∗_{A}} x) y^{∗_{A}} y)
= λ(z^{2} y^{∗_{A}} y) = λ((y z)^{∗_{A}} y z) ≥ 0,

as in (5) on p287 of [286]. This means that λ − λ_{x} is a positive linear functional on A as well. In particular, λ(e_{A}) − λ_{x}(e_{A}) ≥ 0. (24.14.9)

If λ(e_{A}) − λ_{x}(e_{A}) = 0, then λ − λ_{x} ≡ 0 on A, as in Section 18.7, and one can check that (24.14.1) holds for every y ∈ A, using (24.14.2). Alternatively, λ(x^{∗_{A}} x) ≤ λ(e_{A}) ∥x^{∗_{A}} x∥_{A}, (24.14.10) as in Section 18.12, because x^{∗_{A}} x is self-adjoint. This implies that λ_{x}(e_{A}) = λ(x^{∗_{A}} x) < λ(e_{A}), (24.14.11) because of (24.14.5).

Observe that μ_{1} = λ_{x}(e_{A})^{−1} λ_{x}, μ_{2} = (λ(e_{A}) − λ_{x}(e_{A}))^{−1} (λ − λ_{x}) (24.14.12) are positive linear functionals on A with μ_{1}(e_{A}) = μ_{2}(e_{A}) = 1, so that μ_{1}, μ_{2} are elements of P. Of course, λ_{x}(e_{A}) + (λ(e_{A}) − λ_{x}(e_{A})) = λ(e_{A}) = 1 (24.14.13) and λ = λ_{x}(e_{A}) μ_{1} + (λ(e_{A}) − λ_{x}(e_{A})) μ_{2}. (24.14.14) It follows that λ = μ_{1}, (24.14.15) because λ is an extreme point of P, by hypothesis. This means that (24.14.1) holds for every y ∈ A. It is easy to reduce to the case where (24.14.5) holds, so that (24.14.1) holds for all x, y ∈ A.

In the complex case, one can use (24.14.1) to get that λ(x y) = λ(x) λ(y) (24.14.16) for every x, y ∈ A, as in Theorem 11.33 on p286 of [286]. To see this, let n ≥ 3 be an integer, put ω = exp(2πi/n), (24.14.17)

and note that ω^{n} = 1 and ω, ω^{2} ≠ 1. One can check that

∑_{l=1}^{n} ω^{l} = ∑_{l=1}^{n} ω^{2 l} = 0, (24.14.18)

because these sums are not affected by multiplication by ω, ω^{2}, respectively. If x ∈ A and z_{l} = e_{A} + ω^{−l} x for l = 1, …, n, then one can verify that

x = \frac{1}{n} ∑_{l=1}^{n} ω^{l} z_{l}^{∗_{A}} z_{l}. (24.14.19)

In order to get (24.14.16), one can use (24.14.1) with x replaced with z_{l} for each l = 1, …, n.

24.15 Weak^{∗} compactness of P

Let A be an associative algebra over the real or complex numbers with an involution y ↦ y^{∗_{A}} that is conjugate-linear in the complex case. Remember that the set P = P (A) defined in Section 24.9 consists of the λ ∈ A^{sym} that are positive in the strict sense and satisfy

|λ(x)|^{2} ≤ λ(x^{∗_{A}} x) (24.15.1)

for every x ∈ A. One can check that P is a closed set in the space of all real or complex-valued functions on A, as appropriate, with respect to the topology of pointwise convergence. This may be described as the weakest topology with respect to which λ ↦ λ(a) is a continuous real or complex-valued function, as appropriate, for each a ∈ A. More precisely, one can verify that the algebraic dual A^{alg} of A is a closed set in the space of all real or complex-valued functions on A, as appropriate, with respect to this topology. Similarly, A^{sym} is a closed set with respect to this topology, as is the set of all linear functionals on A that are positive in the strict sense. The set of all real or complex-valued functions λ on A, as appropriate, that satisfy (24.15.1) for all x ∈ A is a closed set with respect to this topology as well.

Suppose from now on in this section that A is a Banach algebra with respect to a norm ∥ · ∥_{A}. Suppose also for the moment that the involution on A is continuous, so that ∥y^{∗_{A}}∥_{A} ≤ C ∥y∥_{A} (24.15.2)

for some C ≥ 1 and all y ∈ A. If λ is a positive linear functional on A in the strong sense, then λ is a bounded linear functional, as in Section 24.8. This 24.15. WEAK^{∗} COMPACTNESS OF P 553

means that P may be considered as a subset of the dual space A′ of all bounded linear functionals on A, as in Section 3.1. More precisely, if λ ∈ P, then

∥λ∥_{A′} ≤ C^{1/2}, (24.15.3)

as in Section 24.8.

Under these conditions,

(24.15.4) P is a closed set in A′, with respect to the weak^{∗} topology,

because P is a closed set in the space of all real or complex-valued functions on A, as appropriate, with respect to the topology of pointwise convergence. This implies that

(24.15.5) P is compact, with respect to the weak^{∗} topology

because of (24.15.3) and the Banach–Alaoglu theorem. This basically corresponds to part of Theorem 21.33 on p327 of [151], except that in that result, P is considered as a subset of the dual of the space of self-adjoint elements of A. This also corresponds to part of Remark 4.8.5 on p129 of [11].

Suppose now that A has a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1. If A is commutative, then (24.15.3) holds with C = 1, even if the norm on A is not necessarily invariant under the involution on A, as in Section 18.13. If A is complex and commutative, then P ⊆ A′, and (24.15.3) holds with C = 1, without asking that the involution on A be continuous, as in Section 18.13 again. In this case, (24.15.4) holds for the same reasons as before, so that (24.15.5) holds as well.

In fact, positive linear functionals on A are bounded on A when A is complex, even if A is not necessarily commutative, and without asking that the involution on A be continuous, as in part (e) of Theorem 11.31 on p284 of [286]. The same argument shows that the dual norm os a positive linear functional λ on A is bounded by a constant times λ(e_{A}), where the constant depends on A and not λ. This means that the dual norms of the elements of P are bounded. One can use this to get (24.15.5), as before.

Algebras, matrices, and norms

25.1 Algebras of matrices

Let k be a field, and let A be an algebra in the strict sense over k. Also let n be a positive integer, and let M_{n}(A) be the space of n × n matrices with entries in A. This is a vector space over k with respect to entrywise addition and scalar multiplication. If a, b ∈ M_{n}(A), then a b ∈ M_{n}(A) is defined as usual by

(a b)_{j,r} = ∑_{l=1}^{n} a_{j,l} b_{l,r} (25.1.1)

for j, r = 1, …, n. It is easy to see that matrix multiplication is bilinear, so that M_{n}(A) is an algebra over k in the strict sense too.

If A is an associative algebra, then one can check that M_{n}(A) is an associative algebra with respect to matrix multiplication. If A has a multiplicative identity element e_{A}, then the corresponding identity matrix I = I_{n} ∈ M_{n}(A) is defined to be the matrix whose diagonal entries are equal to e_{A}, and whose other entries are equal to 0. It is easy to see that this is the multiplicative identity element in M_{n}(A).

If x ∈ A, then let μ_{n}(x) ∈ M_{n}(A) be the n×n matrix whose diagonal entries are equal to x, and whose other entries are equal to 0. This defines an algebra homomorphism from A into M_{n}(A). If A has a multiplicative identity element e_{A}, then I_{n} = μ_{n}(e_{A}). (25.1.2)

If a ∈ M_{n}(A), then μ_{n}(x) a and a μ_{n}(x) are the elements of M_{n}(A) whose entries are obtained by multipliying the corresponding entries of a by x on the left and right, respectively.

Let us suppose for the rest of this section that A is associative. Let V be a vector space over k, and let V^{n} be the space of n-tuples of elements of V, considered as a vector space over k with respect to coordinatewise addition and scalar multiplication. This is the same as the direct sum of n copies of V as in Section 21.13, and as in Section 5.12 when n = 2.

Suppose for the moment that V is a left module over A. If a ∈ M_{n}(A) and v ∈ V^{n}, then a · v may be defined as an element of V^{n} as usual by

(a · v)_{j} = ∑_{l=1}^{n} a_{j,l} · v_{l} (25.1.3)

for each j = 1, …, n. One can check that V^{n} is a left module over M_{n}(A) in this way.

We may also consider V^{n} as a left module over A, where x ∈ A acts on elements of V^{n} coordinatewise, as in Section 21.13. This is the same as the action of μ_{n}(x) on V^{n} as in (25.1.3). In particular, if A has a multiplicative identity element e_{A}, and if the action of e_{A} on V is the identity mapping, then the corresponding action of I_{n} on V^{n} is the identity mapping as well.

Suppose for the moment again that V is a right module over A. If a ∈ M_{n}(A) and v ∈ V^{n}, then v · a may be defined as an element of V^{n} by

(v · a)_{l} = ∑_{j=1}^{n} v_{j} · a_{j,l} (25.1.4)

for each l = 1, …, n. One can verify that V^{n} is a right module over M_{n}(A) in this way. We may also consider V^{n} as a right module over A, where x ∈ A acts on elements of V^{n} coordinatewise, as before, which is the same as the action of μ_{n}(x) on V^{n} as in (25.1.4). If A has a multiplicative identity element e_{A}, and the action of e_{A} on V is the identity mapping, then the action of I_{n} = μ_{n}(e_{A}) on V^{n} is the identity mapping too.

25.2 Matrices and linear mappings

Let k be a field, and let A be an algebra in the strict sense over k. If m and n are positive integers, then let M_{m,n}(A) be the space of m × n matrices with entries in A. This is a vector space over k with respect to entrywise addition and scalar multiplication, as before. If n_{1} is another positive integer, a ∈ M_{m,n}(A), and b ∈ M_{n,n_{1}}(A), then a b ∈ M_{m,n_{1}}(A) (25.2.1)

may be defined as in (25.1.1), as usual. Note that

(a, b) ↦ a b (25.2.2)

is bilinear as a mapping from M_{m,n}(A) × M_{n,n_{1}}(A) into M_{m,n_{1}}(A). If n_{2} is another positive integer and c ∈ M_{n_{1},n_{2}}(A), then one can verify that

(a b) c = a (b c), (25.2.3) as elements of M_{m,n_{2}}(A), when A is an associative algebra over k. If A has a multiplicative identity element e_{A}, and I_{m} ∈ M_{m}(A), I_{n} ∈ M_{n}(A) are as in the previous section, then it is easy to see that

I_{m} a = a I_{n} = a. (25.2.4)

Let A^{n} be the space of n-tuples of elements of A, as before, and similarly for A^{m}, A^{n_{1}}. If a ∈ M_{m,n}(A) and v ∈ A^{n}, then

(T_{a}(v))_{j} = ∑_{l=1}^{n} a_{j,l} v_{l} (25.2.5)

defines an element of A^{m}. This defines a linear mapping T_{a} from A^{n} into A^{m}, as vector spaces over k. If b ∈ M_{n,n_{1}}(A), then one can define T_{b}, T_{a b} in the same way, as linear mappings from A^{n_{1}} into A^{n}, A^{m}, respectively. One can check that T_{a}(T_{b}(w)) = T_{a b}(w) (25.2.6)

for every w ∈ A^{n_{1}}, as elements of A^{m}, when A is an associative algebra over k, so that T_{a} ∘ T_{b} = T_{a b}, (25.2.7)

as mappings from A^{n_{1}} into A^{m}.

If A is an associative algebra over k, then A may be considered as a left and right module over itself, as in Section 20.2, so that A^{n}, A^{m} may be considered as left and right modules over A, as in Section 21.13. More precisely, if x ∈ A, then x acts on A^{n}, A^{m} by coordinatewise multiplication on the left and right, as appropriate. If v ∈ A^{n}, then it is easy to see that

(T_{a}(v)) · x = T_{a}(v · x). (25.2.8)

This means that T_{a} is a homomorphism from A^{n} into A^{m}, as right modules over A, as in Section 20.7. If m = n, then (25.2.5) is the same as (25.1.3), and (25.2.8) is the same as saying that

(a · v) · x = a · (v · x). (25.2.9)

Suppose for the moment that A has a multiplicative identity element e_{A}, and for each l = 1, …, n, let u(l) be the element of A^{n} whose lth coordinate is equal to e_{A}, and whose other coordinates are equal to 0. Observe that

(T_{a}(u(l)))_{j} = a_{j,l} (25.2.10)

for all j = 1, …, m and l = 1, …, n, as in (25.2.5). Let T be a linear mapping from A^{n} into A^{m}, as vector spaces over k, and suppose that A is an associative algebra over k, and that T is also a homomorphism from A^{n} into A^{m}, as right modules over A. Consider the matrix a(T) ∈ M_{m,n}(A) defined by

(a(T))_{j,l} = (T (u(l)))_{j} (25.2.11) for j = 1, …, m and l = 1, …, n. One can check that

T = T_{a(T)}, (25.2.12)

where the right side is as in (25.2.5), by expressing each v ∈ A^{n} as

v = ∑_{l=1}^{n} u(l) · v_{l}. (25.2.13)

Similarly, if b ∈ M_{n,n_{1}}(A) and v ∈ A^{n}, then

(R_{b}(v))_{p} = ∑_{l=1}^{n} v_{l} b_{l,p} (25.2.14)

defines an element of A^{n_{1}}. This defines a linear mapping R_{b} from A^{n} into A^{n_{1}}, as vector spaces over k. If a ∈ M_{m,n}(A), then one can define R_{a}, R_{a b} in the same way, as linear mappings from A^{m} into A^{n}, A^{n_{1}}, respectively. One can verify that R_{b}(R_{a}(z)) = R_{a b}(z) (25.2.15) for every z ∈ A^{m}, as elements of A^{n_{1}}, when A is an associative algebra over k. This means that R_{b} ∘ R_{a} = R_{a b}, (25.2.16) as mappings from A^{m} into A^{n_{1}}, in this case.

If A is an associative algebra over k, v ∈ A^{n}, and x ∈ A, then

x · R_{b}(v) = R_{b}(x · v). (25.2.17)

Thus R_{b} is a homomorphism from A^{n} into A^{n_{1}}, as left modules over A. If n = n_{1}, then R_{b}(v) is the same as the action v · b of b on v on the right, as in (25.1.4). In this case, (25.2.17) is the same as saying that

x · (v · b) = (x · v) · b. (25.2.18)

Suppose that A has a multiplicative identity element e_{A} again, and let u(l) be the element of A^{n} defined earlier for each l = 1, …, n. It is easy to see that

(R_{b}(u(l)))_{p} = b_{l,p} (25.2.19)

for every l = 1, …, n and p = 1, …, n_{1}, by (25.2.14). Suppose that A is an associative algebra over k, and let R be a linear mapping from A^{n} into A^{n_{1}}, as vector spaces over k, that is a homomorphism from A^{n} into A^{n_{1}}, as left modules over A. Consider the matrix b(R) ∈ M_{n,n_{1}}(A) defined by

(b(R))_{l,p} = (R(u(l)))_{p} (25.2.20)

for l = 1, …, n and p = 1, …, n_{1}. One can verify that

R = R_{b(R)}, (25.2.21)

where the right side is as in (25.2.14), by expressing every v ∈ A^{n} as

v = ∑_{l=1}^{n} v_{l} · u(l). (25.2.22)

25.3 More on mappings and matrices

Let k be a field, let A be an associative algebra over k, and let m, n be positive integers. Also let V be a vector space over k, and let V^{m}, V^{n} be as in Section 25.1. Suppose for the moment that V is a left module over A. If a ∈ M_{m,n}(A) and v ∈ V^{n}, then

(T_{a}^{V} (v))_{j} = ∑_{l=1}^{n} a_{j,l} · v_{l} (25.3.1)

defines an element of V^{m}. This defines a linear mapping T_{a}^{V} from V^{n} into V^{m}, as vector spaces over k. This is the same as (25.1.3) when m = n. If V = A, considered as a left module over itself, then (25.3.1) is the same as (25.2.5).

Note that a ↦ T_{a}^{V} (25.3.2)

is a linear mapping from M_{m,n}(A) into the space L(V^{n}, V^{m}) of linear mappings from V^{n} into V^{m}. Let n_{1} be another positive integer, let V^{n_{1}} be as in Section 25.1, and suppose that b ∈ M_{n,n_{1}}(A). Thus T_{b}^{V}, T_{a b}^{V} may be defined as linear mappings from V^{n_{1}} into V^{n}, V^{m}, respectively, as in the preceding paragraph. One can check that T_{a}^{V} (T_{b}^{V} (w)) = T_{a b}^{V}(w) (25.3.3)

for every w ∈ V^{n_{1}}, as elements of V^{m}, so that

T_{a}^{V} ∘ T_{b}^{V} = T_{a b}^{V}, (25.3.4)

as mappings from V^{n_{1}} into V^{m}. This corresponds to analogous statements in the previous two sections when m = n = n_{1} and when V = A, as a left module over itself, as before.

Suppose for the moment again that V is a right module over A. If v ∈ V^{n}, then

(R_{b}^{V} (v))_{p} = ∑_{l=1}^{n} v_{l} · b_{l,p} (25.3.5)

defines an element of V^{n}. This defines a linear mapping R_{b}^{V} from V^{n_{1}} into V^{n}, as vector spaces over k. If n = n_{1}, then R_{b}^{V} (v) is the same as the action v · b of b on v on the right, as in (25.1.4). If V = A, considered as a right module over itself, then (25.3.5) is the same as (25.2.14).

As before, b ↦ R_{b}^{V} (25.3.6)

is a linear mapping from M_{n,n_{1}}(A) into L(V^{n_{1}}, V^{n}). Of course, R_{a}^{V}, R_{a b}^{V} may be defined as linear mappings from V^{m} into V^{n}, V^{n_{1}}, respectively, in the same way. One can verify that

R_{b}^{V} (R_{a}^{V} (z)) = R_{a b}(z) (25.3.7)

for every z ∈ V^{m}, as elements of V^{n_{1}}, so that

R_{b}^{V} ∘ R_{a}^{V} = R_{a b}^{V}, (25.3.8)

25.4. SOME NORMS ON V^{N} 559

as linear mappings from V^{m} into V^{n_{1}}. This corresponds to analogous statements in the previous two sections when m = n = n_{1} and when V = A, as a right module over itself.

If V is any vector space over k, then V map be considered as a left module over the algebra L(V) of all linear mappings from V into itself, as in Section 20.2. Thus we can take A = L(V), to get a linear mapping (25.3.2) from M_{m,n}(L(V)) into L(V^{n}, V^{m}). More precisely, one can check that this is a oneto-one mapping from M_{m,n}(L(V)) onto L(V^{n}, V^{m}). If m = n, then this is an algebra isomorphism from M_{n}(L(V)) onto L(V^{n}).

25.4 Some norms on V^{n}

Let V be a vector space over the real or complex numbers with a norm ∥ · ∥_{V}, let n be a positive integer, and let V^{n} be the space of n-tuples of elements of V, considered as a vector space over the real or complex numbers, as appropriate, as in Section 25.1. If 1 ≤ p < ∞, then

∥v∥_{V^{n},p} = ( ∑_{l=1}^{n} ∥v_{l}∥_{V}^{p} )^{1/p} (25.4.1)

defines a norm on V^{n}, as in Section 21.14, and as in Section 5.12 when n = 2. Similarly,

∥v∥_{V^{n},∞} = max_{1≤l≤n} ∥v_{l}∥_{V} (25.4.2)

defines a norm on V^{n}, as before. If V is complete with respect to the metric associated to ∥·∥_{V}, then V^{n} is complete with respect to the metric associated to ∥ · ∥_{V^{n}}, as in Section 5.11. This is also mentioned in Section 21.15 for arbitrary ℓ^{p} direct sums.

Suppose for the moment that ∥·∥_{V} is the norm associated to an inner product ⟨·, ·⟩_{V} on V. In this case, it is easy to see that

⟨v, w⟩_{V^{n}} = ∑_{l=1}^{n}⟨v_{l}, w_{l}⟩_{V} (25.4.3)

defines an inner product on V^{n}, as in Section 22.1, and Section 5.12 when n = 2. The norm on V^{n} associated to this inner product is the same as (25.4.2), with p = 2. If V is a Hilbert space with respect to ⟨·, ·⟩_{V}, then V^{n} is a HIlbert space with respect to (25.4.3).

If 1 ≤ p_{1} ≤ p_{2} ≤ ∞, then

∥v∥_{V^{n},p_{2}} ≤ ∥v∥_{V^{n},p_{1}} (25.4.4)

for every v ∈ V^{n}, as in Sections 2.5 and 21.14. We also have that

∥v∥_{V^{n},p} ≤ n^{1/p} ∥v∥_{V^{n},∞} (25.4.5)

for every v ∈ V^{n} and 1 ≤ p < ∞, as in Section 1.3.

Let m be another positive integer, and let 1 ≤ q ≤ ∞ be given, so that ∥ · ∥_{V^{m},q} defines a norm on V^{m} as before. Using this norm and ∥ · ∥_{V^{n},p} on V^{n}, we can define the space BL(V^{n}, V^{m}) of bounded linear mappings from V^{n} into V^{m} as in Section 2.2. Note that the boundedness of a linear mapping T from V^{n} into V^{m} does not depend on p or q, so that BL(V^{n}, V^{m}) does not really depend on p or q. Thus we may say that T is a bounded linear mapping from V^{n} into V^{m} if it is bounded with respect to ∥ · ∥_{V^{n},p} and ∥ · ∥_{V^{m},q} for some p, q ≥ 1.

However, the corresponding operator norm of T may depend on p and q, and it may be expressed as ∥T ∥_{op,V^{n}V^{m},p q} (25.4.6)

to indicate the roles of p and q. If 1 ≤ p_{1} ≤ p_{2} ≤ ∞, then

∥T ∥_{op,V^{n}V^{m},p_{1} q} ≤ ∥T ∥_{op,V^{n}V^{m},p_{2} q} (25.4.7)

for all q ≥ 1, because of (25.4.4). Similarly, if 1 ≤ q_{1} ≤ q_{2} ≤ ∞, then

∥T ∥_{op,V^{n}V^{m},p q_{2}} ≤ ∥T ∥_{op,V^{n}V^{m},p q_{1}} (25.4.8)

for every p ≥ 1. One can check that

∥T ∥_{op.V^{n}V^{m},∞ q} ≤ n^{1/p} ∥T ∥_{op,V^{n}V^{m},p q} (25.4.9)

for all q ≥ 1 and 1 ≤ p < ∞, using (25.4.5). Similarly,

∥T ∥_{op,V^{n}V^{m},p q} ≤ m^{1/q} ∥T ∥_{op,V^{n}V^{m},p ∞} (25.4.10)

for all p ≥ 1 and 1 ≤ q < ∞.

25.5 Some norms on matrices

Let A be a vector space over the real or complex numbers, and let ∥ · ∥_{A} be a norm on A. Also let m, n be positive integers, and let M_{m,n}(A) be the space of m × n matrices with entries in A. This is a vector space over the real or complex numbers, as appropriate, with respect to entrywise addition and scalar multiplication, as usual. This may also be denoted M_{n}(A) when m = n, as before. We would like to consider some norms on M_{m,n}(A) in this section.

If a ∈ M_{m,n}(A), then put

∥a∥_{M_{m,n}(A),p} = ( ∑_{j=1}^{m} ∑_{l=1}^{n} ∥a_{j,l}∥_{A}^{p})^{1/p} (25.5.1)

when 1 ≤ p < ∞, and

∥a∥_{M_{m,n}(A),∞} = max{∥a_{j,l}∥_{A} : 1 ≤ j ≤ m, 1 ≤ l ≤ n}. (25.5.2)

We may also use the notation ∥a∥_{M_{n}(A).p}, 1 ≤ p ≤ ∞, when m = n. It is easy to see that (25.5.1) and (25.5.2) define norms on M_{m,n}(A). More precisely, we 25.5. SOME NORMS ON MATRICES 561 can list the entries of an m × n matrix in a sequence of length m n, to get a oneto-one linear mapping from M_{m,n}(A) onto the space A^{m n} of all (m n)-tuples of elements of A. Using this, (25.5.1) and (25.5.2) correspond exactly to the analogoue norms on A^{m n} considered in the previous section.

In particular, if 1 ≤ p_{1} ≤ p_{2} ≤ ∞, then ∥a∥_{M_{m,n}(A),p_{2}} ≤ ∥a∥_{M_{m,n}(A),p_{1}}, (25.5.3) as in (25.4.4). Similarly, ∥a∥_{M_{m,n}(A),p} ≤ (m n)^{1/p} ∥a∥_{M_{m,n}(A),∞} (25.5.4) for 1 ≤ p < ∞, as in (25.4.5).

It will be convenient to consider some other norms on M_{m,n}(A), as follows. Put

(25.5.5)  ∥a∥_{M_{m,n}(A),∞ p} = max_{1≤j≤m} ( ∑_{l=1}^{n} ∥a_{j,l}∥_{A}^{p})^{1/p},
(25.5.6)  ∥a∥_{M_{m,n}(A),p ∞} = max_{1≤l≤n} ( ∑_{j=1}^{m} ∥a_{j,l}∥_{A}^{p})^{1/p}

when 1 ≤ p < ∞, and ∥a∥_{M_{m,n}(A),∞ ∞} = ∥a∥_{M_{m,n}(A),∞}, (25.5.7) as in (25.5.2). We may also use the notation ∥a∥_{M_{n}(A),∞ p} and ∥a∥_{M_{n}(A).p ∞}, as appropriate, when m = n, as before. One can check that (25.5.5) and (25.5.6) satisfy the triangle inequality on M_{m,n}(A), using the triangle inequalities for the ℓ^{p} and ℓ^{∞} norms on R^{n}, R^{m}, as appropriate. Using this, it is easy to see that (25.5.5) and (25.5.6) are norms on M_{m,n}(A).

If 1 ≤ p_{1} ≤ p_{2} ≤ ∞, then ∥a∥_{M_{m,n}(A),∞ p_{2}} ≤ ∥a∥_{M_{m,n}(A),∞ p_{1}} (25.5.8) and ∥a∥_{M_{m,n}(A),p_{2} ∞} ≤ ∥a∥_{M_{m,n}(A),p_{1} ∞}, (25.5.9) as in (25.4.4). If 1 ≤ p < ∞, then ∥a∥_{M_{m,n}(A),∞ p} ≤ n^{1/p} ∥a∥_{M_{m,n}(A),∞} (25.5.10) and ∥a∥_{M_{m,n}(A),p ∞} ≤ m^{1/p} ∥a∥_{M_{m,n}(A),∞}, (25.5.11) as in (25.4.5). One can also check that ∥a∥_{M_{m,n}(A),∞ p}, ∥a∥_{M_{m,n}(A),p ∞} ≤ ∥a∥_{M_{m,n}(A),p}. (25.5.12)

25.6 Some estimates for T_{a}

Let A be an algebra in the strict sense over the real or complex numbers, and let ∥ · ∥_{A} be a submultiplicative norm on A. Also let m, n be positive integers, and let a ∈ M_{m,n}(A) be given. If v ∈ A^{n}, then T_{a}(v) ∈ A^{m} may be defined as in Section 25.2, and

∥(T_{a}(v))_{j}∥_{A} ≤ ∑_{l=1}^{n} ∥a_{j,l}∥_{A} ∥v_{l}∥_{A} (25.6.1)

for each j = 1, …, m. One can use this to get that T_{a} is bounded as a linear mapping from A^{n} into A^{m} with respect to the norms ∥·∥_{A^{n},p} and ∥·∥_{A^{m},q} defined in Section 25.4 for any p, q ≥ 1. In this section, we would like to mention some extimates for the corresponding operator norms of T_{a}, in terms of norms of a as in the previous section.

In particular,

∥(T_{a}(v))_{j}∥_{A} ≤ ( ∑_{l=1}^{n} ∥a_{j,l}∥_{A}) ∥v∥_{A^{n},∞} (25.6.2)

for each j = 1, …, m, by (25.6.1). This implies that

∥T_{a}(v)∥_{A^{m},1} ≤ ∥a∥_{M_{m,n}(A),1} ∥v∥_{A^{n},∞}, (25.6.3)

where ∥a∥_{M_{m,n}(A),1} is as in (25.5.1). This means that

∥T_{a}∥_{op,A^{n}A^{m},∞ 1} ≤ ∥a∥_{M_{m,n}(A),1}. (25.6.4)

One can also use (25.6.2) to get that

∥T_{a}(v)∥_{A^{m},∞} ≤ ∥a∥_{M_{m,n}(A),∞ 1} ∥v∥_{A^{n},∞}, (25.6.5)

where ∥a∥_{M_{m,n}(A),∞ 1} is as in (25.5.5). It follows that

∥T_{a}∥_{op,A^{n}A^{m},∞ ∞} ≤ ∥a∥_{M_{m,n}(A),∞ 1}. (25.6.6)

If 1 ≤ p ≤ ∞, then one can check that

∥T_{a}(v)∥_{A^{m},p} ≤ ∥a∥_{M_{m,n}(A),p ∞} ∥v∥_{A^{n},1}, (25.6.7)

where ∥a∥_{M_{m,n}(A),p ∞} is as in (25.5.6). This is similar to an argument in Section 2.3, and is simpler when p = 1, ∞. Indeed, if p = ∞, then this can be seen directly from (25.6.1). If 1 ≤ p < ∞, then

∥T_{a}(v)∥_{A^{m},p} ≤ ( ∑_{j=1}^{m} ( ∑_{l=1}^{n} ∥a_{j,l}∥_{A} ∥v_{l}∥_{A})^{p})^{1/p}, (25.6.8)

25.7. SOME ESTIMATES FOR R_{B} 563

because of (25.6.1). One can verify that the right side is less than or equal to

∑_{l=1}^{n} ( ∑_{j=1}^{m} ∥a_{j,l}∥_{A}^{p} ∥v_{l}∥_{A}^{p})^{1/p}, (25.6.9)

using Minkowski’s inequality for sums, which is to say the triangle inequality for the ℓ^{p} norm on R^{m}. More precisely, the right side of (25.6.8) is equal to (25.6.9) when p = 1, by interchanging the order of summation. Of course, (25.6.9) is equal to

∑_{l=1}^{n} ( ∑_{j=1}^{m} ∥a_{j,l}∥_{A}^{p})^{1/p} ∥v_{l}∥_{A}. (25.6.10)

It is easy to see that this is less than or equal to the right side of (25.6.7), by the definition of ∥a∥_{M_{m,n}(A),p ∞}. This shows that

∥T_{a}∥_{op,A^{n}A^{m},1 p} ≤ ∥a∥_{M_{m,n}(A),p ∞}. (25.6.11)

Suppose for the moment that A has a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1. If u(l) is as in Section 25.2 for l = 1, …, n, then

∥a∥_{M_{m,n}(A),p ∞} = max_{1≤l≤n} ∥T_{a}(u(l))∥_{A^{m},p}, (25.6.12)

because of (25.2.10). This implies that

∥a∥_{M_{m,n}(A),p ∞} ≤ ∥T_{a}∥_{op,A^{n} A^{m},1 p} (25.6.13)

because ∥u(l)∥_{A^{n},1} = 1 (25.6.14)

for each l. It follows that

∥T_{a}∥_{op,A^{n}A^{m},1 p} = ∥a∥_{M_{m,n}(A),p ∞}. (25.6.15)

25.7 Some estimates for R_{b}

Let A be an algebra over the real or complex numbers in the strict sense with a submultiplicative norm ∥ · ∥_{A} again, and let n, n_{1} be positive integers. If b ∈ M_{n,n_{1}}(A) and v ∈ A^{n}, then R_{b}(v) ∈ A^{n_{1}} may be defined as in Section 25.2, and

∥(R_{b}(v))_{r}∥_{A} ≤ ∑_{l=1}^{n} ∥v_{l}∥_{A}∥b_{l,r}∥_{A} (25.7.1)

for each r = 1, …, n_{1}. One can use this to get that R_{b} is a bounded linear mapping from A^{n} into A^{n_{1}} with respect to the norms ∥ · ∥_{A^{n},p} and ∥ · ∥_{A^{m},q} defined in Section 25.4 for any p, q ≥ 1, as before. We would like to mention some estimates for the corresponding operator norms for R_{b} that are analogous to those discussed in the previous section.

564 CHAPTER 25. ALGEBRAS, MATRICES, AND NORMS

Note that

∥(R_{b}(v))_{r}∥_{A} ≤ ∥v∥_{A^{n},∞} ∑_{l=1}^{n} ∥b_{l,r}∥_{A} (25.7.2)

for each r = 1, …, n_{1}. This implies that ∥R_{b}(v)∥_{A^{n_{1}} ,1} ≤ ∥b∥_{M_{n,n_{1}} (A),1} ∥v∥_{A^{n},∞}, (25.7.3) so that ∥R_{b}∥_{op,A^{n}A^{n_{1}} ,∞ 1} ≤ ∥b∥_{M_{n,n_{1}} (A),1}, (25.7.4) as before. One can also use (25.7.2) to get that ∥R_{b}(v)∥_{A^{n_{1}} ,∞} ≤ ∥v∥_{A^{n},∞} ∥b∥_{M_{n,n_{1}} (A),1 ∞}, (25.7.5) where ∥b∥_{M_{n,n_{1}} (A),1 ∞} is as in (25.5.6). This means that ∥R_{b}∥_{op,A^{n}A^{n_{1}} ,∞ ∞} ≤ ∥b∥_{M_{n,n_{1}} (A),1 ∞}. (25.7.6)

If 1 ≤ p ≤ ∞, then ∥R_{b}(v)∥_{A^{n_{1}} ,p} ≤ ∥b∥_{M_{n,n_{1}} (A),∞ p} ∥v∥_{A^{n},1}, (25.7.7) where ∥b∥_{M_{n,n_{1}} (A),∞ p} is as in (25.5.5). This is analogous to obtaining (25.6.7) from (25.6.1). Thus ∥R_{b}∥_{op,A^{n}A^{n_{1}} ,1 p} ≤ ∥b∥_{M_{n,n_{1}} (A),∞ p}. (25.7.8)

Suppose for the moment again that A has a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1. If u(l) is as in Section 25.2, then

∥b∥_{M_{n,n_{1}} (A),∞ p} = max_{1≤l≤n} ∥R_{b}(u(l))∥_{A^{n_{1}} ,p}, (25.7.9)

because of (25.2.19). This means that ∥b∥_{M_{n,n_{1}} (A),∞ p} ≤ ∥R_{b}∥_{op,A^{n}A^{n_{1}} ,1 p}, (25.7.10) because of (25.6.14). It follows that ∥R_{b}∥_{op,A^{n}A^{n_{1}} ,1 p} = ∥b∥_{M_{n,n_{1}} (A),∞ p}, (25.7.11) as before.

25.8 Some norms of products

Let A be an algebra in the strict sense over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}, and let m, n, and n_{1} be positive integers. Also let a ∈ M_{m,n}(A) and b ∈ M_{n,n_{1}}(A) be given, so that a b ∈ M_{m,n_{1}}(A) may be defined as in Sections 25.1 and 25.2. Note that

∥(a b)_{j,r}∥_{A} ≤ ∑_{l=1}^{n} ∥a_{j,l}∥_{A} ∥b_{l,r}∥_{A} (25.8.1)

for each j = 1, …, m and r = 1, …, n_{1}.

If 1 ≤ p ≤ ∞, then

∥a b∥_{M_{m,n_{1}} (A),∞ p} ≤ ∥a∥_{M_{m,n}(A),∞ 1} ∥b∥_{M_{n,n_{1}} (A),∞ p}. (25.8.2)

To see this, it suffices to check that

( ∑_{r=1}^{n_{1}} ( ∑_{l=1}^{n} ∥a_{j,l}∥_{A} ∥b_{l,r}∥_{A})^{p})^{1/p} ≤ ∥a∥_{M_{m,n}(A),∞ 1} ∥b∥_{M_{n,n_{1}} (A),∞ p} (25.8.3)

for j = 1, …, m when p < ∞, and that

∑_{l=1}^{n} ∥a_{j,l}∥_{A} ∥b_{l,r}∥_{A} ≤ ∥a∥_{M_{m,n}(A),∞,1} ∥b∥_{M_{n,n_{1}} (A),∞ ∞} (25.8.4)

for j = 1, …, m and r = 1, …, n_{1}, for the p = ∞ case. One can verify (25.8.4) directly from the definitions. If p < ∞, then the left side of (25.8.3) is less than or equal to

∑_{l=1}^{n} ( ∑_{r=1}^{n_{1}} ∥a_{j,l}∥_{A}^{p} ∥b_{l,r}∥_{A}^{p})^{1/p} = ∑_{l=1}^{n} ∥a_{j,l}∥_{A} ( ∑_{r=1}^{n_{1}} ∥b_{l,r}∥_{A}^{p})^{1/p}, (25.8.5)

as in Section 25.6. If p = 1, then the left side of (25.8.3) is equal to (25.8.5), by interchanging the order of summation. It is easy to see that (25.8.5) is less than or equal to

( ∑_{l=1}^{n} ∥a_{j,l}∥_{A}) ∥b∥_{M_{n,n_{1}} (A),∞ p} (25.8.6)

for each j = 1, …, m, by the definition of ∥b∥_{M_{n,n_{1}} (A),∞ p}. This is less than or equal to the right side of (25.8.3) for each j, by the definition of ∥a∥_{M_{m,n}(A),∞ 1}.

Similarly,

∥a b∥_{M_{m,n_{1}} (A),p ∞} ≤ ∥a∥_{M_{m,n}(A),p ∞} ∥b∥_{M_{n,n_{1}} (A),1 ∞} (25.8.7)

for every p ≥ 1. In this case, one would like to check that

( ∑_{j=1}^{m} ( ∑_{l=1}^{n} ∥a_{j,l}∥_{A} ∥b_{l,r}∥_{A})^{p})^{1/p} ≤ ∥a∥_{M_{m,n}(A),p ∞} ∥b∥_{M_{n,n_{1}} (A),1 ∞} (25.8.8)

for r = 1, …, n_{1} when p < ∞, and that

∑_{l=1}^{n} ∥a_{j,l}∥_{A} ∥b_{l,r}∥_{A} ≤ ∥a∥_{M_{m,n}(A),∞ ∞} ∥b∥_{M_{n,n_{1}} (A),1 ∞} (25.8.9)

for j = 1, …, m and r = 1, …, n_{1} when p = ∞. One can verify (25.8.4) directly from the definitions, as before. If p < ∞, then the left side of (25.8.8) is less than or equal to

∑_{l=1}^{n} ( ∑_{j=1}^{m} ∥a_{j,l}∥_{A}^{p} ∥b_{l,r}∥_{A}^{p})^{1/p} = ∑_{l=1}^{n} ( ∑_{j=1}^{m} ∥a_{j,l}∥_{A}^{p})^{1/p} ∥b_{l,r}∥_{A}, (25.8.10)

as in Section 25.6, as before. If p = 1, then the left side of (25.8.8) is equal to (25.8.10), as before. Observe that (25.8.10) is less than or equal to

∥a∥_{M_{m,n}(A),p ∞} ( ∑_{l=1}^{n} ∥b_{l,r}∥_{A}) (25.8.11)

for each r = 1, …, n_{1}, by definition of ∥a∥_{M_{m,n}(A),p ∞}. This is less than or equal to the right side of (25.8.8), by the definition of ∥b∥_{M_{n,n_{1}} (A),1 ∞}.

In particular, ∥·∥_{M_{n}(A),∞ 1} and ∥·∥_{M_{n}(A),1 ∞} are submultiplicative on M_{n}(A). Suppose now that A has a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1. If I_{n} is as in Section 25.1, then it is easy to see that

∥I_{n}∥_{M_{n}(A),∞ p} = ∥I_{n}∥_{M_{n}(A),p ∞} = 1 (25.8.12)

for each p ≥ 1.

If A is associative, then (25.8.2) can be obtained from (25.6.15), and analogous statements for b and a b. Similarly, (25.8.7) can be obtained from (25.7.11) and its analogues for a and a b under these conditions.

25.9 Some estimates for T_{a}^{V}

Let V be a vector space over the real or complex numbers with a norm ∥ · ∥_{V}, and let m, n be positive integers. We may consider V as a left module over the algebra BL(V) of bounded linear mappings from V into itself, as in Section 20.2. If a ∈ M_{m,n}(BL(V)), then we can define a linear mapping T_{a}^{V} from V^{n} into V^{m} as in (25.3.1). More precisely,

(T_{a}^{V} (v))_{j} = ∑_{l=1}^{n} a_{j,l}(v_{l}) (25.9.1)

for each j = 1, …, m in this case, where a_{j,l} ∈ BL(V) for all j = 1, …, m and l = 1, …, n. This implies that

∥(T_{a}^{V} (v))_{j}∥_{V} ≤ ∑_{l=1}^{n} ∥a_{j,l}∥_{op,V V} ∥v_{l}∥_{V} (25.9.2)

for each j = 1, …, m.

One can use this to get that T_{a}^{V} is bounded as a linear mapping from V^{n} into V^{m} with respect to the norms ∥ · ∥_{V^{n},p} and ∥ · ∥_{V^{m},q} defined in Section 25.9. SOME ESTIMATES FOR T_{A}^{V} 567

25.4 for any p, q ≥ 1, as before. We would like to mention some estimates for the corresponding operator norm of T_{a}^{V}, as in Section 25.6. Let us begin by observing that

∥(T_{a}^{V} (v))_{j}∥_{V} ≤ ( ∑_{l=1}^{n} ∥a_{j,l}∥_{op,V V} ) ∥v∥_{V^{n},∞} (25.9.3)

for each j = 1, …, m, by (25.9.2). This implies that

∥T_{a}^{V} (v)∥_{V^{m},1} ≤ ∥a∥_{M_{m,n}(BL(V)),1} ∥v∥_{V^{n},∞}, (25.9.4)

so that ∥T_{a}^{V} ∥_{op,V^{n}V^{m},∞ 1} ≤ ∥a∥_{M_{m,n}(BL(V)),1}. (25.9.5) One can also use (25.9.3) to get that

∥T_{a}^{V} (v)∥_{V^{m},∞} ≤ ∥a∥_{M_{m,n}(BL(V)),∞ 1} ∥v∥_{V^{n},∞}, (25.9.6)

so that ∥T_{a}^{V} ∥_{op,V^{n}V^{m},∞ ∞} ≤ ∥a∥_{M_{m,n}(BL(V)),∞ 1}. (25.9.7)

If 1 ≤ p ≤ ∞, then

∥T_{a}^{V} (v)∥_{V^{m},p} ≤ ∥a∥_{M_{m,n}(B(V)),p ∞} ∥v∥_{V^{n},1}, (25.9.8)

as in Section 25.6. More precisely, this can be obtained from (25.9.2) in essentially the same way that (25.6.7) was obtained from (25.6.1). This means that ∥T_{a}^{V} ∥_{op,V^{n}V^{m},1 p} ≤ ∥a∥_{M_{m,n}(BL(V)),p ∞}. (25.9.9)

Remember that every linear mapping from V^{n} into V^{m} is of the form T_{a}^{V} for some a ∈ M_{m,n}(L(V)), as in Section 25.3. If T_{a}^{V} is bounded with respect to ∥ · ∥_{V^{n},p_{0}} and ∥ · ∥_{V^{m},p} for any p_{0}, p ≥ 1, then it is easy to see that

a ∈ M_{m,n}(BL(V)). (25.9.10)

More precisely, if v ∈ V^{n}, 1 ≤ l ≤ n, and only the lth component of v is nonzero, then (T_{a}^{V} (v))_{j} = a_{j,l}(v_{l}) (25.9.11) for each j = 1, …, m. In this case,

∥a_{j,l}(v_{l})∥_{V} ≤ ∥T_{a}^{V} (v)∥_{V^{m},p} ≤ ∥T_{a}^{V} ∥_{op,V^{n}V^{m},p_{0} p} ∥v∥_{V^{n},p_{0}}
(25.9.12)  = ∥T_{a}^{V} ∥_{op,V^{n}V^{m},p_{0} p} ∥v_{l}∥_{V}

for each j = 1, …, m.

This implies that a_{j,l} ∈ BL(V) for each j = 1, …, m, with

∥a_{j,l}∥_{op,V V} ≤ ∥T_{a}^{V} ∥_{op,V^{n}V^{m},p_{0} p}. (25.9.13)

Thus ∥a∥_{M_{m,n}(BL(V)),∞} ≤ ∥T_{a}^{V} ∥_{op,V^{n}V^{m},p_{0} p} (25.9.14) where the left side is as in (25.5.2). It follows that

∥T ∥_{op,V^{n}V^{m},1 ∞} = ∥a∥_{M_{m,n}(BL(V)),∞}, (25.9.15)

by (25.9.9), with p = ∞.

25.10 Transposes and involutions

Let k be a field, and let A be an algebra in the strict sense over k. If a ∈ M_{m,n}(A) for some positive integers m, n, then the corresponding transpose matrix a^{t} in M_{n,m}(A) is defined as usual by

a_{l,j}^{t} = a_{j,l} (25.10.1)

for each j = 1, …, m and l = 1, …, n. Note that

a ↦ a^{t} (25.10.2)

defines a one-to-one mapping from M_{m,n}(A) onto M_{n,m}(A), and that this is a linear mapping from M_{m,n}(A) into M_{n,m}(A), as vector spaces over k. Of course, (a^{t})^{t} = a. (25.10.3)

Let n_{1} be another positive integer, and let b ∈ M_{n,n_{1}}(A) be given, so that a b ∈ M_{m,n_{1}}(A) may be defined as in Section 25.2. If A is commutative, then

(a b)^{t} = b^{t} a^{t}. (25.10.4)

In particular, (25.10.2) defines an algebra involution on M_{n}(A) when A is commutative.

Remember that the opposite algebra A^{op} is the same as A as a vector space over k, with the order of multiplication exchanged, as in Section 20.3. If x ∈ A, then we may use x^{op} to refer to x as an element of A^{op}, as before. Equivalently, x ↦ x^{op} is the identity mapping on A, considered as a mapping from A onto A^{op}, and it is an opposite algebra isomorphism from A onto A^{op}, by construction. Of course, the opposite algebra (A^{op})^{op} of A^{op} is the same as A.

Let us use ã to refer to a as an element of M_{m,n}(A^{op}), so that

ã_{j,l} = (a_{j,l})^{op} (25.10.5)

for each j = 1, …, m and l = 1, …, n, as an element of A^{op}. Similarly, b̃ is an element of M_{n,n_{1}}(A^{op}), and ã b̃ ∈ M_{m,n_{1}}(A^{op}) is defined by

(ã b̃)_{j,r} = ∑_{l=1}^{n} ã_{j,l} b̃_{l,r} = ∑_{l=1}^{n}(a_{j,l})^{op} (b_{l,r})^{op} = ∑_{l=1}^{n}(b_{l,r} a_{j,l})^{op} (25.10.6)

for j = 1, …, m and r = 1, …, n_{1}.

Let a^{T} be the transpose of a considered as an element of M_{n,m}(A^{op}), so that

a^{T} = ˜(a^{t}), (25.10.7)

which is to say that a_{l,j}^{T} = (a_{j,l})^{op} = ã_{j,l}. (25.10.8)

Equivalently, a^{T} = a^{t}, the ordinary transpose of ã. Note that

(a^{T})^{T} = a. (25.10.9)

It is easy to see that (a b)^{T} = b^{T} a^{T} , (25.10.10)

as elements of M_{n_{1},m}(A^{op}).

It follows that a ↦ a^{T} (25.10.11)

is an opposite algebra isomorphism from M_{n}(A) onto M_{n}(A^{op}). This means that this defines an isomorphism from the opposite algebra M_{n}(A)^{op} of M_{n}(A) onto M_{n}(A^{op}).

Let x ↦ x^{∗_{A}} be an opposite algebra isomorphism from A onto itself, which may be conjugate-linear when k = C. If a ∈ M_{m,n}(A), then define a^{∗} = a^{∗_{Mm,n(A)}} ∈ M_{n,m}(A) by a_{l,j}^{∗} = (a_{j,l})^{∗_{A}} (25.10.12)

for j = 1, …, m and l = 1, …, n. This defines a one-to-one linear mapping from M_{m,n}(A) onto M_{n,m}(A), unless k = C and x ↦ x^{∗_{A}} is conjugate-linear, in which case a ↦ a^{∗} is conjugate-linear as well. If x ↦ x^{∗_{A}} is an involution on A, then (a^{∗})^{∗} = a. (25.10.13)

One can check that (a b)^{∗} = b^{∗} a^{∗}. (25.10.14)

In particular, a ↦ a^{∗} = a^{∗_{Mn(A)}} is an opposite algebra isomorphism from M_{n}(A) onto itself, which is conjugate-linear when k = C and x ↦ x^{∗_{A}} is conjugatelinear on A. If x ↦ x^{∗_{A}} is an involution on A, then a ↦ a^{∗} is an involution on M_{n}(A), as in (25.10.13).

25.11 Traces of matrices

Let k be a field, let A be a vector space over k, and let n be a positive integer. If a ∈ M_{n}(A), then the trace of a is defined as usual by

tr a = ∑_{j=1}^{n} a_{j,j}. (25.11.1)

This defines a linear mapping from M_{n}(A) into A, as vector spaces over k. Note that tr a^{t} = tr a. (25.11.2)

Suppose now that A is an algebra in the strict sense over k, and let m be another positive integer. If a ∈ M_{m,n}(A) and b ∈ M_{n,m}(A), then a b ∈ M_{m}(A), b a ∈ M_{n}(A),

tr(a b) = ∑_{j=1}^{m} ∑_{l=1}^{n} a_{j,l} b_{l,j}, (25.11.3)

570 CHAPTER 25. ALGEBRAS, MATRICES, AND NORMS

and

tr(b a) = ∑_{l=1}^{n} ∑_{j=1}^{m} b_{l,j} a_{j,l}. (25.11.4)

If A is commutative, then it follows that tr(a b) = tr(b a). (25.11.5)

Let x ↦ x^{∗_{A}} be an involution on A, which may be conjugate-linear when k = C. If a ∈ M_{n}(A) and a^{∗} ∈ M_{n}(A) is as in (25.10.12), then tr a^{∗} = (tr a)^{∗_{A}}. (25.11.6) If a, b ∈ M_{m,n}(A) and a^{∗}, b^{∗} ∈ M_{n,m}(A) are as in (25.10.12), then a b^{∗}, b a^{∗} are elements of M_{m}(A), a^{∗} b, b^{∗} a ∈ M_{n}(A), and (a b^{∗})^{∗} = b a^{∗}, (a^{∗} b)^{∗} = b^{∗} a. (25.11.7) Thus (25.11.8) tr(b a^{∗}) = (tr(a b^{∗}))^{∗_{A}}, tr(b^{∗} a) = (tr(a^{∗} b))^{∗_{A}}. If A is commutative, then (25.11.9) tr(a b^{∗}) = tr(b^{∗} a), tr(a^{∗} b) = tr(b a^{∗}), as in (25.11.5).

Suppose that A is commutative, so that we can take x ↦ x^{∗_{A}} to be the identity mapping on A, and (25.10.12) is the same as the transpose of a matrix. If a, b ∈ M_{m,n}(A), then we get that

(25.11.10) tr(b a^{t}) = tr(a b^{t}) = tr(b^{t} a) = tr(a^{t} b) = ∑_{j=1}^{m} ∑_{l=1}^{n} a_{j,l} b_{j,l}.

If A = k = R, then this defines an inner product on M_{m,n}(R). The norm associated to this inner product is the same as the norm ∥ · ∥_{M_{m,n}(R),2} defined in Section 25.5, using the absolute value of a real number as the norm on R.

Similarly, if A = k = C, then we can use complex-conjugation as the involution on A. If a ∈ M_{m,n}(C), then a^{∗} ∈ M_{n,m}(C) is defined by a_{l,j}^{∗} = \overline{a_{j,l}} (25.11.11) for j = 1, …, m and l = 1, …, n, as in (25.10.12). If a ∈ M_{n}(C), then tr a^{∗} = \overline{tr a}, (25.11.12) as in (25.11.6). In this case,

tr(a b^{∗}) = tr(b^{∗} a) = ∑_{j=1}^{m} ∑_{l=1}^{n} a_{j,l} \overline{b_{j,l}} (25.11.13)

defines an inner product on M_{n}(C). The norm associated to this inner product is the same as the norm ∥ · ∥_{M_{m,n}(C),2} defined in Section 25.5 again, using the absolute value of a complex number as the norm on C.

25.12 Traces of linear mappings

Let k be a field, and let V be a vector space over k of positive finite dimension. If T is a linear mapping from V into itself, then the trace tr T ∈ k is defined by choosing a basis for V, and taking the trace of the matrix corresponding to T with respect to this basis. If R is another linear mapping from V into itself, then tr(R ∘ T) = tr(T ∘ R), (25.12.1)

as in (25.11.5). One can use this to show that the trace does not depend on the choice of basis for V, by a standard argument. Of course, the trace defines a linear mapping from L(V) into k, as before.

Suppose now that k = R or C, and that ⟨·, ·⟩_{V} is an inner product on V. If u_{1}, …, u_{n} is an orthonormal basis for V with respect to this inner product, then

v = ∑_{j=1}^{n}⟨v, u_{j}⟩_{V} u_{j} (25.12.2)

for each v ∈ V. If T is a linear mapping from V into itself, then

T (u_{l}) = ∑_{j=1}^{n}⟨T (u_{l}), u_{j}⟩_{V} u_{j} (25.12.3)

for each j = 1, …, n. Thus the n × n matrix corresponding to T and this basis is given by ⟨T (u_{l}), u_{j}⟩_{V} . (25.12.4)

This implies that

tr T = ∑_{j=1}^{n}⟨T (u_{j}), u_{j}⟩_{V} . (25.12.5)

If T^{∗_{V}} ∈ L(V) is the adjoint of T with respect to this inner product, then the matrix corresponding to T^{∗_{V}} and this basis is

⟨T^{∗_{V}} (u_{l}), u_{j}⟩_{V} = ⟨u_{l}, T (u_{j})⟩_{V} . (25.12.6)

This is the transpose of the matrix (25.12.4) associated to T in the real case, and in the complex case it is the same as taking the complex conjugates of the entries of the transpose of the matrix (25.12.4) associated to T. In particular,

tr T^{∗_{V}} = tr T (25.12.7)

in the real case, and tr T^{∗_{V}} = \overline{tr T} (25.12.8)

in the complex case. If T is self-adjoint, then it follows that

tr T ∈ R (25.12.9)

572 CHAPTER 25. ALGEBRAS, MATRICES, AND NORMS in the complex case.

In both cases, ⟨R, T ⟩_{L(V)} = tr(R ∘ T^{∗_{V}}) (25.12.10) defines an inner product on L(V). More precisely,

⟨T, T ⟩_{L(V)} = tr(T ∘ T^{∗_{V}}) = tr(T^{∗_{V}} ∘ T)
(25.12.11)  = ∑_{j=1}^{n}⟨T^{∗_{V}} (T (u_{j})), u_{j}⟩_{V} = ∑_{j=1}^{n}⟨T (u_{j}), T (u_{j})⟩_{V} .

This inner product corresponds to the one defined on M_{n}(R) or M_{n}(C), as appropriate, in the previous section, using the correspondence between linear mappings from V into itself and n × n matrices of real numbers obtained from this orthonormal basis for V.

Remember that if T is self-adjoint, then T is said to be nonnegative with respect to the inner product when ⟨T (v), v⟩_{V} ≥ 0 (25.12.12) for every v ∈ V, as in Section 8.10. In this case, tr T ≥ 0, (25.12.13) by (25.12.5).

If T is self-adjoint and R is any linear mapping from V into itself, then it is easy to see that R^{∗V} ∘ T ∘ R is self-adjoint too. Note that ⟨R^{∗_{V}} (T (R(v))), v⟩_{V} = ⟨T (R(v)), R(v)⟩_{V} (25.12.14) for every v ∈ V. If T is nonnegative, then it follows that R^{∗_{V}} ∘ T ∘ R is nonnegative as well. In particular, this means that tr(R^{∗_{V}} ∘ T ∘ R) ≥ 0. (25.12.15) Thus tr(T ∘ R ∘ R^{∗_{V}}) ≥ 0, (25.12.16) by (25.12.1).

If B and T are self-adjoint linear mappings from V into itself, then tr(B ∘ T) ∈ R (25.12.17) in the complex case, because (25.12.18) \overline{tr(B ∘ T)} = tr(B ∘ T)^{∗_{V}} = tr(T^{∗_{V}} ∘ B^{∗_{V}}) = tr(T ∘ B) = tr(B ∘ T). If B and T are nonnegative, then tr(B ∘ T) ≥ 0. (25.12.19)

This can be seen using a self-adjoint square root of B and (25.12.16), or using an orthonormal basis of eigenvectors for T.

If u ∈ V and ∥u∥_{V} = 1, where ∥ · ∥_{V} is the norm associated to the inner product, then P_{u}(v) = ⟨v, u⟩_{V} u (25.12.20)

is the orthogonal projection of V onto the one-dimensional subspace spanned by u. Note that P_{u} is self-adjoint on V, and that

⟨P_{u}(v), v⟩_{V} = |⟨v, u⟩_{V} |^{2} (25.12.21)

for every v ∈ V, so that P_{u} is nonnegative on V. If B is any linear mapping from V into itself, then

tr(B ∘ P_{u}) = ⟨B(u), u⟩_{V} , (25.12.22)

as one can see using an orthonormal basis that contains u. If B is self-adjoint, then it follows that B is nonnegative when (25.12.19) holds for all nonnegative self-adjoint linear mappings T from V into itself.

25.13 Traces and linear functionals

Let k be a field, let m, n be positive integers, and let b ∈ M_{n,m}(k) be given. Observe that λ_{b}(a) = tr(a b) (25.13.1)

defines a linear functional on M_{m,n}(k), as a vector space over k. It is easy to see that every linear functional on M_{m,n}(k) corresponds to a unique b ∈ M_{n,m}(k) in this way, using (25.11.3).

Suppose now that m = n, and observe that

(25.13.2) λ_{b}(a^{t}) = tr(a^{t} b) = tr(a^{t} b)^{t} = tr(b^{t} a) = tr(a b^{t}) = λ_{b^{t}}(a)

for every a ∈ M_{n}(k). It follows that

λ_{b}(a^{t}) = λ_{b}(a) (25.13.3)

for every a ∈ M_{n}(k) if and only if

b^{t} = b. (25.13.4)

Suppose for the moment that k = C, and let a^{∗} be as in (25.11.11) for each a ∈ M_{n}(C). Observe that

(25.13.5) λ_{b}(a^{∗}) = tr(a^{∗} b) = \overline{tr(a^{∗} b)^{∗}} = \overline{tr(b^{∗} a)} = \overline{tr(a b^{∗})} = \overline{λ_{b^{∗}}(a)}

for every a ∈ M_{n}(C). Thus λ_{b}(a^{∗}) = \overline{λ_{b}(a)} (25.13.6) for every a ∈ M_{n}(C) if and only if

b^{∗} = b. (25.13.7)

Let (V, ⟨·, ·⟩_{V}) be an inner product space over the real or complex numbers of dimension n. If B is a linear mapping from V into itself, then

λ_{B}(T) = tr(B ∘ T) (25.13.8)

defines a linear functional on the space L(V) of all linear mappings from V into itself. Every linear functional on L(V) corresponds to a unique B ∈ L(V) in this way, as before.

If T ∈ L(V), then let T^{∗_{V}} be the adjoint of T with respect to the inner product, as before. This defines an involution on L(V) that is conjugate-linear in the complex case, as usual. Using the earlier remarks, we get that λ_{B} is symmetric or Hermitian symmetric on L(V) in the strong sense, as in Section 23.4, if and only if B is self-adjoint. This also uses an orthonormal basis for V, to get a correspondence between L(V) and M_{n}(R) or M_{n}(C), as appropriate, such that the adjoint of a linear mapping corresponds to the transpose or complexconjugate of the transpose of the associated matrix, as appropriate.

If B is self-adjoint and nonnegative on V, then

(25.13.9) λ_{B} is positive on L(V),

with respect to the involution T ↦ T^{∗_{V}} on L(V), in the sense of Section 18.7, as in the previous section. Conversely, if B is self-adjoint on V, then one can check that (25.13.9) implies that B is nonnegative on V.

If u_{1}, …, u_{n} is an orthonormal basis for V and b_{1}, …, b_{n} are real numbers, then

B(v) = ∑_{j=1}^{n} b_{j} ⟨v, u_{j}⟩_{V} u_{j} (25.13.10)

defines a self-adjoint linear mapping from V into itself. It is well known that every self-adjoint linear mapping from V into itself can be expressed in this way. One can check that

λ_{B}(T) = tr(B ∘ T) = ∑_{j=1}^{n} b_{j} ⟨T (u_{j}), u_{j}⟩_{V} (25.13.11)

for every T ∈ L(V) in this case. Note that

⟨B(v), v⟩_{V} = ∑_{j=1}^{n} b_{j} |⟨v, u_{j}⟩_{V} |^{2} (25.13.12)

for every v ∈ V. This implies that B is nonnegative if and only if b_{j} ≥ 0 for each j.

Clearly

λ_{B}(I_{V}) = tr B = ∑_{j=1}^{n} b_{j}, (25.13.13)

where I_{V} is the identity mapping on V. Remember that P = P (L(V)) is the set of positive linear functionals on L(V) whose value at I_{V} is less than or equal to 1, as in Section 24.9. This corresponds to the set of nonnegative self-adjoint linear mappings from V into itself with trace less than or equal to 1, as in (25.13.8). A nonzero extreme point of this set satisfies

λ_{B}(I_{V}) = tr B = 1, (25.13.14)

as in Section 24.10. If B is a nonzero extreme point of this set, and B is as in (25.13.10), then one can check that b_{l} = 0 for all but one l, so that B has rank one. Of course, (25.13.14) implies that the nonzero b_{l} is equal to 1 in this case. Conversely, one can verify that the nonnegative self-adjoint linear mappings on V of this type correspond to extreme points of P.

25.14 A continuity result

An associative algebra A over a field k is said to be simple if A and {0} are the only two-sided ideals in A, as on p21 of [11], p469 of [151], and p335 of [226]. More precisely, this formulation corresponds to part (c) of Exercise 5 on p337 of [226].

If A is an associative algebra over the real or complex numbers with a submultiplicative norm ∥ · ∥_{A}, then A is said to be topologically simple if A and {0} are the only two-sided ideals in A that are closed sets with respect to the metric associated to the norm, as on p22 of [11]. Of course, if A is simple, then A is topologically simple. If A is a Banach algebra with a multiplicative identity element, and if A is topologically simple, then A is simple, as in Corollary 1 on p24 of [11]. In this case, the closure of a proper two-sided ideal in A is a proper two-sided ideal in A, as in Section 6.11.

Let k be a field again, and let B be a division algebra over k, as in Section 21.3. Also let n be a positive integer, and consider the corresponding associative algebra M_{n}(B), as in Section 25.1. The elements of M_{n}(B) correspond to linear mappings from B^{n} into itself, as a vector space over k, that are homomorphisms from B^{n} into itself, as a right module over B, as in Section 25.2.

It is well known that M_{n}(B) is simple, as mentioned on p335 of [226]. This is probably more familiar when B is a field, in which case one might as well take B = k. This case corresponds to Exercise 6 on p337 of [226]. Using suitable bases for B^{n} as a right module over itself, one can check that a nonzero onesided ideal in M_{n}(B) contains a nonzero element that corresponds to a module homomorphism from B^{n} into itself that has rank one, which is to say that it maps B^{n} onto the submodule generated by a single nonzero element. One can use this to get that a nonzero two-sided ideal in M_{n}(B) contains all elements of M_{n}(B).

Suppose now that B = k = C, or k = R and B = R, C, or the algebra H of quaternions, as in Section 7.1. Let (A, ∥ · ∥_{A}) be a Banach algebra over the real or complex numbers, as appropriate, with a multiplicative identity element e_{A} with ∥e_{A}∥_{A} = 1. Also let ϕ be a homomorphism from A into M_{n}(A), as algebras over the real or complex numbers, as appropriate. If

ϕ(A) = M_{n}(B), (25.14.1)

then (25.14.2) ϕ is continuous,

as in Exercise (5) on p25 of [11]. Although this is stated in [11] in the complex case, the same argument works in the real cases.

Of course, in each of these cases, B may be considered as a Banach algebra over the real or complex numbers, as appropriate, using the standard absolute value of a real or complex number, or the standard norm of a quaternion. We may consider M_{n}(B) as a Banach algebra with respect to a suitable norm as well. Using a suitable norm on B^{n}, as in Section 25.4, we get the correspoonding operator norm on linear mappings from B^{n} into itself, which leads to a norm on M_{n}(B), by taking the operator norm of a linear mapping from B^{n} into itself associated to an element of M_{n}(B). In particular, one could use ∥ · ∥_{M_{n}(B),∞ 1}, or one could use ∥ · ∥_{M_{n}(B),1 ∞}, as in Section 25.8. The continuity property (25.14.2) is the same as saying that ϕ is bounded as a linear mapping from A into M_{n}(B) with respect to such a norm on M_{n}(B).

Note that (25.14.3) ker ϕ is a two-sided ideal in A,

as mentioned in Section 19.2. We also have that

ker ϕ ≠ A, (25.14.4)

by (25.14.1). More precisely,

(25.14.5) ker ϕ is a maximal proper two-sided ideal in A,

because M_{n}(B) has no nonzero proper two-sided ideals, as before. This implies that (25.14.6) ker ϕ is a closed set in A

with respect to the metric associated to the norm, because A is a Banach algebra with a multiplicative identity element, as in Section 6.11.

Using ϕ, we get an isomorphism ψ from A/ ker ϕ onto M_{n}(B), as algebras over the real or complex numbers, as appropriate. In particular, this means that

(25.14.7) dim A/ ker ϕ = dim M_{n}(B),

as vector spaces over the real or complex numbers, as appropriate.

25.15. SOME NORMS ON C^{2} 577

Because ker ϕ is a closed linear subspace of A, the quotient norm ∥ · ∥_{A/ ker ϕ} may be defined on A/ ker ϕ as in Section 6.10. Remember that the natural quotient mapping q from A onto A/ ker ϕ is a bounded linear mapping with respect to the quotient norm, by construction. In order to get that ϕ is a bounded linear mapping, it suffices to show that

(25.14.8) ψ is a bounded linear mapping

with respect to the quotient norm, because

ϕ = ψ ∘ q, (25.14.9)

by construction. The boundedness of ψ can be obtained using the fact that M_{n}(B) has finite dimension as a vector space over the real or complex numbers, as appropriate, as in Section 1.11.

25.15 Some norms on C^{2}

Let t be a positive real number, and put

∥w∥_{1,t} = |w_{1}| + t |w_{2}| (25.15.1)

for every w ∈ C^{2}, as in Exercise 13 on p289 of [286]. This defines a norm on C^{2}, as a vector space over the complex numbers. This leads to a corresponding operator norm ∥ · ∥_{op,t} on the algebra L(C^{2}) of linear mappings from C^{2} into itself, as a vector space over C. If R ∈ L(C^{2}), then we let R^{∗} be the adjoint of R with respect to the standard inner product on C^{2}. Thus R^{∗} corresponds to the 2×2 matrix of complex numbers obtained by taking the complex conjugates of the entries of the transpose of the matrix associated to R in the usual way.

If R ∈ L(C^{2}), then r_{L(C^{2})}(R) may be defined as in Section 6.13, with respect to ∥ · ∥_{op,t} on L(C^{2}), or with respect to the operator norm associated to the standard norm on C^{2}, for instance. This is the same as the maximum of the absolute values of the elements of the spectrum σ(R) = σ_{L(V)}(R) of R with respect to L(C^{2}), as in Section 6.14. Of course, σ(R) does not depend on the norm that we use on L(V), so that r_{L(C^{2})}(R) does not depend on the choice of norm either. This is also related to some remarks in Section 7.2.

Let X be the linear mapping from C^{2} into itself corresponding to the matrix \binom{0 t^{2}}{1 0}, as in [286], so that X(w) = (t^{2} w_{2}, w_{1}) (25.15.2)

for every w ∈ C^{2}. Observe that

∥X(w)∥_{1,t} = t^{2} |w_{2}| + t |w_{1}| = t ∥w∥_{1,t} (25.15.3)

for every w ∈ C^{2}. In particular,

∥X∥_{op,t} = t, (25.15.4)

as in part (a) of Exercise 13 on p289 of [286]. The operator norm of X with respect to the standard Euclidean norm on C^{2}, or with respect to the norm ∥ · ∥_{p} defined in Section 1.3 for any p ≥ 1, is equal to

max(1, t^{2}). (25.15.5)

578 CHAPTER 25. ALGEBRAS, MATRICES, AND NORMS

Of course, X^{∗} corresponds to the matrix \binom{0 1}{t^{2} 0}, so that X^{∗}(w) = (w_{2}, t^{2} w_{1}) (25.15.6) for every w ∈ C^{2}. Thus ∥X^{∗}(w)∥_{1,t} = |w_{2}| + t^{3} |w_{1}| (25.15.7) for every w ∈ C^{2}. One can use this to get that

(25.15.8)  ∥X^{∗}∥_{op,t} = t^{3} when t ≥ 1
= t^{−1} when t ≤ 1.

The operator norm of X^{∗} with respect to the standard Euclidean norm on C^{2}, or with respect to the norm ∥ · ∥_{p} on C^{2} for any p ≥ 2, is equal to (25.15.5) again.

Clearly X((t, 1)) = (t^{2}, t) = t (t, 1) (25.15.9) and X((t, −1)) = (−t^{2}, t) = −t (t, −1). (25.15.10) This implies that the spectrum of X is given by σ_{L(C^{2})}(X) = {t, −t}, (25.15.11) as in part (b) of Exercise 13 on p289 of [286]. It is easy to see directly that X(X(w)) = X((t^{2} w_{2}, w_{1})) = t^{2} w (25.15.12) for every w ∈ C^{2}. Using (25.15.11), we get that r_{L(C^{2})}(X) = t, (25.15.13) which could also be verified in other ways.

It follows that σ_{L(C^{2})}(X^{∗}) = {t, −t}, (25.15.14) as in [286]. In fact, X^{∗}((1, t)) = (t, t^{2}) = t (1, t) (25.15.15) and X^{∗}((−1, t)) = (t, −t^{2}) = −t (−1, t). (25.15.16) It is easy to see directly again that X^{∗}(X^{∗}(w)) = X^{∗}((w_{2}, t^{2} w_{1})) = t^{2} w (25.15.17) for every w ∈ C^{2}. One can use (25.15.14) to get that r_{L(C^{2})}(X^{∗}) = t, (25.15.18) which could be verified in other ways too. In particular, this is related to a remark in Section 17.11.

Observe that

X^{∗}(X(w)) = X^{∗}((t^{2} w_{2}, w_{1})) = (w_{1}, t^{4} w_{2}) (25.15.19)

and X(X^{∗}(w)) = X((w_{2}, t^{2} w_{1})) = (t^{4} w_{1}, w_{2}) (25.15.20)

for every w ∈ C^{2}. Thus

σ_{L(C^{2})}(X^{∗} ∘ X) = σ_{L(C^{2})}(X ∘ X^{∗}) = {1, t^{4}}, (25.15.21)

as in part (c) of Exercise 13 on p289 of [286]. This implies that

r_{L(C^{2})}(X^{∗} ∘ X) = r_{L(C^{2})}(X ∘ X^{∗}) = max(1, t^{4}), (25.15.22)

which could be verified in other ways as well. This is the same as the operator norms of X^{∗} ∘ X and X ∘ X^{∗} with respect to ∥ · ∥_{1,t}, or with respect to ∥ · ∥_{p} on C^{2} for any p ≥ 1.

Note that (25.15.22) is not less than or equal to the product of (25.15.13) and (25.15.18) when t > 1. Similarly, (25.15.21) is not contained in the set of products of elements of (25.15.11) and (25.15.14) when t ≠ 1. This corresponds to part of part (e) of Exercise 13 on p289 of [286]. More precisely, this shows that the hypothesis that two elements of a Banach algebra commute are needed for some statements in Sections 7.2 and 14.6, as mentioned in [286].

By construction,

(X^{∗} + X)(w) = (1 + t^{2}) (w_{2}, w_{1}) (25.15.23)

for every w ∈ C^{2}, so that

(X + X^{∗})((1, 1)) = (1 + t^{2}) (1, 1) (25.15.24)

and (X + X^{∗})((1, −1)) = −(1 + t^{2}) (1, −1). (25.15.25)

This means that σ_{L(C^{2})}(X + X^{∗}) = {1 + t^{2}, −1 − t^{2}}, (25.15.26)

as in part (d) of Exercise 13 on p289 of [286]. It follows that

r_{L(C^{2})}(X + X^{∗}) = 1 + t^{2}, (25.15.27)

which could be verified in other ways, as susual. This is the same as the operator norm of X + X^{∗} with respect to ∥ · ∥_{1,t}, or with respect to ∥ · ∥_{p} on C^{2} for any p ≥ 1.

If t ≠ 1, then (25.15.27) is not less than or equal to the sum of (25.15.13) and (25.15.18), because (t − 1)^{2} > 0. We also have that (25.15.26) is not contained in the set of sums of elements of (25.15.11) and (25.15.14) when t ≠ 1. This corresponds to another part of part (e) of Exercise 13 on p289 of [286]. This shows that the hypothesis that two elements of a Banach algebra commute are needed for some statements in Sections 9.7 and 14.6, as mentioned in [286].

Let B be the linear mapping from C^{2} into itself defined by

B(w) = (w_{1} + w_{2}, w_{1} + w_{2}), (25.15.28)

which corresponds to the matrix \binom{1 1}{1 1}. One can check that B is self-adjoint and nonnegative with respect to the standard inner product on C^{2}. This implies that λ_{B}(R) = tr(B ∘ R) (25.15.29)

defines a positive linear functional on L(C^{2}), as in Section 25.13. This corresponds to part (f) of Exercise 13 on p289 of [286]. More precisely, λ_{B}(R) is equal to the sum of the four entries of the matrix associated to R, as in Section 25.11.

Note that λ_{B}(I_{C^{2}}) = tr B = 2, (25.15.30)

where I_{C^{2}} is the identity mapping on C^{2}. We also have that

λ_{B}(X) = 1 + t^{2}. (25.15.31)

This shows that the dual norm of λ_{B} with respect to ∥ · ∥_{op,t} on L(V) is at least t^{−1} + t, because of (25.15.4). This is larger than (25.15.30) when t ≠ 1, so that (t − 1)^{2} > 0, as in part (g) of Exercise 13 on p289 of [286]. This is related to some of the statements in Sections 18.11 and 18.13, as in [286].

Part (h) of Exercise 13 on p290 of [286] deals with positive linear functionals λ on L(V), or equivalently M_{2}(C), with λ(I_{C^{2}}) ≤ 1. This does not involve the norm (25.15.1), and is related to some of the remarks in Section 25.13, with V = C^{2} and n = 2. This is also related to the necessity of commutativity in Section 24.14, as in [286].

Appendix A

Absolute values on fields

A.1 Metrics and ultrametrics

Let X be a set, and let d(x, y) be a nonnegative real-valued function defined for x, y ∈ X. As usual, d(x, y) is said to be a metric on X if it satisfies the following three conditions. First,

(A.1.1) d(x, y) = 0 if and only if x = y.

Second, (A.1.2) d(x, y) = d(y, x) for every x, y ∈ X.

Third, (A.1.3) d(x, z) ≤ d(x, y) + d(y, z) for every x, y, z ∈ X,

which is known as the triangle inequality.

The condition

(A.1.4) d(x, z) ≤ max(d(x, y), d(y, z)) for every x, y, z ∈ X

is the ultrametric version of the triangle inequality. If d(x, y) satisfies (A.1.1), (A.1.2), and (A.1.4) on X, then d(x, y) is said to be an ultrametric on X. Of course, (A.1.4) implies (A.1.3), so that an ultrametric on X is a metric in particular.

The discrete metric on X is defined by putting d(x, y) equal to 0 when x = y, and to 1 otherwise. It is easy to see that this is an ultrametric on X.

If d(·, ·) is any metric on X, then the open ball in X centered at x ∈ X with radius r > 0 with respect to d(·, ·) is defined as usual by

B(x, r) = B_{d}(x, r) = {y ∈ X : d(x, y) < r}. (A.1.5)

Similarly, the closed ball in X centered at x with radius r ≥ 0 with respect to d(·, ·) is defined by

\overline{B}(x, r) = \overline{B}_{d}(x, r) = \{y ∈ X : d(x, y) ≤ r\}. (A.1.6)

It is well known that open balls are open sets in X, and that closed balls are closed sets. If d(·, ·) is an ultrametric on X, then one can check that open balls in X are closed sets. One can also verify that closed balls in X of positive radius are open sets in this case.

Let a be a positive real number. If a ≤ 1, then it is well known that

(r + t)^{a} ≤ r^{a} + t^{a} (A.1.7)

for all nonnegative real numbers r, t, as in Section 1.12. If d(·, ·) is a metric on X, then it follows that d(x, y)^{a} (A.1.8)

is a metric on X as well. If d(·, ·) is an ultrametric on X, then it is easy to see that (A.1.8) is an ultrametric on X for every a > 0. In both cases, we have that

B_{d^{a}}(x, r^{a}) = B_{d}(x, r) (A.1.9)

for every x ∈ X and r > 0, and

\overline{B}_{d^{a}}(x, r^{a}) = \overline{B}_{d}(x, r) (A.1.10)

for every x ∈ X and r ≥ 0.

In particular, one can use (A.1.9) to get that d(·, ·) and (A.1.8) determine the same topology on X. This means that convergence of sequences in X is the same for d(·, ·) and (A.1.8), which is easy to see directly anyway. These metrics also determine the same Cauchy sequences in X, for instance. Using this, one can check that X is complete as a metric space with respect to d(·, ·) if and only if X is complete with respect to (A.1.8).

If d(·, ·) is a metric on X and Y is a subset of X, then the restriction of d(x, y) to x, y ∈ Y defines a metric on Y. Similarly, if d(·, ·) is an ultrametric on X, then the restriction of d(x, y) to x, y ∈ Y is an ultrametric on Y.

See [59, 156, 340] for some topics related to ultrametrics.

A.2 Absolute value functions

Let k be a field. A nonnegative real-valued function | · | on k is said to be an absolute value function on k if it satisfies the following three conditions. First,

(A.2.1) |x| = 0 if and only if x = 0.

Second, (A.2.2) |x y| = |x| |y| for every x, y ∈ k.

Third, (A.2.3) |x + y| ≤ |x| + |y| for every x, y ∈ k.

Let us say that |·| is an ultrametric absolute value function on k if it satisfies (A.2.1), (A.2.2), and

(A.2.4) |x + y| ≤ max(|x|, |y|) for every x, y ∈ k.

Clearly (A.2.4) implies (A.2.3), so that an ultrametric absolute value function on k is an absolute value function in the previous sense.

It is well known that the standard absolute values on R and C are absolute value functions in this sense. The trivial absolute value function may be defined on any field k by putting |x| equal to 0 when x = 0, and to 1 otherwise. It is easy to see that this defines an ultrametric absolute value function on k.

If | · | is an absolute value function on a field k and k_{0} is a subfield of k, then the restriction of |x| to x ∈ k_{0} defines an absolute value function on k_{0}. If | · | is an ultrametric absolute value function on k, then its restriction to k_{0} is an ultrametric absolute value function as well.

If | · | is an absolute value function on a field k, then

d(x, y) = |x − y| (A.2.5)

defines a metric on k. More precisely, in order to check that this is symmetric in x and y, one needs to verify that

| − 1_{k}| = 1, (A.2.6)

where 1_{k} is the multiplicative identity element in k, and 1 = 1_{R} is the multiplicative identity element in R. To do this, one should first get that

|1_{k}| = 1. (A.2.7)

One can obtain (A.2.7) using the facts that 1_{k}^{2} = 1_{k} and 1_{k} ≠ 0 in k, and one can obtain (A.2.6) from this and the fact that (−1_{k})^{2} = 1_{k}. If | · | is an ultrametric absolute value function on k, then (A.2.5) is an ultrametric on k.

Let p be a prime number. The p-adic absolute value |x|_{p} of a rational number x is defined as follows. Of course, we put |0|_{p} = 0, and so we may suppose that x ≠ 0. In this case, x = p^{j} (a/b) (A.2.8)

for some integers a, b, and j, where a, b ≠ 0 and neither a nor b is a multiple of p, and we put |x|_{p} = p^{−j}. (A.2.9)

It is easy to see that this is well defined, even though the expression (A.2.8) for x is not unique.

One can check that this defines an ultrametric absolute value function on the field Q of rational numbers. This implies that

d_{p}(x, y) = |x − y|_{p} (A.2.10)

defines an ultrametric on Q, as before. This is known as the p-adic metric on Q.

The field Q_{p} of p-adic numbers can be obtained by completing Q with respect to the p-adic metric, in much the same way that the real numbers can be obtained by completing Q with respect to the standard Euclidean metric.

Similarly, if k is a field with an absolute value function | · |, and if k is not complete with respect to the associated metric (A.2.5), then one can pass to a completion. It is well known that the completion is also a field, and that | · | extends to an absolute value function on the completion in a natural way. If | · | is an ultrametric absolute value function on k, then its extension to the completion of k is an ultrametric absolute value function on the completion.

A.3 The archimedean property

Let k be a field. If x ∈ k and n is a positive integer, then let n · x be the sum of n x’s in k. It is easy to see that

m · (n · x) = (m n) · x (A.3.1)

for all m, n ∈ Z_{+} and x ∈ k. Note that

n · x = (n · 1_{k}) x (A.3.2)

for every n ∈ Z_{+} and x ∈ k. We also have that

(m n) · 1_{k} = m · (n · 1_{k}) = (m · 1_{k}) (n · 1_{k}) (A.3.3)

for every m, n ∈ Z_{+}.

Let | · | be an absolute value function on k. We say that | · | is archimedean on k if there are positive integers n such that

(A.3.4) |n · 1_{k}| is arbitrarily large.

Thus | · | is non-archimedean on k if there is a positive real number C such that

|n · 1_{k}| ≤ C (A.3.5)

for all n ∈ Z_{+}.

If there is a positive integer n_{0} such that

|n_{0} · 1_{k}| > 1, (A.3.6)

then | · | is archimedean on k. Indeed, in this case,

|n_{0}^{j} · 1_{k}| = |(n_{0} · 1_{k})^{j}| = |n_{0} · 1_{k}|^{j} → ∞ as j → ∞. (A.3.7)

If | · | is non-archimedean on k, then it follows that (A.3.5) holds with C = 1.

If |·| is an ultrametric absolute value function on k, then it is easy to see that (A.3.5) holds with C = 1, so that |·| is non-archimedean on k. Conversely, if |·| is non-archimedean on k, then it is well known that |·| is an ultrametric absolute value function on k. This corresponds to Lemma 1.5 on p16 of [62], and Theorem 2.2.2 on p28 of [131]. Note that the term “non-archimedean” is sometimes used directly for absolute value functions that satisfy the ultrametric version of the triangle inequality, or something equivalent to it in a straightforward way, as in [62, 131]. This is equivalent to the terminology being used here, because of the result that was just mentioned.

A.4 Equivalent absolute value functions

Let k be a field, and let | · | be an absolute value function on k. If a is a positive real number, then |x|^{a} (A.4.1)

satisfies the first two conditions in the definition of an absolute value function. If a ≤ 1, then (A.4.1) satisfies the triangle inequality on k, because of (A.1.7). and thus defines an absolute value function on k. If |·| is an ultrametric absolute value function on k, then it is easy to see that (A.4.1) is an ultrametric absolute value function on k for every a > 0.

A pair | · |_{1}, | · |_{2} of absolute value functions on k are said to be equivalent on k if there is a positive real number a such that

|x|_{2} = |x|_{1}^{a} (A.4.2)

for every x ∈ k. Of course, this implies that

|x − y|_{2} = |x − y|_{1}^{a} (A.4.3)

for every x, y ∈ k. In particular, this means that the metrics on k associated to | · |_{1} and | · |_{2} determine the same topology on k, as in Section A.1.

Conversely, if |·|_{1} and |·|_{2} are absolute value functions on k whose associated metrics determine the same topology on k, then it is well known that | · |_{1} and | · |_{2} are equivalent in the sense of (A.4.2). This corresponds to Lemma 3.2 on p20 of [62], and to Lemma 3.1.2 on p42 of [131].

A famous theorem of Ostrowski states that any absolute value function on Q is either equivalent to the standard Euclidean absolute value function, trivial, or equivalent to the p-adic absolute value function for some p. This corresponds to Theorem 2.1 on p16 of [62], and to Theorem 3.1.3 on p44 of [131].

Suppose that k is a field with an archimedean absolute value function | · |, and that k is complete with respect to the metric associated to | · |. Another famous theorem of Ostrowski states that k is isomorphic to R or C, in such a way that | · | corresponds to an absolute value function that is equivalent to the standard Euclidean absolute value function. See Theorem 1.1 on p33 of [62].

A.5 Norms and seminorms

Let k be a field with an absolute value function | · |, and let V be a vector space over k. A nonnegative real-valued function N on V is said to be a seminorm or pseudonorm with respect to | · | on k if

N(t v) = |t| N(v) (A.5.1)

for every t ∈ k and v ∈ V, and

N(v + w) ≤ N(v) + N(w) (A.5.2) for every v, w ∈ V. Of course, if k = R or C, equipped with the standard Euclidean absolute value function, then this is the same as in Section 1.2.

If N is a nonnegative real-valued function on V that satisfies (A.5.1) and

N(v + w) ≤ max(N(v), N(w)) (A.5.3)

for every v, w ∈ V, then we say that N is a semi-ultranorm or pseudo-ultranorm on V with respect to | · | on k. As usual, (A.5.3) implies (A.5.2), so that a semiultranorm on V is a seminorm in particular. If N is a semi-ultranorm on V, and N(v) > 0 for some v ∈ V, then one can check that | · | is an ultrametric absolute value function on k.

If N is a seminorm on V with respect to | · | on k, and if

(A.5.4) N(v) > 0 for every v ∈ V with v ≠ 0,

then we say that N is a norm on V with respect to | · | on k. This is the same as in Section 1.1 when k = R or C with the standard absolute value function, as before. If N is a semi-ultranorm on V with respect to | · | on k that satisfies (A.5.3), then we say that N is an ultranorm on V with respect to |·| on k. Note that | · | is a norm on k, as a one-dimensional vector space over itself, and an ultranorm on k when | · | is ultrametric absolute value function on k.

If N is a norm on V, then it is easy to see that

d_{N}(v, w) = N(v − w) (A.5.5)

is a metric on V. This uses (A.2.7) to get that (A.5.5) is symmetric in v and w. If N is an ultranorm on V, then (A.5.5) is an ultrametric on V.

Suppose for the moment that | · | is the trivial absolute value function on k. The trivial ultranorm on V is defined by taking N(v) = 1 when v ∈ V and v ≠ 0, and N(0) = 0. It is easy to see that this defines an ultranorm on V, for which the corresponding metric is the discrete metric.

Let a be a positive real number, and suppose that | · |^{a} is an absolute value function on k too. Suppose that N is a nonnegative real-valued function on V that satisfies (A.5.1), so that

N(t v)^{a} = |t|^{a} N(v)^{a} (A.5.6)

for every t ∈ k and v ∈ V. If we also have that

N(v + w)^{a} ≤ N(v)^{a} + N(w)^{a}, (A.5.7)

then N^{a} is a seminorm on V with respect to | · |^{a} on k. If N is a seminorm on V with respect to | · | on k and a ≤ 1, then (A.5.7) follows from (A.5.2), because of (A.1.7). If N satisfies (A.5.4) and (A.5.7), then N^{a} is a norm on V with respect to | · |^{a} on k.

Similarly, if | · | is an ultrametric absolute value function on k, and N is a semi-ultranorm on V with respect to | · | on k, then N^{a} is a semi-ultranorm on V with respect to | · |^{a} on k for every a > 0. In this case, if N is an ultranorm on V with respect to | · | on k, then N^{a} is an ultranorm on V with respect to | · |^{a} on k for every a > 0.

A.6. SOME NORMS ON K^{N} 587

A.6 Some norms on k^{n}

Let k be a field with an absolute value function | · |, and let n be a positive integer. Of course, the space k^{n} of n-tuples of elements of k is a vector space over k, with respect to coordinatewise addition and scalar multiplication. If v ∈ k^{n}, then put

∥v∥_{∞} = max_{1≤j≤n} |v_{j}|, (A.6.1)

as in Section 1.3. One can check that this defines a norm on k^{n} with respect to | · | on k, as before. If | · | is an ultrametric absolute value function on k, then one can verify that (A.6.1) defines an ultranorm on k^{n} with respect to | · | on k.

Let r be a positive real number, and put

∥v∥_{r} = ( ∑_{j=1}^{n} |v_{j}|^{r})^{1/r} (A.6.2)

for every v ∈ k^{n}. It is easy to see that this defines a norm on k^{n} with respect to | · | on k when r = 1. If 1 < r < ∞, then one can check that this defines a norm on k^{n} with respect to | · | on k, using Minkowski’s inequality for finite sums, as in Section 1.3. We also have that

∥v∥_{∞} ≤ ∥v∥_{r} ≤ n^{1/r} ∥v∥_{r} (A.6.3)

for every v ∈ k^{n} and 0 < r < ∞, as before.

Let e_{1}, …, e_{n} be the standard basis vectors for k^{n}, so that the lth coordinate of e_{j} is equal to 1 when j = l, and to 0 otherwise. If N is a seminorm on k^{n} with respect to | · | on k, then it is easy to see that

N(v) ≤ ∑_{j=1}^{n} N(e_{j}) |v_{j}| (A.6.4)

for every v ∈ k^{n}, as in Section 1.11. This implies that

N(v) ≤ ( max_{1≤j≤n} N(e_{j})) ∥v∥_{1} (A.6.5)

for every v ∈ k^{n}. Similarly,

N(v) ≤ ( ∑_{j=1}^{n} N(e_{j})) ∥v∥_{∞} (A.6.6)

for every v ∈ k^{n}. If N is a semi-ultranorm on k^{n} with respect to | · | on k, then we get that

N(v) ≤ max_{1≤j≤n}(N(e_{j})|v_{j}|) ≤ ( max_{1≤j≤n} N(e_{j})) ∥v∥_{∞} (A.6.7)

for every v ∈ k^{n}.

Suppose that k is complete with respect to the metric associated to |·|. This implies that k^{n} is complete with respect to the metric associated to (A.6.1), by standard arguments. Similarly, k^{n} is complete with respect to the metric associated to (A.6.2) when r ≥ 1.

Let N be a norm on k^{n} with respect to | · | on k. Suppose that there is a positive real number c such that

c ∥v∥_{∞} ≤ N(v) (A.6.8)

for every v ∈ k^{n}. Under these conditions, one can check that k^{n} is also complete with respect to the metric associated to N.

In fact, if k is complete with respect to the metric associated to | · |, and N is any norm on k^{n} with respect to | · | on k, then it is well known that there is a c > 0 such that (A.6.8) holds for every v ∈ k^{n}. This corresponds to Lemma 2.1 on p116 of [62], and to Proposition 5.2.3 on p138 of [131].

If V is a vector space over k of positive finite dimension n, then V is isomorphis to k^{n}, as a vector space over k. If N_{V} is a norm on V with respect to | · | on k, and if k is complete with respect to the metric associatived to | · |, then V is complete with respect to the metric associated to N_{V}, because of the analogous statement for k^{n} in the preceding paragraphs.

Let W be a vector space over k with a norm N_{W} with respect to | · | on k. Of course, if V is a linear subspace of W, then the restriction of N_{W} to V defines a norm on V. If V has finite dimension as a vector space over k, and if k is complete with respect to the metric associated to | · |, then V is complete with respect to the metric associated to the restriction of N_{W} to V. This is the same as the restriction to V of the metric on W associated to N_{W}. This implies that V is a closed set in W with respect to the metric associated to N_{W}, as in Section 1.6.

A.7 Sequences and series

Let k be a field with an absolute value function | · |, let V be a vector space over k, and let N be a norm on V with respect to | · | on k. If {v_{j}}_{j=1}^{∞} and {w_{j}}_{j=1}^{∞} are sequences of elements of V that converge to v, w ∈ V, respectively, with respect to the metric d_{N} associated to N, then it is easy to see that

(A.7.1) {v_{j} + w_{j}}_{j=1}^{∞} converges to v + w

with respect to d_{N}. Similarly, if {t_{j}}_{j=1}^{∞} is a sequence of elements of k that converges to t ∈ k with respect to the metric associated to | · |, then

(A.7.2) {t_{j} v_{j}}_{j=1}^{∞} converges to t v

in V with respect to N, as in Section 1.5.

An infinite series ∑_{j=1}^{∞} v_{j} with terms in V is said to converge in V with respect to N if the sequence of partial sums ∑_{j=1}^{n} v_{j} converges in V with respect to d_{N}, in which case we put

∑_{j=1}^{∞} v_{j} = lim_{n→∞} ∑_{j=1}^{n} v_{j}, (A.7.3)

as usual. Under these conditions, if t ∈ k, then ∑_{j=1}^{∞} t v_{j} converges in V as well, with

∑_{j=1}^{∞}(t v_{j}) = t ∑_{j=1}^{∞} v_{j}. (A.7.4)

If ∑_{j=1}^{∞} w_{j} is another convergent series of elements of V, then ∑_{j=1}^{∞}(v_{j} + w_{j}) converges, with

∑_{j=1}^{∞}(v_{j} + w_{j}) = ∑_{j=1}^{∞} v_{j} + ∑_{j=1}^{∞} w_{j}, (A.7.5)

as before.

If ∑_{j=1}^{∞} v_{j} is any infinite series with terms in V, then the corresponding sequence of partial sums is a Cauchy sequence with respect to d_{N} if and only if for every ϵ > 0 there is a positive integer L(ϵ) such that

N( ∑_{j=l}^{n} v_{j}) < ϵ (A.7.6)

for all integers n ≥ l ≥ L(ϵ), as in Section 1.5 again. This implies that

lim_{j→∞} N(v_{j}) = 0, (A.7.7)

as before. Note that

N( ∑_{j=l}^{n} v_{j}) ≤ ∑_{j=l}^{n} N(v_{j}) (A.7.8)

for all positive integers l ≤ n, as in Section 1.7. If N is an ultranorm on V, then

N( ∑_{j=l}^{n} v_{j}) ≤ max_{l≤j≤n} N(v_{j}) (A.7.9)

for all l ≤ n.

We say that ∑_{j=1}^{∞} v_{j} converges absolutely with respect to N if

∑_{j=1}^{∞} N(v_{j}) (A.7.10)

converges as an infinite series of nonnegative real numbers, as in Section 1.7. This implies that the corresponding sequence of partial sums is a Cauchy sequence with respect to d_{N}, as before.

If V is complete with respect to d_{N}, then V is said to be a Banach space with respect to N, as in Section 1.5. In this case, if ∑_{j=1}^{∞} v_{j} converges absolutely with respect to N, then ∑_{j=1}^{∞} v_{j} converges in V, with

N( ∑_{j=1}^{∞} v_{j}) ≤ ∑_{j=1}^{∞} N(v_{j}), (A.7.11)

as in Section 1.7. We also have that the completeness of V with respect to d_{N} is characterized by the condition that absolute convergence of an infinite series in V with respect to N implies that the series converges, as before.

If N is an ultranorm on V, then (A.7.7) implies that the sequence of partial sums ∑_{j=1}^{n} v_{j} is a Cauchy sequence in V with respect to d_{N}, because of (A.7.9). If V is a Banach space with respect to N too, then it follows that ∑_{j=1}^{∞} v_{j} converges in V. One can check that

N( ∑_{j=1}^{∞} v_{j}) ≤ max_{j≥1} N(v_{j}) (A.7.12)

in this case. More precisely, one can verify that the maximum on the right side is attained, by reducing to the maximum of finitely many terms when at least one of the terms is not zero.

If V is not already complete with respect to d_{N}, then V has a completion that is a Banach space over k, as in Section 1.15. If N is an ultranorm on V, then its extension to the completion of V is an ultranorm on the completion of V.

A.8 Discrete absolute value functions

Let k be a field with an absolute value function | · | again. Observe that

{|t| : t ∈ k, t ≠ 0} (A.8.1)

is a subgroup of the multiplicative group R_{+} of positive real numbers. This may be called the valuation group associatied to | · | on k, as on p42 of [62]. Of course, this is the trivial subgroup {1} of R_{+} if and only if | · | is the trivial absolute value function on k.

Put ρ_{1} = sup{|t| : t ∈ k, |t| < 1}, (A.8.2)

so that 0 ≤ ρ_{1} ≤ 1, (A.8.3)

by construction. It is easy to see that

ρ_{1} = 0 (A.8.4)

if and only if | · | is the trivial absolute value function on k.

A.9. SUPREMUM SEMINORMS AND C_{0} SPACES 591

If ρ_{1} = 1, (A.8.5)

then (A.8.6) 1 is a limit point of (A.8.1),

with respect to the standard Eulidean metric on R. Conversely, one can check that (A.8.6) implies (A.8.5), because the valuation group (A.8.2) contains the multiplicative inverses of its elements. In this case, one can verify that (A.8.1) is dense in R_{+}, with respect to the standard Euclidean metric.

If ρ_{1} < 1, (A.8.7)

then |·| is said to be discrete on k, as on p42 of [62]. If |·| is nontrivial on k too, so that ρ_{1} > 0, then it is not too difficult to show that (A.8.1) consists exactly of the integer powers of ρ_{1}. In particular, this means that the supremum on the right side of (A.8.2) is attained. This corresponds to Lemma 1.2 on p42 of [62], and to part of Problem 63 on p39 of [131]. More precisely, in these statements in [62, 131], |·| is asked to be an ultrametric absolute value function on k, and this is used to consider other versions of these properties. However, this is not really needed for the versions of these properties mentioned here. If | · | is discrete on k, then one can show that | · | is an ultrametric absolute value function on k, as in the next paragraph.

If | · | is archimedean on k, then it is easy to see that k has characteristic 0, so that k contains a copy of Q. This implies that | · | on k induces an archimedean absolute value function on Q. This induced absolute value function on Q is equivalent to the standard Euclidean absolute value function on Q, by Ostrowski’s theorem, as in Section A.4. One can use this to get that | · | is not discrete on k, because the standard Euclidean absolute value function on Q is not discrete. If | · | is a discrete absolute value function on k, then it follows that | · | is non-archimedean on k, so that | · | is an ultrametric absolute value function on k, as in Section A.3.

A.9 Supremum seminorms and c_{0} spaces

Let X be a nonempty set, let k be a field, and let W be a vector space over k. The space of all functions on X with values in W is a vector space over k with respect to pointwise addition and scalar multiplication, as in Section 2.1.

Let | · | be an absolute value function on k, and let N_{W} be a seminorm on W with respect to | · | on k. Let us say that a W -valued function f on X is bounded with respect to N_{W} on W if

(A.9.1) N_{W} (f(x)) is bounded as a real-valued function on X.

Let ℓ^{∞}(X, W) = ℓ_{N_{W}}^{∞} (X, W) (A.9.2) be the space of all W -valued functions on X that are bounded with respect to N_{W}. This is a linear subspace of the space of all W -valued functions on X, as in Section 2.3.

If f ∈ ℓ_{N_{W}}^{∞} (X, W), then put

∥f∥_{∞} = ∥f∥_{sup} = ∥f∥_{ℓ^{∞}(X,W)} = ∥f∥_{ℓ_{N_{W}}^{∞} (X,W)} = sup_{x∈X} N_{W} (f(x)), (A.9.3)

as before. One can check that this defines a seminorm on ℓ_{N_{W}}^{∞} (X, W), which is the supremum seminorm associated to N_{W}. If N_{W} is a semi-ultranorm on W, then (A.9.3) is a semi-ultranorm on ℓ_{N_{W}}^{∞} (X, W).

If N_{W} is a norm on W, then (A.9.3) is a norm on ℓ_{N_{W}}^{∞} (X, W), which is the supremum norm associated to N_{W} on W. The metric on ℓ_{N_{W}}^{∞} (X, W) corresponding to the supremum norm is the same as the supremum metric corresponding to the metric on W associated to N_{W}, as in Section 2.3. If N_{W} is an ultranorm on W, then (A.9.3) is an ultranorm on ℓ_{N_{W}}^{∞} (X, W).

If N_{W} is a norm on W, and W is a Banach space with respect to N_{W}, then

(A.9.4) ℓ_{N_{W}}^{∞} (X, W) is a Banach space

with respect to (A.9.3). This is analogous to the statements in Sections 1.6 and 2.3 for vector spaces over the real and complex numbers. In particular, if k is complete with respect to the metric associated to | · |, then one can take W = k.

If f is a function on X with values in W, then the support of f may be defined as the set supp f of x ∈ X such that f(x) ≠ 0, as in Sections 1.12 and 2.3. Let

c_{00}(X, W) (A.9.5)

be the space of W -valued functions f on X such that supp f has only finitely many elements, as before. This is a linear subspace of the space of all W -valued functions on X, as a vector space over k.

Let N_{W} be a seminorm on W with respect to | · | on k again. A W -valued function f on X is said to vanish at infinity on X with respect to N_{W} if

(A.9.6) N_{W} (f(x)) vanishes at infinity on X,

as a nonnegative real-valued function on X, as in Section 2.6. Let

c_{0}(X, W) = c_{0,N_{W}} (X, W) (A.9.7)

be the space of W -valued functions on X that vanish at infinity with respect to N_{W}. It is easy to see that this is a linear subspace of ℓ_{N_{W}}^{∞} (X, W), as before. If N_{W} is a norm on W, then c_{0,N_{W}} (X, W) is the same as the closure of c_{00}(X, W) in ℓ_{N_{W}}^{∞} (X, W) with respect to the supremum metric, as in Sections 1.13 and 2.6.

A.10 Lipschitz and operator seminorms

Let (X, d_{X}) be a nonempty metric space, let k be a field with an absolute value function | · |, and let W be a vector space over k with a norm N_{W} with respect to | · | on k. Thus W may also be considered as a metric space with respect to the corresponding metric d_{N_{W}}, so that the space Lip(X, W) of all Lipschitz mappings from X into W may be defined as in Section 2.1. One can check that

(A.10.1) Lip(X, W) is a linear subspace of the space of all W -valued functions on X,

as a vector space over k with respect to pointwise addition and scalar multiplication on X, as before.

If f ∈ Lip(X, W), then the corresponding Lipschitz constant Lip(f) = Lip_{X,W} (f) may be defined as in Section 2.1. One can verify that

(A.10.2) Lip(f) is a seminorm on Lip(X, W)

with respect to | · |, on k, as before. Remember that Lip(f) = 0 if and only if f is constant on X. Similarly, if N_{W} is an ultranorm on W, then one can check that (A.10.3) Lip(f) is a semi-ultranorm on Lip(X, W).

Let V be another vector space over k, and let us denote the space of all linear mappings from V into W as L(V, W), as in Section 2.2. This is a linear subspace of the space of all W -valued functions on V, as a vector space over k.

Let N_{V}, N_{W} be seminorms on V, W, respectively, with respect to | · | on k. A linear mapping T from V into W is said to be bounded with respect to these seminorms if there is a nonnegative real number C such that

N_{W} (T (v)) ≤ C N_{V} (v) (A.10.4)

for every v ∈ V, as in Section 2.2. This implies that

N_{W} (T (u) − T (v)) = N_{W} (T (u − v)) ≤ C N_{V} (u − v) (A.10.5)

for every u, v ∈ V, as before. If N_{V}, N_{W} are norms on V, W, respectively, then (A.10.5) says that T is Lipschitz with constant C with respect to the associated metrics d_{N_{V}}, d_{N_{W}}. Otherwise, d_{N_{V}} and d_{N_{W}} may be considered as semimetrics on V and W, respectively, as in Section 1.2.

Let BL(V, W) be the space of all bounded linear mappings from V into W with respect to N_{V}, N_{W}, respectively. One can check that this is a linear subspace of L(V, W), as in Section 2.2.

If T ∈ BL(V, W), then put

(A.10.6) ∥T ∥_{op} = ∥T ∥_{op,V W} = inf{C ≥ 0 : (A.10.4) holds}.

One can verify that

(A.10.7) ∥ · ∥_{op} defines a seminorm on BL(V, W) with respect to |·| on k, which is the operator seminorm associated to N_{V}, N_{W}. It is easy to see that the infimum on the right side of (A.10.6) is automatically attained, so that N_{W} (T (v)) ≤ ∥T ∥_{op,V W} N_{V} (v) (A.10.8) for every v ∈ V. If N_{W} is a norm on W, then

(A.10.9) ∥ · ∥_{op} is a norm on BL(V, W),

which is the operator norm associated to N_{V}, N_{W}.

Similarly, if N_{W} is a semi-ultranorm on W, then one can check that

(A.10.10) ∥ · ∥_{op} is a semi-ultranorm on BL(V, W).

If N_{W} is an ultranorm on W, then

(A.10.11) ∥ · ∥_{op} is an ultranorm on BL(V, W).

Let Z be another vector space over k, with a seminorm N_{Z} with respect to |·| on k. If T_{1} is a bounded linear mapping from V into W, and T_{2} is a bounded linear mapping from W into Z, then T_{2} ∘ T_{1} is a bounded linear mapping from V into Z, with ∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}, (A.10.12) as in Section 2.2.

If N_{W} is a norm on W, and W is complete with respect to the metric associated to N_{W}, then (A.10.13) BL(V, W) is complete with respect to the metric associated to the operator norm. This follows from the same type of argument as in Section 2.2.

Let us suppose from now on in this section that N_{V} and N_{W} are norms on V and W, respectively. Note that (A.10.6) is the same as the Lipschitz constant Lip(T) of T with respect to the metrics associated to N_{V}, N_{W}, as in Section 2.2.

Let T be a linear mapping from V into W, and suppose for the moment that N_{W} (T (v)) is bounded on a ball of positive radius in V centered at 0 with respect to the metric associated to N_{V}. Of course, this condition holds in particular when T is continuous at 0 with respect to the metrics associated to N_{V}, N_{W}. If | · | is not the trivial absolute value function on k, then one can check that

(A.10.14) T is a bounded linear mapping from V into W.

Let V_{0} be a linear subspace of V that is dense in V with respect to the metric associated to N_{V}, and let T_{0} be a bounded linear mapping from V_{0} into W, with respect to the restriction of N_{V} to V_{0}. If W is complete with respect to the metric associated to N_{W}, then

(A.10.15) there is a unique extension of T_{0} to a bounded linear mapping T from V into W,

as in Section 2.2. One can check that the operator norm of T on V is equal to the operator norm of T_{0} on V_{0}, as before.

A.11. ℓ^{R} SPACES 595

A.11 ℓ^{r} Spaces

Let X be a nonempty set, let r be a positive real number, and let k be a field with an absolute value function | · |. Also let W be a vector space over k with a seminorm N_{W} with respect to | · | on k. Consider the space ℓ^{r}(X, W) = ℓ_{N_{W}}^{r} (X, W) (A.11.1) of W -valued functions f on X such that (A.11.2) N_{W} (f(x)) is r-summable as a

nonnegative real-valued function on X,

as in Section 2.5. This was discussed in Section 2.6 for vector spaces over the real or complex numbers with a norm. As before, one can take W = k, with | · | as the norm.

One can check that ℓ_{N_{W}}^{r} (X, W) is a linear subspace of the space of all W - valued functions on X, as in Section 2.6. If f ∈ ℓ_{N_{W}}^{r} (X, W), then put

∥f∥_{r} = ∥f∥_{ℓ^{r}(X,W)} = ∥f∥_{ℓ_{N_{W}}^{r} (X,W)} = ( ∑_{x∈X} N_{W} (f(x))^{r})^{1/r}, (A.11.3)

as before. If r ≥ 1, then this defines a seminorm on ℓ_{N_{w}}^{r}(X, W) with respect to | · |, and a norm when N_{W} is a norm on W. If 0 < r ≤ 1, then this satisfies the usual homogeneity property of a seminorm, and ∥f + g∥_{r}^{r} ≤ ∥f∥_{r}^{r} + ∥g∥_{r}^{r} (A.11.4) for all f, g ∈ ℓ_{N_{W}}^{r} (X, W). If N_{W} is a norm on W, then we get that ∥f − g∥_{r}^{r} (A.11.5) is a metric on ℓ_{N_{W}}^{r} (X, W) when r ≤ 1, as usual.

If 0 < r_{1} ≤ r_{2} ≤ +∞, then ℓ_{N_{W}}^{r_{1}} (X, W) ⊆ ℓ_{N_{W}}^{r_{2}} (X, W), (A.11.6) as in Section 2.6. If f ∈ ℓ_{N_{W}}^{r_{1}} (X, W), then ∥f∥_{r_{2}} ≤ ∥f∥_{r_{1}}, (A.11.7) as before. If 0 < r < ∞, then c_{00}(X, W) ⊆ ℓ_{N_{W}}^{r} (X, W) ⊆ c_{0,N_{W}} (X, W), (A.11.8) as before, where c_{0,N_{W}} (X, W) is as in Section A.9. If N_{W} is a norm on W, then one can check that (A.11.9) c_{00}(X, W) is dense in ℓ_{N_{W}}^{r} (X, W) when r < ∞, as in Section 2.6. This uses the metric associated to (A.11.3) when r ≥ 1, and the metric (A.11.5) when r ≤ 1.

If N_{W} is a norm on W and W is complete with respect to the associated metric, then (A.11.10) ℓ_{N_{W}}^{r} (X, W) is complete

with respect to the metric associated to (A.11.3) when r ≥ 1, and with respect to the metric (A.11.5) when r ≤ 1, as in Section 2.6.

A.12 Some bounded linear mappings

Let X be a nonempty set, let k be a field, and let W be a vector space over k. If f ∈ c_{00}(X, W), then

∑_{x∈X} f(x) (A.12.1)

may be defined as an element of W as in Section 2.3. This defines a linear mapping from c_{00}(X, W) into W, as before.

If a is any W -valued function on X and f ∈ c_{00}(X, k), then

a f ∈ c_{00}(X, W). (A.12.2)

This means that

T_{a}(f) = ∑_{x∈X} a(x) f(x) (A.12.3)

defines an element of W, and that T_{a} defines a linear mapping from c_{00}(X, k) into W, as in Section 2.3.

If y ∈ X, then let δ_{y} be the k-valued function on X equal to 1 at y at to 0 at every other point in X, as in Section 1.12. It is easy to see that

(A.12.4) the collection of δ_{y}’s, y ∈ X, is a basis for c_{00}(X, k),

as a vector space over k, as before. If 0 < r ≤ ∞, then

∥δ_{y}∥_{ℓ^{r}(X,k)} = 1 (A.12.5)

for every y ∈ X, using | · | as the norm on k on the left side.

Of course, T_{a}(δ_{y}) = a(y) (A.12.6)

for every y ∈ X, as in Section 2.3. If T is any linear mapping from c_{00}(X, k) into W, then a(y) = a_{T} (y) = T (δ_{y}) (A.12.7)

defines a W -valued function on X. It is easy to see that this is the unique W -valued function on X such that

T = T_{a}, (A.12.8) as before.

Let N_{W} be a seminorm on W with respect to | · | on k, and suppose that 1 ≤ r ≤ ∞, so that the restriction of the ℓ^{r} norm to c_{00}(X, k) defines a norm on c_{00}(X, k) with respect to | · | on k. Also let a be a W -valued function on X again, and suppose that

(A.12.9) T_{a} is bounded as a linear mapping from c_{00}(X, k) into W, with respect to the ℓ^{r} norm on c_{00}(X, k) and N_{W} on W.

Under these conditions, we get that

N(a(y)) = N_{W} (T_{a}(δ_{y})) ≤ ∥T_{a}∥_{op,r} ∥δ_{y}∥_{ℓ^{r}(X,k)} = ∥T_{a}∥_{op,r} (A.12.10)

for every y ∈ X, where ∥T_{a}∥_{op,r} is the operator seminorm of T_{a} corresponding to the restriction of the ℓ^{r} norm to c_{00}(X, k) and N_{W} on W. This means that a is bounded on X with respect to N_{W} on W, with

∥a∥_{ℓ_{N_{W}}^{∞} (X,W)} ≤ ∥T_{a}∥_{op,r}. (A.12.11)

If a is any element of ℓ_{N_{W}}^{∞} (X, W) and f ∈ c_{00}(X, k), then

N_{W} (T_{a}(f)) ≤ ∑_{x∈X} N_{W} (a(x)) |f(x)| ≤ ∥a∥_{ℓ_{Nw}^{∞} (X,W)} ∥f∥_{ℓ^{1}(X,k)}, (A.12.12)

as in Section 2.3. This implies that

(A.12.13) T_{a} is a bounded linear mapping from c_{00}(X, k) into W, with respect to the ℓ^{1} norm on c_{00}(X, k) and N_{W} on W,

with ∥T_{a}∥_{op,1} ≤ ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)}. (A.12.14)

Thus ∥T_{a}∥_{op,1} = ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)}, (A.12.15)

because of (A.12.11). If N_{W} is a norm on W, and W is complete with respect to the associated metric, then

(A.12.16) T_{a} has a unique extension to a bounded linear mapping from ℓ^{1}(X, k) into W, with the same operator norm,

as in Section A.10.

If N_{W} is a semi-ultranorm on W with respect to | · | on k, then

N_{W} (T_{a}(f)) ≤ max_{x∈X}(N_{W} (a(x)) |f(x)|) ≤ ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)} ∥f∥_{ℓ^{∞}(X,k)} (A.12.17)

for every f ∈ c_{00}(X, k). In this case,

(A.12.18) T_{a} is a bounded linear mapping from c_{00}(X, k) into W, with respect to the ℓ^{∞} norm on c_{00}(X, k) and N_{W} on W, with ∥T_{a}∥_{op,∞} ≤ ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)}. (A.12.19)

This means that ∥T_{a}∥_{op,∞} = ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)}, (A.12.20)

by (A.12.11). If N_{W} is an ultranorm on W, and W is complete with respect to the associated ultrametric, then

(A.12.21) T_{a} has a unique extension to a bounded linear mapping from c_{0}(X, k) into W, with the same operator norm,

as in Section A.10 again.

A.13 Sums of vectors

Let X be a nonempty set, and let k be a field with an absolute value function | · |. Also let W be a vector space over k with a norm N_{W} with respect to | · | on k, and let f be a function on X with values in W. As in Section 2.7, we say that

(A.13.1)  ∑_{x∈X} f(x) converges in the generalized sense

with respect to the metric associated to N_{W} if there is a w ∈ W with the following property: for every ϵ > 0 there is a finite set A(ϵ) ⊆ X such that for every finite set A ⊆ X with A(ϵ) ⊆ A, (A.13.2)

we have that

N_{W} ( ∑_{x∈A} f(x) − w) < ϵ. (A.13.3)

One can check that such a w ∈ W is unique when it exists, as before, in which case it is considered to be the value of the sum. This is equivalent to the convergence of the corresponding net of sums over finite subsets of X, as before.

Let us say that

(A.13.4)  ∑_{x∈X} f(x) satisfies the generalized Cauchy condition

with respect to N_{W} if for every ϵ > 0 there is a finite subset A_{0}(ϵ) of X with the following property, as before: if A, B are finite subsets of X such that

A_{0}(ϵ) ⊆ A, B, (A.13.5)

then

N_{W} ( ∑_{x∈A} f(x) − ∑_{x∈B} f(x)) < ϵ. (A.13.6)

This is the same as saying that the corresponding net of sums over finite subsets of X is a Cauchy net with respect to the metric associated to N_{W}, as before. If the sum converges in the generalized sense, then it satisfies the generalized Cauchy condition with A_{0}(ϵ) = A(ϵ/2) (A.13.7)

for every ϵ > 0, as before. If N_{W} is an ultranorm on W, then one can take

A_{0}(ϵ) = A(ϵ) (A.13.8)

for each ϵ > 0.

As in Section 2.7, the generalized Cauchy condition is the same as saying that for every ϵ > 0 there is a finite subset A_{1}(ϵ) of X with the following property: if C is a finite subset of X such that

A_{1}(ϵ) ∩ C = ∅, (A.13.9)

then

N_{W} ( ∑_{x∈C} f(x)) < ϵ. (A.13.10)

The previous version implies this one, with

A_{1}(ϵ) = A_{0}(ϵ) (A.13.11)

for each ϵ > 0, as before. This version also implies the previous one, with

A_{0}(ϵ) = A_{1}(ϵ/2) (A.13.12)

for every ϵ > 0, as before. If N_{W} is an ultranorm on W, then one can take

A_{0}(ϵ) = A_{1}(ϵ) (A.13.13)

for every ϵ > 0.

If C is any nonempty finite subset of X, then

N_{W} ( ∑_{x∈C} f(x)) ≤ ∑_{x∈C} N_{W} (f(x)), (A.13.14)

as in Section 2.7. If f ∈ ℓ_{N_{W}}^{1} (X, W), then one can use this to get that the sum satisfies the generalized Cauchy condition, as before. If the sum converges in the generalized sense, then one can verify that

N_{W} ( ∑_{x∈X} f(x)) ≤ ∑_{x∈X} N_{W} (f(x)), (A.13.15)

as before.

The generalized Cauchy condition implies that

(A.13.16) f vanishes at infinity on X with respect to N_{W} , as before. If N_{W} is an ultranorm on W, then

N_{W} ( ∑_{x∈C} f(x)) ≤ max_{x∈C} N_{W} (f(x)) (A.13.17)

for every nonempty finite subset C of X. In this case, if f vanishes at infinity on X with respect to N_{W}, then it is easy to see that the sum satisfies the generalized Cauchy condition, using the remarks in the preceding paragraph. If the sum converges in the generalized sense, then one can check that

N_{W} ( ∑_{x∈X} f(x)) ≤ max_{x∈X} N_{W} (f(x)). (A.13.18)

Note that the maximum on the right is attained on X, because N_{W} (f(x)) vanishes at infinity on X, by hypothesis.

Let {x_{j}}_{j=1}^{∞} be a sequence of distinct elements of X such that

supp f ⊆ {x_{j} : j ∈ Z_{+}}. (A.13.19)

If ∑_{x∈X} f(x) converges in the generalized sense, then it is easy to see that

∑_{j=1}^{∞} f(x_{j}) = ∑_{x∈X} f(x), (A.13.20)

including the convergese of the series on the left, as in Section 2.7. If ∑_{x∈X} f(x) satisfies the generalized Cauchy condition, and if the sum on the left side of (A.13.20) converges, then one can check that ∑_{x∈X} f(x) converges in the generalized sense, with sum as in (A.13.20), as before.

If ∑_{x∈X} f(x) satisfies the generalized Cauchy condition, then

(A.13.21) { ∑_{j=1}^{n} f(x_{j})}_{n=1}^{∞} is a Cauchy sequence in W

with respect to the metric associated to N_{W}, as in Section 2.7. If W is a Banach space with respect to N_{W}, then it follows that the series on the left side of (A.13.20) converges in W. This implies that ∑_{x∈W} f(x) converges in the generalized sense, with sum as in (A.13.20), as in the preceding paragraph.

A.14 Some related spaces of functions

Let us continue with the same notation and hypotheses as at the beginning of the previous section. Let Sum_{N_{W}} (X, W) (A.14.1)

be the space of W -valued functions f on X such that ∑_{x∈X} f(x) converges in the generalized sense, as in Section 2.8. This is a linear subspace of the space A.14. SOME RELATED SPACES OF FUNCTIONS 601 of all W -valued functions on X, and the sum defines a linear mapping from Sum_{N_{W}} (X, W) into W, as before.

Let GCC_{N_{W}} (X, W) (A.14.2) be the space of all W -valued functions f on X such that ∑_{x∈X} f(x) satisfies the generalized Cauchy condition, as in Section 2.8 again. This is a linear subspace of the space of all W -valued functions on X too, with c_{00}(X, W) ⊆ Sum_{N_{W}} (X, W) ⊆ GCC_{N_{W}} (X, W) ⊆ c_{0,N_{W}} (X, W), (A.14.3) as before. If W is a Banach space, then Sum_{N_{W}} (X, W) = GCC_{N_{W}} (X, W), (A.14.4) as in the previous section.

We have seen that ℓ_{N_{W}}^{1} (X, W) ⊆ GCC_{N_{W}} (X, W) (A.14.5) so that ℓ_{N_{W}}^{1} (X, W) ⊆ Sum_{N_{W}} (X, W) (A.14.6) when W is a Banach space. If N_{W} is an ultranorm on W, then GCC_{N_{W}} (X, W) = c_{0,N_{W}} (X, W), (A.14.7) as in the previous section. If W is also a Banach space, then it follows that Sum_{N_{W}} (X, W) = c_{0,N_{W}} (X, W) (A.14.8) in this case.

As in Section 2.9, we say that a W -valued function f on X has bounded finite sums on X with respect to N_{W} if the norms

N_{W} ( ∑_{x∈A} f(x)) (A.14.9)

of the sums of f over all finite subsets A of X are bounded. The space BF S_{N_{W}} (X, W) (A.14.10) of all W -valued functions on X with bounded finite sums is a linear subspace of the space of all W -valued functions on X, as before. One can check that GCC_{N_{W}} (X, W) ⊆ BF S_{N_{W}} (X, W) ⊆ ℓ_{N_{W}}^{∞} (X, W), (A.14.11) as before.

If f ∈ BF S_{N_{W}} (X, W), then we put

∥f∥_{BF S} = ∥f∥_{BF S_{N_{W}} (X,W)}
(A.14.12)  = sup {N_{W} ( ∑_{x∈A} f(x)) : A is a finite subset of X}.

This defines a norm on BF S_{N_{W}} (X, W) with respect to | · | on k, as before. Clearly ∥f∥_{ℓ_{N_{W}}^{∞} (X,W)} ≤ ∥f∥_{BF S_{N_{W}} (X,W)} (A.14.13)

for every f ∈ BF S_{N_{W}} (X, W). If f ∈ ℓ_{N_{W}}^{1} (X, W), then f ∈ BF S_{N_{W}} (X, W), with ∥f∥_{BF S_{N_{W}} (X,W)} ≤ ∥f∥_{ℓ_{N_{W}}^{1} (X,W)}, (A.14.14)

as before. If f ∈ Sum_{N_{W}} (X, W), then f ∈ BF S_{N_{W}} (X, W), by (A.14.3) and (A.14.11), and

N_{W} ( ∑_{x∈X} f(x)) ≤ ∥f∥_{BF S_{N_{W}} (X,W)}, (A.14.15)

as before.

Suppose for the moment that N_{W} is an ultranorm on W. If f ∈ ℓ_{N_{W}}^{∞} (X, W), then it is easy to see that f ∈ BF S_{N_{W}} (X, W), with

∥f∥_{BF S_{N_{W}} (X,W)} ≤ ∥f∥_{ℓ_{N_{W}}^{∞} (X,W)}. (A.14.16)

This implies that BF S_{N_{W}} (X, W) = ℓ_{N_{W}}^{∞} (X, W), (A.14.17)

with ∥f∥_{BF S_{N_{W}} (X,W)} = ∥f∥_{ℓ_{N_{W}}^{∞} (X,W)} (A.14.18)

for every f ∈ ℓ_{N_{W}}^{∞} (X, W).

If N_{W} is any norm on W, then

(A.14.19) GCC_{N_{W}} (X, W) is the same as the closure of c_{00}(X, W) in BF S_{N_{W}} (X, W),

as in Section 2.9. If W is a Banach space, then

(A.14.20) BF S_{N_{W}} (X, W) is complete

with respect to the metric associated to (A.14.12), as before.

Appendix B

More on metrics and norms

B.1 q-Metrics and q-semimetrics

Let X be a set, and let d(x, y) be a nonnegative real-valued function defined for x, y ∈ X such that d(x, x) = 0 (B.1.1)

for every x ∈ X, and d(x, y) = d(y, x) (B.1.2)

for every x, y ∈ X. Also let q be a positive real number, and suppose that

d(x, z)^{q} ≤ d(x, y)^{q} + d(y, z)^{q} (B.1.3)

for every x, y, z ∈ X. Under these conditions, we say that d(·, ·) is a q-semimetric or q-pseudometric on X. If we also have that

d(x, y) > 0 (B.1.4)

when x ≠ y, then we say that d(·, ·) is a q-metric on X. Thus q-semimetrics and q-metrics are the same as ordinary semimetrics and metrics when q = 1, as in Sections 1.2 and A.1.

Of course, (B.1.3) is the same as saying that

d(x, z) ≤ (d(x, y)^{q} + d(y, z)^{q})^{1/q}. (B.1.5)

The right side is monotonically decreasing in q, as in Section 1.12. This implies that the property of being a q-semimetric or q-metric is more restrictive as q increases.

If d(·, ·) satisfies (B.1.1), (B.1.2), and

d(x, z) ≤ max(d(x, y), d(y, z)) (B.1.6)

for every x, y, z ∈ X, then we say that d(·, ·) is a semi-ultrametric or pseudoultrametric on X. If d(·, ·) satisfies (B.1.4) as well, then d(·, ·) is an ultrametric on X, as in Section A.1. It is easy to see that a semi-ultrametric on X is a q-semimetric on X for every q > 0, and similarly an ultrametric on X is a q-metric for every q > 0.

Note that

lim_{q→∞}(d(x, y)^{q} + d(y, z)^{q})^{1/q} = max(d(x, y), d(y, z)) (B.1.7)

for every x, y, z ∈ X, as in Section 1.3. Because of this, we shall consider semiultrametrics and ultrametrics as being q-semimetrics and q-metrics, respectively, with q = ∞.

Of course, if d(·, ·) is a q-semimetric on X and Y is a subset of X, then the restriction of d(x, y) to x, y ∈ Y is a q-semimetric on Y. Similarly, if d(·, ·) is a q-metric on X, then the restriction of d(x, y) to x, y ∈ Y is a q-metric on Y.

B.2 More on q-metrics and q-semimetrics

Let X be a set, and let a be a positive real number. If d(·, ·) is a q-semimetric on X for some q > 0, then it is easy to see that

d(x, y)^{a} (B.2.1)

is a (q/a)-semimetric on X. Similarly, if d(·, ·) is a q-metric on X, then (B.2.1) is a (q/a) metric on X. If q = ∞, then this means that if d(·, ·) is a semiultrametric or ultrametric on X, then (B.2.1) has the same property, as in Section A.1.

If d(·, ·) is a q-semimetric or q-metric on X and a ≤ 1, then it follows that (B.2.1) has the same property, as in Section A.1. This uses the fact that the property of being a q-semimetric of q-metric is more restrictive as q increases, as in the previous section.

If d(·, ·) is a q-semimetric on X, then one can define open and closed balls in X with respect to d(·, ·) in the usual way, as in Section A.1. The open and closed balls in X with radius r with respect to d(·, ·) are the same as the open and closed balls in X with radius r^{a} with respect to (B.2.1), respectively, and with the same center, as before.

One can use open balls with respect to d(·, ·) to define what it means for a subset of X to be an open set with respect to d(·, ·) in the usual way. The open subsets of X with respect to d(·, ·) are the same as the open sets with respect to (B.2.1), because of the remark about the corresponding open balls in the preceding paragraph.

It is easy to see that this defines a topology on X, as usual. If q ≥ 1, then d(·, ·) is a semimetric on X, and otherwise

d(x, y)^{q} (B.2.2)

is a semimetric on X that determines the same topology as d(·, ·). The topology determined by a semimetric has many of the same properties as the topology B.3. Q-ABSOLUTE VALUE FUNCTIONS 605

determined by a metric. In particular, one can check that open balls with respect to a semimetric are open sets. This implies that open balls with respect to d(·, ·) are open sets, which could also be verified more directly when q < 1.

Similarly, closed balls with respect to a semimetric are closed sets, by standard arguments. This implies that closed balls with respect to d(·, ·) are closed sets, which could be verified more directly when q < 1 as well.

If d(·, ·) is a semi-ultrametric on X, then open balls in X with respect to d(·, ·) are closed sets too, and closed balls of positive radius are open sets, as in Section A.1.

It is well known and easy to see that the topology determined by a metric is Hausdorff. This implies that the topology determined by a q-metric is Hausdorff, which could be verified more directly, as usual. If d(·, ·) is a q-semimetric on X, and the topology determined on X by d(·, ·) satisfies the first or zeroth separation condition, then d(·, ·) is a q-metric on X.

It is well known that the topology determined by a metric is normal in the strong sense, and regular in the strong sense in particular, as mentioned in Section 5.5. This implies that the topology determined by a q-metric has the same properties, as before. It is not too difficult to show that the topology determined by a semimetric is regular and normal in the strict sense, which implies the analogous statements for q-semimetrics.

If d(·, ·) is a q-metric on X, then one can define the notions of Cauchy sequences and completeness with respect to d(·, ·) in the usual way. It is easy to see that d(·, ·) and (B.2.1) determine the same Cauchy sequences in X, as in Section A.1. This implies that X is complete with respect to d(·, ·) if and only if X is complete with respect to (B.2.1), as before.

B.3 q-Absolute value functions

Let k be a field, and let |·| be a nonnegative real-valued function on k such that

(B.3.1) |x| = 0 if and only if x = 0

and |x y| = |x| |y| (B.3.2)

for every x, y ∈ k. This implies that

|1_{k}| = | − 1_{k}| = 1, (B.3.3)

as in Section A.2.

Let q be a positive real number, and suppose that

|x + y|^{q} ≤ |x|^{q} + |y|^{q} (B.3.4)

for every x, y ∈ k. In this case, we say that | · | is a q-absolute value function on k. This is the same as an ordinary absolute value function, as in Section A.2, when q = 1.

Note that (B.3.4) is the same as saying that

|x + y| ≤ (|x|^{q} + |y|^{q})^{1/q}. (B.3.5)

The right side is monotonically decreasing in q, as in Section 1.12, so that this condition becomes more restrictive as q increases.

It is easy to see that an ultrametric absolute value function on k is a qabsolute value function for every q > 0. We also have that

lim_{q→∞}(|x|^{q} + |y|^{q})^{1/q} (B.3.6)

for every x, y ∈ k, as in Section 1.3. Ultrametric absolute value functions on k will be considered as q-absolute value functions with q = ∞.

If | · | is a q-absolute value function on k for some q > 0 and a is a positive real number, then |x|^{a} (B.3.7)

is a (q/a)-absolute value function on k. If q = ∞, then this means that (B.3.7) is an ultrametric absolute value function on k when | · | is an ultrametric absolute value function on k, as in Section A.4.

If |·| is a q-absolute value function on k, then |·| is an absolute value function on k when q ≥ 1, and otherwise |x|^{q} (B.3.8)

is an absolute value function on k. This means that q-absolute value functions have the same types of properties as ordinary absolute value functions, as in Sections A.2, A.3, A.4, and A.8.

In particular, if | · | is a q-absolute value function on k, then

d(x, y) = |x − y| (B.3.9)

is a q-metric on k.

B.4 Quasimetric absolute value functions

Let k be a field again, and let | · | be a nonnegative real-valued function on k that satisfies (B.3.1) and (B.3.2). Suppose that there is a real number C ≥ 1 such that |1_{k} + z| ≤ C (B.4.1)

for every z ∈ k with |z| ≤ 1, (B.4.2)

where 1_{k} is the multiplicative identity element in k. In this case, | · | may be called a valuation on k, as in Definition 1.1 on p12 of [62]. This term is also sometimes used in other but related ways.

Suppose that x, y ∈ k satisfy

|x| ≤ |y|. (B.4.3)

If y ≠ 0, then z = x/y satisfies (B.4.2), and (B.4.1) implies that

|x + y| = |1_{k} + z| |y| ≤ C |y|. (B.4.4)

Let us say that | · | is a quasimetric absolute value function on k if it satisfies (B.3.1) and (B.3.2), as before, and if there is a real number C_{1} ≥ 1 such that

|x + y| ≤ C_{1} (|x| + |y|) (B.4.5)

for every x, y ∈ k. This is the same as saying that | · | is an absolute value function on k when C_{1} = 1.

Alternatively, one may ask that there be a real number C_{0} ≥ 1 such that

|x + y| ≤ C_{0} max(|x|, |y|) (B.4.6)

for every x, y ∈ k. This means that | · | is an ultrametric absolute value function on k when C_{0} = 1.

Clearly (B.4.6) implies (B.4.5), with C_{1} = C_{0}. Conversely, (B.4.5) implies (B.4.6), with C_{0} = 2 C_{1}.

Similarly, (B.4.6) implies (B.4.1), with C = C_{0}. Conversely, (B.4.1) implies (B.4.6), with C_{0} = C, as in (B.4.4). Thus a quasimetric absolute value function on k is the same as a valuation on k in the sense of [62].

If a is a positive real number, then (B.3.7) also satisfies (B.3.1) and (B.3.2). If | · | is a valuation on k in the sense of [62] with constant C, then

|1_{k} + z|^{a} ≤ C^{a} (B.4.7)

for every z ∈ k with |z|^{a} ≤ 1, (B.4.8)

so that (B.3.7) is a valuation on k in the sense of [62] with constant C^{a}. This corresponds to Lemma 1.1 on p13 of [62]. Equivalently, if | · | is a quasimetric absolute value function on k satisfying (B.4.6) for some C_{0}, then

|x + y|^{a} ≤ C_{0}^{a} max(|x|^{a}, |y|^{a}) (B.4.9)

for every x, y ∈ k.

Lemma 1.2 on p13 of [62] says that | · | is an absolute value function on k if and only if | · | is a valuation on k in the sense of [62] with constant C = 2. Of course, the “only if” part of this statement is very easy to see. This means that | · | is an absolute value function on k if and only if | · | is a quasimetric absolute value function on k that satisfies (B.4.6) with C_{0} = 2.

If | · | is a valuation on k in the sense of [62], or equivalently a quasimetric absolute value function, then it follows that (B.3.7) is an absolute value function on k when a > 0 is sufficiently small, as in the corollary on p14 of [62].

Let q be a positive real number, and remember that

(B.4.10) | · | is a q-absolute value function on k if and only if |x|^{q} is an absolute value function on k. Lemma 1.2 on p13 of [62] implies that this happens if and only if |·| is a valuation on k in the sense of [62] with constant C = 2^{1/q}, which is the same as saying that | · | is a quasimetric absolute value function on k that satisfies (B.4.6) with C_{0} = 2^{1/q}. If | · | is a valuation on k in the sense of [62], or equivalently a quasimetric absolute value function, then we get that (B.4.10) holds when q > 0 is sufficiently small.

B.5 q-Norms and q-seminorms

Let k be a field, and let |·| be a q_{k}-absolute value function on k for some q_{k} > 0. Also let V be a vector space over k, let N be a nonnegative real-valued function on V, and let q_{N} be a positive real number. We say that N is a q_{N}-seminorm or q_{N}-pseudonorm on V with respect to | · | on k if it satisfies the following two conditions. The first condition is the usual homogeneity property

N(t v) = |t| N(v) (B.5.1)

for every t ∈ k and v ∈ V. The second condition is the q_{N}-seminorm version of the triangle inequality, which is that

N(v + w)^{q_{N}} ≤ N(v)^{q_{N}} + N(w)^{q_{N}} (B.5.2)

for every v, w ∈ V.

If we also have that N(v) > 0 when v ≠ 0, then N is said to be a q_{N}-norm on V with respect to |·| on k. If q_{k} = q_{N} = 1, then q_{N}-seminorms and q_{N}-norms on V with respect to | · | on k are the same as ordinary seminorms and norms, as in Section A.5. Of course, | · | is a q_{k}-norm on k, as a one-dimensional vector space over itself, as before.

As usual, (B.5.2) is the same as saying that

N(v + w) ≤ (N(v)^{q_{N}} + N(w)^{q_{N}})^{1/q_{N}} (B.5.3)

for every v, w ∈ V. The right side is monotonically decreasing in q_{N}, as in Section 1.12, so that this condition is more restrictive as q_{N} increases, as before.

If N satisfies (B.5.1) and

N(v + w) ≤ max(N(v), N(w)) (B.5.4)

for every v, w ∈ V, then we say that N is a semi-ultranorm or pseudo-ultranorm on V with respect to | · | on k, as in Section A.5. If N(v) > 0 when v ≠ 0 too, then N is said to be an ultranorm on V with respect to | · | on k, as before. A semi-ultranorm on V is a q_{N}-seminorm for every q_{N} > 0, as usual, and similarly an ultranorm is a q_{N}-norm for every q_{N} > 0.

As in Section 1.3,

lim_{q_{N} →∞}(N(v)^{q_{N}} + N(w)^{q_{N}})^{1/q_{N}} = max(N(v), N(w)) (B.5.5)

B.6. ℓ^{R} SPACES AND Q-SEMINORMS 609

for every v, w ∈ V. Semi-ultranorms and ultranorms on V may be considered as q_{N}-seminorms and q_{N}-norms, respectively, with q_{N} = ∞, as before.

If N is a q_{N}-seminorm on V with respect to | · | on k, and if N(v) > 0 for some v ∈ V, then it is easy to see that | · | is a q_{N}-absolute value function on k. In particular, if N is a q_{N}-norm on V, and V ≠ {0}, then | · | is a q_{N}-absolute value function on k.

If N is a q_{N}-seminorm on V with respect to | · | on k, then

d_{N}(v, w) = N(v − w) (B.5.6)

is a q_{N}-semimetric on V. If N is a q_{N}-norm on V, then this is a q_{N}-metic on V.

Let a be a positive real number, and remember that | · |^{a} defines a (q_{k}/a)- absolute value function on k, as in Section B.3. If N is a q_{N}-seminorm on V with respect to | · | on k, then it is easy to see that

N(v)^{a} (B.5.7)

is a (q_{N}/a)-semimetric on V with respect to | · |^{a} on k. Similarly, if N is a q_{N}-norm on V with respect to | · | on k, then this is a (q_{N}/a)-norm on V with respect to | · |^{a} on k.

B.6 ℓ^{r} Spaces and q-seminorms

Let X be a nonempty set, and let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0. Also let W be a vector space over k, and let N_{W} be a q_{W} -seminorm on W with respect to | · | on k for some q_{W} > 0. Consider the space ℓ^{∞}(X, W) = ℓ_{N_{W}}^{∞} (X, W) (B.6.1)

of all W -valued functions f on X that are bounded with respect to N_{W}, so that N_{W} (f(x)) is bounded as a real-valued function on X, as in Section A.9. In this case, we put

∥f∥_{∞} = ∥f∥_{sup} = ∥f∥_{ℓ^{∞}(X,W)} = ∥f∥_{ℓ_{N_{W}}^{∞} (X,W)} = sup_{x∈X} N(f(x)), (B.6.2)

as before.

If r is a positive real number, then we let

ℓ^{r}(X, W) = ℓ_{N_{W}}^{r} (X, W) (B.6.3)

be the space of W -valued functions f on X such that N_{W} (f(x)) is r-summable as a nonnegative real-valued function on X, as in Section A.11. Under these conditions, we put

∥f∥_{r} = ∥f∥_{ℓ^{r}(X,W)} = ∥f∥_{ℓ_{N_{W}}^{r} (X,W)} = ( ∑_{x∈X} N_{W} (f(x))^{r})^{1/r}, (B.6.4)

as before.

One can check that ℓ_{N_{W}}^{r} (X, W) is a linear subspace of the space of all W - valued functions on X for each 0 < r ≤ ∞. Note that (B.6.2) and (B.6.4) satisfy the usual homogeneity condition (B.5.1) with respect to | · | on k.

If r ≤ q_{W} , (B.6.5)

then one can verify that

(B.6.6) ∥ · ∥_{ℓ_{N_{W}}^{r} (X,W)} is an r-seminorm on ℓ_{N_{W}}^{r} (X, W),

with respect to | · | on k. This uses the fact that

(B.6.7) N_{W} is an r-seminorm on W

with respect to | · | on k, because of (B.6.5), as in the previous section. If N_{W} is a q_{W} -norm on W, then

(B.6.8) ∥ · ∥_{ℓ_{N_{W}}^{r} (X,W)} is an r-norm on ℓ_{N_{W}}^{r} (X, W).

If q_{W} ≤ r, (B.6.9)

then (B.6.10) ∥ · ∥_{ℓ_{N_{W}}^{r} (X,W)} is a q_{W} -seminorm on ℓ_{N_{W}}^{r} (X, W)

with respect to | · | on k. This can be seen fairly directly when r = ∞, and otherwise it can be obtained using Minkowski’s inequality for sums. If N_{W} is a q_{W} -norm on W, then we get that

(B.6.11) ∥ · ∥_{ℓ_{N_{W}}^{r} (X,W)} is a q_{W} -norm on ℓ_{N_{W}}^{r} (X, W),

as before.

B.7 More on ℓ^{r} spaces

Let us continue with the same notation and hypotheses as in the previous section. We may refer to (B.6.2) as the supremum q_{W} -seminorm on ℓ_{N_{W}}^{∞} (X, W) associated to N_{W}, or as the supremum q_{W} -norm when N_{W} is a q_{W} -norm on W. This leads to the corresponding supremum q_{W} -semimetric or supremum q_{W} -metric on ℓ_{N_{W}}^{∞} (X, W) in the usual way.

If 0 < r_{1} ≤ r_{2} ≤ +∞, then

ℓ_{N_{W}}^{r_{1}} (X, W) ⊆ ℓ_{N_{W}}^{r_{2}} (X, W), (B.7.1)

as in Sections 2.6 and A.11. In this case,

∥f∥_{ℓ_{N_{W}}^{r_{2}} (X,W)} ≤ ∥f∥_{ℓ_{N_{W}}^{r_{1}} (X,W)} (B.7.2)

B.8. Q-ABSOLUTE CONVERGENCE 611

for every f ∈ ℓ_{N_{W}}^{r_{1}} (X, W), as before.

Let us say that a W -valued function f on X vanishes at infinity on X with respect to N_{W} if N_{W} (f(x)) vanishes at infinity as a nonnegative real-valued function on X, as in Section A.9. The space

c_{0}(X, W) = c_{0,N_{W}} (X, W) (B.7.3)

of these functions is a linear subspace of ℓ_{N_{W}}^{∞} (X, W), as before. Remember that c_{00}(X, W) is the space of W valued functions on X with finite support, as in Section A.9. One can check that

(B.7.4) c_{0,N_{W}} (X, W) is the closure of c_{00}(X, W) in ℓ_{N_{W}}^{∞} (X, W)

with respect to the supremum q_{W} -semimetric, as before.

If 0 < r < ∞, then

c_{00}(X, W) ⊆ ℓ_{N_{W}}^{r} (X, W) ⊆ c_{0,N_{W}} (X, W), (B.7.5)

as in Section A.11. In this case, one can verify that

(B.7.6) c_{00}(X, W) is dense in ℓ_{N_{W}}^{r} (X, W)

with respect to the q_{W} or r-semimetric, as appropriate, associated to (B.6.4), as before.

Suppose that N_{W} is a q_{W} -norm on W, and that W is complete with respect to the corresponding q_{W} -metric. One can check that

(B.7.7) ℓ_{N_{W}}^{r} (X, W) is complete

with respect to the q_{W} or r-metric associated to (B.6.2) or (B.6.4), as appropriate, as in Sections A.9 and A.11.

B.8 q-Absolute convergence

Let k be a field with a q_{k}-absolute value function for some q_{k} > 0, let V be a vector space over k, and let N be a q_{N}-norm on V with respect to | · | on k for some q_{N} > 0. Of course, one can define convergence of sequences in V with respect to the q_{N}-metric d_{N} associated to N as in Section B.5 in the usual way, which is the same as convergence of sequences with respect to the topology determined on V by d_{N} as in Section B.2. One can check that convergent sequences in V with respect to d_{N} have the usual properties in terms of sums and scalar multiplication, as in Section A.7.

Convergence of an infinite series ∑_{j=1}^{∞} v_{j} with terms in V with respect to N means that the corresponding sequence of partial sums ∑_{j=1}^{n} v_{j} converges in V with respect to d_{N}, as before. This also has the usual properties in terms of sums and scalar multiplication, because of the analogous statements for sequences.

The sequence of partial sums is a Cauchy sequence with respect to d_{N} if and only if for every ϵ > 0 there is a positive integer L(ϵ) such that

N( ∑_{j=l}^{n} v_{j}) < ϵ (B.8.1)

for all integers n ≥ l ≥ L(ϵ), as in Section A.7. This implies that {v_{j}}_{j=1}^{∞} converges to 0 with respect to d_{N}, as usual.

Suppose that q_{N} < ∞, and observe that

N( ∑_{j=l}^{n} v_{j})^{q_{N}} ≤ ∑_{j=l}^{n} N(v_{j})^{q_{N}} (B.8.2)

for all positive integers l ≤ n. We say that ∑_{j=1}^{∞} converges q_{N}-absolutely with respect to N if

∑_{j=1}^{∞} N(v_{j})^{q_{N}} (B.8.3)

converges as an infinite series of nonnegative real numbers. This is the same as ordinary absolute convergence when q_{N} = 1, as in Section A.7. This implies that the corresponding sequence of partial sums is a Cauchy sequence with respect to d_{N}, because of (B.8.2), as before.

If V is complete with respect to d_{N}, as in Section B.2, then we say that V is a q_{N}-Banach space. This is the same as a Banach space when q_{N} = 1, as in Section A.7. If V is a q_{N}-Banach space with respect to N and ∑_{j=1}^{∞} v_{j} converges q_{N}-absolutely with respect to N, then ∑_{j=1}^{∞} v_{j} converges in V, and one can check that

N( ∑_{j=1}^{∞} v_{j})^{q_{N}} ≤ ∑_{j=1}^{∞} N(v_{j})^{q_{N}} . (B.8.4)

One can also verify that the completeness of V with respect to d_{N} is characterized by the condition that q_{N}-absolutely convergent series in V with respect to N converge in V, as before. If V is not already complete with respect to d_{N}, then V has a completion that is a q_{N}-Banach space over k, as usual.

B.9 Some more Lipschitz conditions

Let X, Y be sets with q_{X}, q_{Y} -semimetrics d_{X}, d_{Y} for some q_{X}, q_{Y} > 0, and let α be a positive real number. A mapping f from X into Y is said to be Lipschitz of order α with respect to d_{X}, d_{Y} if there is a nonnegative real number C such that d_{Y} (f(x), f(w)) ≤ C d_{X}(x, w)^{α} (B.9.1)

for every x, w ∈ X. This is the same as in Section 2.1 when α = 1. As before, we may say that f is Lipschitz of order α with constant C, to be more precise.

One can define uniform continuity of mappings from X into Y with respect to d_{X}, d_{Y} in the same way as for metric spaces. In particular,

(B.9.2) Lipschitz mappings of order α are uniformly continuous,

as in Section 2.1. Constant mappings from X into Y are Lipschitz of order α with C = 0. If f is Lipschitz of order α with constant C = 0, and if d_{Y} is a q_{Y} -metric on Y, then f is constant on X.

Let a_{X}, a_{Y} be positive real numbers, so that d_{X}^{a_{X}}, d_{Y}^{a_{Y}} are (q_{X}/a_{X}), (q_{Y} /a_{Y})- semimetrics on X, Y, respectively, as in Section B.2. It is easy to see that (B.9.1) is the same as saying that

d_{Y} (f(x), f(w))^{a_{Y}} ≤ C^{a_{Y}} (d_{X}(x, w)^{a_{X}})^{α a_{Y} /a_{X}} (B.9.3)

for every x, w ∈ X. This means that f is Lipschitz of order α with constant C with respect to d_{X}, d_{Y} if and only if

(B.9.4) f is Lipschitz of order α a_{Y} /a_{X} with constant C^{a_{Y}}

with respect to d_{X}^{a_{X}}, d_{Y}^{a_{Y}}.

The space of all Lipschitz mappings from X into Y of order α with respect to d_{X}, d_{Y} may be denoted Lip_{α}(X, Y). (B.9.5) If α = 1, then this corresponds to Lip(X, Y), as in Section 2.1.

Suppose that X ≠ ∅, and that f is a Lipschitz mapping of order α from X into Y. We would like to put

(B.9.6) Lip_{α}(f) = Lip_{α,X,Y} (f)
= sup \{\frac{d_{Y} (f(x), f(w))}{d_{X}(x, w)^{α}} : x, w ∈ X, d_{X}(x, w) > 0\},

at least if there are x, w ∈ X such that d_{X}(x, w) > 0. Otherwise, this may be interpreted as being equal to 0. This corresponds to Lip(f) as in Section 2.1 when α = 1, and we may also use this notation here in this case. One can check that Lip_{α}(f) is the smallest nonnegative real number C such that f is Lipschitz of order α with constant C, as before.

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and suppose now that Y is a vector space over k with a q_{Y} -seminorm N_{Y} with respect to | · | on k. In this case, we can take d_{Y} to be the q_{Y} -semimetric d_{N_{Y}} on Y associated to N_{Y}. One can check that

(B.9.7) Lip_{α}(X, Y) is a linear subspace of the space of all functions on X with values in Y,

as in Sections 2.1 and A.10. More precisely,

(B.9.8) Lip_{α}(f) is a q_{Y} -seminorm on Lip_{α}(X, Y)

with respect to | · | on k, as before.

Note that the space of uniformly continuous mappings from X into Y is a linear subspace of the space of all mappings from X into Y under these conditions.

B.10 Operator q-seminorms

Let k be a field with a q_{k}-absolute value function for some q_{k} > 0, and let V, W be vector spaces over k with q_{V}, q_{W} -seminorms N_{V}, N_{W}, respectively, with respect to | · | on k, for some q_{V} , q_{W} > 0. A linear mapping T from V into W is said to be bounded with respect to N_{V}, N_{W} if there is a nonnegative real number C such that N_{W} (T (v)) ≤ C N_{V} (v) (B.10.1)

for every v ∈ V, as in Sections 2.2 and A.10. This means that

N_{W} (T (u) − T (v)) ≤ C N_{V} (u − v) (B.10.2)

for every u, v ∈ V, as before. This says that T is Lipschitz of order 1 with constant C with respect to the q_{V}, q_{W} -semimetrics d_{N_{V}}, d_{N_{W}} associated to N_{V}, N_{W}, respectively, as in the previous section.

Let BL(V, W) be the space of all bounded linear mappings from V into W with respect to N_{V}, N_{W}, respectively, as before. One can verify that this is a linear subspace of the space L(V, W) of all linear mappings from V into W, as before. If T ∈ BL(V, W), then put

(B.10.3) ∥T ∥_{op} = ∥T ∥_{op,V W} = inf{C ≥ 0 : (B.10.1) holds},

as in Section A.10. This is the same as the Lipschitz constant Lip(T) = Lip_{1}(T) of T with respect to the q_{V}, q_{W} -metrics associated to N_{V}, N_{W}, respectively, as before.

In particular,

(B.10.4) ∥ · ∥_{op} is a q_{W} -seminorm on BL(V, W)

with respect to | · | on k, as in the previous section. This is the operator q_{W} - seminorm on BL(V, W) associated to N_{V}, N_{W}. Note that the infimum on the right side of (B.10.3) is automatically attained, so that

N_{W} (T (v)) ≤ ∥T ∥_{op,V W} N_{V} (v) (B.10.5)

for every v ∈ V. If N_{W} is a q_{W} -norm on W, then

(B.10.6) ∥ · ∥_{op} is a q_{W} -norm on BL(V, W),

as in Section A.10. In this case, ∥ · ∥_{op} may be called the operator q_{W} -norm on BL(V, W) associated to N_{V}, N_{W}.

Let Z be another vector space over k, with a q_{Z}-seminorm N_{Z} with respect to | · | on k for some q_{Z} > 0. If T_{1}, T_{2} are bounded linear mappings from V, W into W, Z, respectively, then T_{2} ∘ T_{1} is a bounded linear mapping from V into Z, with ∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}, (B.10.7)

as in Sections 2.2 and A.10.

Let T be a linear mapping from V into W again, and suppose for the moment that N_{W} (T (v)) is bounded on a ball of positive radius in V centered at 0 with respect to the q_{V} -semimetric associated to N_{V}. If | · | is not the trivial absolute value function on k, then one can verify that

(B.10.8) T is a bounded linear mapping from V into W

with respect to N_{V}, N_{W}, as in Section A.10. In particular, the hypothesis on T holds when T is continuous at 0 with respect to the q_{V}, q_{W} -semimetrics associated to N_{V}, N_{W}, respectively, as before.

If N_{W} is a q_{W} -norm on W, and W is complete with respect to the q_{W} -metric associated to N_{W}, then (B.10.9) BL(V, W) is complete

with respect to the q_{W} -metric associated to ∥ · ∥_{op}, as in Sections 2.2 and A.10.

Let V_{0} be a linear subspace of V that is dense in V with respect to the q_{V} -semimetric associated to N_{V}, and let T_{0} be a bounded linear mapping from V_{0} into W, with respect to the restriction of N_{V} to V_{0}. If N_{W} is a q_{W} -norm on W, and W is complete with respect to the q_{W} -metric associated to N_{W}, then

(B.10.10) there is a unique extension of T_{0} to a bounded linear mapping T from V into W,

as in Sections 2.2 and A.10. One can verify that the operator q_{W} -norm of T on V is equal to the operator q_{W} -norm of T_{0} on V_{0}, as before.

B.11 Some more bounded linear mappings

Let X be a nonempty set, let k be a field with a q_{k}-absolute value function for some q_{k} > 0, and let W be a vector space over k with a q_{W} -seminorm N_{W} with respect to | · | on k for some q_{W} > 0. If a is a W -valued function on X and f is a k-valued function on X with finite support, then put

T_{a}(f) = ∑_{x∈X} a(x) f(x), (B.11.1)

as in Section A.12. This defines a linear mapping from c_{00}(X, k) into W, as before.

Let δ_{y} be the k-valued function on X equal to 1 at y ∈ X and to 0 at every other point in X, as before. Remember that ∥δ_{y}∥_{ℓ^{r}(X,k)} = 1 for every r > 0, using | · | as a q_{k}-norm on k, as a one-dimensional vector space over itself. We also have that T_{a}(δ_{y}) = a(y) (B.11.2)

for every y ∈ X, as before.

Suppose that

(B.11.3) T_{a} is bounded as a linear mapping from c_{00}(X, k) into W, with respect to ∥ · ∥_{ℓ^{r}(X,k)} on c_{00}(X, k) and N_{W} on W for some r > 0. This implies that

N(a(y)) ≤ ∥T ∥_{op,r} (B.11.4)

for every y ∈ X, as in Section A.12, where ∥T_{a}∥_{op,r} is the operator q_{W} -seminorm of T_{a} with respect to ∥ · ∥_{ℓ^{r}(X,k)} on c_{00}(X, k) and N_{W} on W. Equivalently, a is bounded on X with respect to N_{W} on W, with

∥a∥_{ℓ_{N_{W}}^{∞} (X,W)} ≤ ∥T_{a}∥_{op,r}. (B.11.5)

If a is any element of ℓ_{N_{W}}^{∞} (X, W), f ∈ c_{00}(X, W), and q_{W} < ∞, then

(B.11.6)  N_{W} (T_{a}(f))^{q_{W}} ≤ ∑_{x∈X} N_{W} (a(x))^{q_{W}} |f(x)|^{q_{W}}
≤ ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)}^{q_{W}} ∥f∥_{ℓ^{q_{W}} (X,k)}^{q_{W}},

as before. This means that

(B.11.7) T_{a} is a bounded linear mapping from c_{00}(X, k) into W, with respect to ∥ · ∥_{ℓ^{q_{W}} (X,k)} on c_{00}(X, k) and N_{W} on W,

with ∥T_{a}∥_{op,q_{W}} ≤ ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)}. (B.11.8)

More precisely, we have that

∥T_{a}∥_{op,q_{W}} = ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)}, (B.11.9)

because of (B.11.5). If N_{W} is a q_{W} -norm on W, and W is complete with respect to the associated q_{W} -metric, then it follows that

(B.11.10) T_{a} has a unique extension to a bounded linear mapping from ℓ^{q_{W}} (X, k) into W, with the same operator q_{W} -norm,

as in the previous section.

If q_{W} = ∞, then

N_{W} (T_{a}(f)) ≤ ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)} ∥f∥_{ℓ^{∞}(X,k)} (B.11.11)

for every f ∈ c_{00}(X, k), so that T_{a} is a bounded linear mapping from c_{00}(X, k) into W, with respect to ∥ · ∥_{ℓ^{∞}(X,k)} on c_{00}(X, k) and N_{W} on W, as in Section A.12. We also have that

∥T_{a}∥_{op,∞} = ∥a∥_{ℓ_{N_{W}}^{∞} (X,W)} (B.11.12)

in this case, because of (B.11.5), as before. If N_{W} is an ultranorm on W, and W is complete with respect to the associated ultrametric, then T_{a} has a unique extension to a bounded linear mapping from c_{0}(X, k) into W, with the same operator norm, as before.

B.12 More on sums of vectors

Let X be a nonempty set, let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0 again, and let W be a vector space over k with a q_{W} -norm N_{W} with respect to | · | on k for some q_{W} > 0. If f is a W -valued function on X, then the convergence of the sum ∑_{x∈X} f(x) in the generalized sense with respect to the q_{W} -metric associated to N_{W} can be defined in the same way as in Section A.13. Similarly, the generalized Cauchy condition for the sum can be defined in the same way as before. These have essentially the same properties as before, with ϵ/2 replaced with

ϵ/2^{q_{W}} (B.12.1)

in a couple of places when q_{W} < ∞.

If q_{W} < ∞, then

N_{W} ( ∑_{x∈C} f(x))^{q_{W}} ≤ ∑_{x∈C} N_{W} (f(x))^{q_{W}} (B.12.2)

for every nonempty finite subset C of X. This implies that the sum satisfies the generalized Cauchy condition when

f ∈ ℓ_{N_{W}}^{q_{W}} (X, W), (B.12.3)

as before. If the sum converges in the generalized sense, then one can check that

N_{W} ( ∑_{x∈X} f(x))^{q_{W}} ≤ ∑_{x∈X} N_{W} (f(x))^{q_{W}} , (B.12.4)

as before.

One can define Sum_{N_{W}} (X, W) and GCC_{N_{W}} (X, W) in the same was as in Section A.14, and with basically the same properties as before. If q_{W} < ∞, then ℓ_{N_{W}}^{q_{W}} (X, W) ⊆ GCC_{N_{W}} (X, W), (B.12.5)

as in the preceding paragraph. This implies that

ℓ_{N_{W}}^{q_{W}} (X, W) ⊆ Sum_{N_{W}} (X, W) (B.12.6)

when W is a q_{W} -Banach space, as before.

One can also define BF S_{N_{W}} (X, W) and ∥f∥_{BF S_{N_{W}} (X,W)} in the same way as before, and with essentially the same properties as before. More precisely,

(B.12.7) ∥f∥_{BF S_{N_{W}} (X,W)} is a q_{W} -norm on BF S_{N_{W}} (X, W),

with respect to | · | on k. It is easy to see that

ℓ_{N_{W}}^{q_{W}} (X, W) ⊆ BF S_{N_{W}} (X, W), (B.12.8)

with ∥f∥_{BF S_{N_{W}} (X,W)} ≤ ∥f∥_{ℓ_{N_{W}}^{q_{W}} (X,W)} (B.12.9)

for every f ∈ ℓ_{N_{W}}^{q_{W}} (X, W).

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Index

A[[T]], 341 A[T], 340 A^{op}, 449 A^{sym}, 543 A_{asa}, 543 A_{h}, 543 A_{r}, 528 A_{sa}, 528 absolute convergence, 10, 589, 612 absolute value functions, 582, 605

archimedean, 584 discrete, 591 equivalent, 585 non-archimedean, 584 quasimetric, 607 ultrametric, 582, 606 valuation group, 590

absorbing sets, 64 adjoint operators, 59, 438, 444 algebra automorphisms, 251

∗-automorphisms, 277, 474 algebra homomorphisms, 127, 423

∗-homomorphisms, 277, 474 opposite, 128, 424

algebra involutions, 128, 424

conjugate linear, 128 algebra isomorphisms, 127, 424

∗-isomorphisms, 277, 474 opposite, 128, 424

algebraic dual spaces, 52, 433 algebras in the strict sense, 124, 423 anti-self-adjoint elements, 151, 441 anti-self-adjoint linear operators, 180 antisymmetric linear operators, 179 approximate eigenvalues, 200 approximate point spectrum, 200 associative algebras, 124, 423 invertible elements, 129, 425 simple, 575

B(X, Y), 155 B^{∗} algebras, 275 Baire category theorem, 77 Banach ∗-algebras, 275 Banach algebras, 129

topologically simple, 575 Banach spaces, 9, 590, 612 Banach -algebra, 275 Banach–Alaoglu theorem, 83 Banach–Steinhaus theorem, 80 bases for topologies, 21, 296 BF S(X, W), 38 BF S_{N_{W}} (X, W), 601, 617 bilinear forms, 120, 178, 436 alternating, 437 antisymmetric, 178, 436 kernels, 508 nondegenerate, 437, 508 skew-symmetric, 436 symmetric, 178, 436 bilinear functionals, 120, 436 bilinear functions, 436 bilinear mappings, 117, 423

kernels, 505 nondegenerate on the left, 505 on the right, 505 bilipschitz mappings, 160 binomial coefficients, 374 binomial series, 376 binomial theorem, 374 BL(V), 126 BL(V, W), 25, 593, 614 BL_{W} (V), 451 Borel measures, 108 Borel sets, 107 bounded bilinear mappings, 118 bounded finite sums, 38, 601 bounded functions, 3, 28, 155, 591 bounded linear functionals, 52 bounded linear mappings, 25, 593, 614 bounded sesquilinear forms, 121

C, 2 C(X, A), 158 C(X, C), 6 C(X, R), 6 C(X, Y), 155 C[T_{1}, …, T_{n}], 297 C^{∗} algebras, 275 C^{∗} algebras of operators, 276 C^{∗} identity, 60, 154 C^{lai}(X, C), 397 c^{lai}(X, C), 56 C^{lai}(X, R), 397 c^{lai}(X, R), 56 (C)_{r}^{p}[[T]], 345 C_{0}(X, C), 100 c_{0}(X, C), 19 C_{0}(X, R), 100 c_{0}(X, R), 19 c_{0}(X, W), 33, 592, 611 c_{0,N_{W}} (X, W), 592, 611 (C)_{0,r}[[T]], 345 C_{0,w}(X, C), 400 C_{0,w}(X, R), 400 c_{0,w}(X, Z), 339 c_{00}(X, C), 17 c_{00}(X, R), 17 c_{00}(X, W), 27, 487, 592 C_{b}(X, C), 7 C_{b}(X, R), 7 C_{b}(X, Y), 155 C_{b,w}(X, C), 400 C_{b,w}(X, R), 400 C_{com}(X, C), 99 C_{com}(X, R), 99 (C)_{ρ}[[T]], 345 Cartesian products, 114, 487 Cauchy nets, 271 Cauchy products, 211, 212, 215 Cauchy–Schwarz inequality, 13, 91 semidefinite case, 186, 187

centralizers, 321, 428

chain rule, 359 closed balls, 4, 581 closed graph theorem, 169 codimensions of linear subspaces, 432 commutative algebras, 124, 423 compact support, 99 complete regularity, 104 complete separability, 21 completely Hausdorff spaces, 105 completions, 22 complex measures, 111 complex-linear mappings, 12 complexifications

inner products, 268 linear mappings, 268 vector spaces, 267 conjugate exponents, 57 conjugate-linear mappings, 13 contraction mapping theorem, 162 contractions, 162 convergence in measure, 96 convergent nets, 270 convergent series, 8, 588 convex cones, 63 convex sets, 3 countable compactness, 296

degrees of polynomials, 341

dense sets, 77 derivations, 227 derivatives, 348 direct products of modules

over associative algebras, 487 of representations

over associative algebras, 487 of vector spaces, 487

direct sums of Hilbert spaces, 492 of modules

over associative algebras, 455, 487 of representations

over associative algebras, 455,

487 of vector spaces, 116, 487

algebraic, 488

c_{0}, 488 ℓ^{∞}, 488 ℓ^{p}, 489

directed systems, 270 discrete metric, 581 disk algebra, 310, 421 division algebras, 473 division rings, 473 dual linear mappings, 72, 433 dual norms, 52 dual spaces, 52

eigenvalues, 197, 426 eigenvectors, 197, 426 equicontinuity, 238 essential range, 145 exponential function, 226, 245, 363 extreme points, 544

F_{σ} sets, 107 first category, 78 first countability condition, 102 formal polynomials, 296, 340, 430 formal power series, 341

G(A), 131, 426 G_{1}(A), 235 G_{δ} sets, 107 GCC(X, W), 36 GCC_{N_{W}} (X, W), 601, 617 Gelfand map, 283 Gelfand spectrum, 282 Gelfand transform, 282, 283 Gelfand–Mazur theorem, 136 general linear groups, 432 generalized Cauchy condition, 34, 598,

617 generalized convergence, 33, 598, 617 GL(V), 432 graphs of mappings, 167, 168

H, 146 H(U), 12 H^{∞}(U), 12 Hahn decomposition theorem, 112 Hahn–Banach theorem, 65, 67 Hermitian antisymmetry, 179 Hermitian elements, 151 Hermitian forms, 178 Hermitian linear operators, 180 Hermitian symmetry, 178 Hilbert spaces, 14 Hölder’s inequality, 57, 92 holomorphic functions, 311 Hom_{A}(V_{1}, V_{2}), 471 homogeneous functions, 61 homotopies, 239

ideals in commutative algebras, 139 left, 425 right, 425 two-sided, 138, 425 idempotent elements, 148, 152 identity matrix, 554 indicator functions, 95 inner derivations, 229 inner products, 13 intertwining mappings, 453 invariant linear subspaces, 464 irreducibility, 464

algebraically irreducible, 464 norm-irreducibility, 465 topologically irreducibile, 465

isometric linear mappings, 42 isometric mappings, 22

Jacobson radical, 140 Jordan decomposition, 112

k[X_{1}, …, X_{n}], 430

L(V), 124, 423

L(V, W), 25, 433, 593

ℓ^{2}(X, C), 38 ℓ^{2}(X, R), 38 ℓ^{2}(X, W), 37 ℓ^{∞}(X, A), 157 L^{∞}(X, C), 91 ℓ^{∞}(X, C), 3 L^{∞}(X, R), 91 ℓ^{∞}(X, R), 3 ℓ^{∞}(X, W), 28, 591, 609 ℓ_{N_{W}}^{∞} (X, W), 591, 609 ℓ_{w}^{∞}(X, Z), 338 L^{p}(X, C), 90 L^{p}(X, R), 90 ℓ^{p}(X, W), 32 ℓ_{w}^{p}(X, Z), 338 ℓ^{r}(X, W), 595, 609 ℓ_{N_{W}}^{r} (X, W), 595, 609 L_{k}(V_{1}, V_{2}), 469 L_{W} (V), 431 left inverses, 246, 247 left topological divisors of zero, 248 Leibniz rule, 228 limit point property, 295 limit points, 295 limits at infinity, 56, 397 Lindelöf’s theorem, 296 Lindelöf property, 296 linear orderings, 270 linear functionals, 52, 433

antisymmetric, 515

Hermitian symmetric, 515 symmetric, 515

Lip(X, Y), 23 Lip_{α}(X, Y), 613 Lip_{b}(X, W), 158 Lipschitz mappings, 23, 612

of order α, 612 local compactness, 100 logarithms, 363, 364, 373

M(X, C), 111 M(X, R), 111 M_{m,n}(A), 555, 560 M_{n}(A), 554, 560 maximal ideal space, 282 meager sets, 78 metrics, 581, 603 Minkowski functional, 64 Minkowski’s inequality, 5, 31, 91 modules over associative algebras, 447 algebraically cyclic, 462 algebraically irreducible, 464 algebraicaly simple, 464 cyclic, 462 cyclic vectors, 462 direct products, 487 direct sums, 455, 487 dual modules, 459 free, 487 homomorphisms, 454 irreducible, 464 isomorphisms, 454 left, 447 nondegenerate, 456 norm-cyclic, 463 norm-cyclic vectors, 463 norm-irreducible, 465 quotient modules, 453 right, 447 simple, 464 submodules, 450 topologically irreducible, 465 topologically cyclic, 463 topologically cyclic vectors, 463 unitarily equivalent, 454

multi-indices, 296

multiplication operators, 88, 94, 102 left, 127, 427, 447 right, 128, 427, 448

multiplicative identity elements, 124,

423 adjoining, 145, 395, 397, 405, 407,

412, 425, 457 left, 402, 427 right, 402, 427 multiplicative inverses, 129

N = N (A), 140

negative variation measures, 112

nets, 270

nilpotent elements, 140, 141, 148, 426 nilradical, 140 non-meager sets, 78 nondegenerate collections, 483 nonnegative bilinear forms, 186 nonnegative elements, 327 nonnegative Hermitian forms, 186 nonnegative linear functionals, 413 nonnegative linear operators, 187 normal operators, 188 normal elements, 152 normal topological spaces, 104 norms, 2, 586 nowhere dense sets, 78

open balls, 4, 581 open mapping theorem, 165 open mappings, 164 operator norms, 25, 594, 614 operator q-norms, 614 operator q-seminorms, 614 operator seminorms, 594, 614 opposite algebra, 449 orthogonal complements, 51 orthogonal projections, 45, 48, 49, 51,

174 orthogonal transformations, 42 orthogonal vectors, 14 orthonormal bases, 48 orthonormal vectors, 44, 46 Ostrowski’s theorems, 585

p-adic absolute value, 583 p-adic metric, 583 p-adic numbers, 583 p-summable functions, 31 parallelogram law, 50 partial isometries, 261 partial orderings, 270 perfectly normal spaces, 110 point spectrum, 197, 427 pointwise boundedness, 79 pointwise convergence, 272 polar decompositions, 259, 260 polarization, 42, 178, 179, 183, 184 polydisk algebras, 312 polynomial convexity, 303 positive functionals, 409–411 positive linear functionals, 409–411 positive variation measures, 112 pre-directed system, 270 pre-orderings, 270 product topology, 114 projections, 172 pseudo-ultrametrics, 603 pseudo-ultranorms, 586, 608 pseudocompactness, 296 pseudometrics, 4, 603 pseudonorms, 4, 585

Q, 583

q-absolute convergence, 612 q-absolute value functions, 605 q-Banach spaces, 612 q-metrics, 603 q-norms, 608 q-pseudometrics, 603 q-pseudonorms, 608 q-semimetrics, 603 q-seminorms, 608 Q_{p}, 583 quasinilpotent elements, 148 quaternions, 146 quotient algebras, 138 quotient mappings, 136 quotient spaces, 136

R, 2

R = R(A), 140 R[T_{1}, …, T_{n}], 297 (R)_{r}^{p}[[T]], 345 R_{+}, 363, 590 (R)_{0,r}[[T]], 345 r_{A}(x), 142, 147 (R)_{ρ}[[T]], 345 rad(A), 140 rad_{Sp}(A), 317 radial at a point, 64 radicals, 140, 317 radius of convergence, 214 real measures, 111 real-linear mappings, 13 real-linear subspaces, 151 reducibility, 464

algebraically reducible, 464 norm-reducibility, 465 topologically reducible, 465 reflexive Banach spaces, 75 regular topological spaces, 105 representations

of associative algebras, 446 ∗-representations, 475, 476 algebraically irreducible, 464 algebraically simple, 464 cyclic, 462 cyclic vectors, 462 direct products, 487 direct sums, 455, 487 equivalent, 454 homomorphisms, 453 invariant linear subspaces, 449 irreducible, 464 isomorphisms, 454 nondegenerate, 456 norm irreducible, 465 norm-cyclic, 463 norm-cyclic vectors, 463 quotient representations, 453 simple, 464 subrepresentations, 450 topologically cyclic, 463 topologically cyclic vectors, 463 topologically irreducible, 465 unitarily equivalent, 454

resolvent set of an element, 134 Riesz representation theorems, 53, 416,

525 right inverses, 247 right topological divisors of zero, 248 roots, 382, 384

Schur’s lemma, 466, 467 second category, 78 second countability condition, 21 second dual mappings, 75, 76 second dual spaces, 74 self-adjoint elements, 151, 441 self-adjoint linear operators, 180 semi-ultrametrics, 603 semi-ultranorms, 586, 608 semimetrics, 4, 603 seminorms, 4, 585 semisimple Banach algebras, 141 separable topological spaces, 20, 296 separate continuity, 119 sequential continuity, 101, 102 sesquilinear forms, 121, 506

Hermitian antisymmetric, 179 Hermitian symmetric, 178 kernels, 506, 508 nondegenerate, 442, 508 on the left, 506 on the right, 506

σ_{A}(x), 133, 426 σ_{ap}(T), 200 σ_{p}(T), 197, 427 signed measures, 111 Sp(A), 282, 289, 426 spectral mapping theorem, 192, 430 spectral permanence theorem, 149 spectrum of an element, 133, 426 square roots, 258, 266, 382, 384 standard Euclidean norm, 6 starlike sets, 64 strong limit point property, 295 strong limit points, 295 strong operator topology, 82, 273 subadditive functions, 61 subalgebras, 125, 423 sublinear functions, 61 submultiplicative norms, 125 submultiplicative sequences, 142 Sum(X, W), 36 Sum_{N_{W}} (X, W), 600, 617 summable functions, 30 supports of functions, 16, 27, 99, 592 supremum metrics, 3, 28, 155, 592, 610 supremum norms, 3, 28, 592, 610 supremum semimetrics, 610 supremum seminorms, 592, 610 symmetric functions, 61 symmetric linear operators, 179 symmetric sets, 62

T -cyclic subspaces, 473

topologically T -cyclic closed sub-

spaces, 474 tending to infinity, 107 topological divisors of zero, 249 total orderings, 270 total variation measures, 111 totally bounded sets, 84 traces

of linear mappings, 571 of matrices, 569

transpose matrices, 568 triangle inequality, 2, 581, 608

ultrametric version, 581 trivial ultranorm, 586 trivial absolute value function, 583 two-sided ideals, 138

U(A), 152, 337, 441 UC(X, Y), 156 UC_{b}(X, Y), 156 ultrametrics, 581, 604 ultranorms, 586, 608 uniform boundedness, 79 uniform boundedness principle, 80 uniform continuity, 613 uniform convergence, 272 unitary elements, 336 unitary mappings, 42 Urysohn spaces, 104 Urysohn’s lemma, 100, 105 Urysohn’s metrization theorem, 106

V′, 52 V [[T]], 341 V [T], 340 V^{alg}, 52, 433 V_{r}^{p}[[T]], 345 V_{0,r}[[T]], 345 V_{R}, 2 V_{ρ}[[T]], 345 valuations, 606 vanishing at infinity, 19, 32, 100, 592,

611

weak convergence, 86, 273 weak topologies, 287 weak topology, 86 weak^{∗} topology, 82, 273 weakly bounded sets, 87

Z_{+}, 19