Some aspects of analysis on circles, disks, and annuli

Stephen Semmes Rice University

Abstract These informal notes deal with some spaces of power series and Laurent

series over fields with absolute value functions.

Contents

I Preliminaries 4

1 Vector spaces and algebras 4 2 Polynomials and power series 5 3 Laurent polynomials and series 7 4 Some inequalities 10 5 q-Metrics and q-semimetrics 11 6 q-Absolute value functions 13 7 q-Norms and q-seminorms 15 8 Absolute values on k((T)) 16 9 q-Banach spaces and infinite series 18 10 Bounded linear mappings 19 11 Submultiplicative q-seminorms 21 12 Multiplication operators 22 13 Supremum q-seminorms 23 14 More on absolute value functions 24

II Sums and norms 25

15 ℓ^{r} q-Seminorms 25 16 Comparing ℓ^{r} spaces 27 17 Doubly-infinite series 28 18 Sums over sets 31 19 Iterated sums 32 20 Cauchy products on Z_{+} ∪ {0} 34 21 Cauchy products on Z 37 22 Hilbert spaces 41 23 Hilbert space adjoints 43 24 Isometric linear mappings 44 25 C^{∗} Algebras 45 26 Some weighted conditions 47 27 Lebesgue spaces 49

III Power series 51

28 Some spaces of power series 51 29 Submultiplicativity 53 30 Radius of convergence 54 31 A multiplicativity condition 55 32 Some continuity properties 57 33 Functions on Banach algebras 58 34 Holomorphic functions 59 35 Rescaling power series 60 36 Hardy spaces 62 37 Geometric series 63 38 Boundary values 65 39 Some remarks about invertibility 66

IV Laurent series 67

40 Some spaces of Laurent series 68 41 Multiplying Laurent series 70 42 Another multiplicativity condition 72 43 Some subsets of algebras 73 44 Functions on these subsets 74 45 Rescaling Laurent series 76 46 Fourier series 77 47 Pairs of radii 78 48 Products with two radii 80 49 Related sets and functions 81 50 Radii of convergence 83 51 The complex case 84

V Some additional topics 86

58 Hilbert space contractions 96

52 Cesaro means 86 53 Abel sums 88 54 Abel sums and Cauchy products 90 55 Summing Fourier series 92 56 Continuous functions and unitary operators 93 57 Analytic type 94 References 97

Index 98

Part I

Preliminaries

1 Vector spaces and algebras

Let k be a field, let V be a vector space over k, and let X be a nonempty set. The space of all V -valued functions on X will be denoted c(X, V). This is a vector space over k with respect to pointwise addition and scalar multiplication of functions.

The support of a V -valued function f on X is defined to be the set

supp f = {x ∈ X : f(x) ≠ 0}. (1.1)

Let c_{00}(X, V) be the set of V -valued functions f on X whose support has only finitely many elements. It is easy to see that this is a linear subspace of c(X, V). Of course, c_{00}(X, V) is the same as c(X, V) when X has only finitely many elements.

If f ∈ c_{00}(X, V), then the sum

∑_{x∈X} f(x) (1.2)

can be defined as an element of V by reducing to the sum over any nonempty finite subset of X that contains the support of f. The mapping from f to the sum (1.2) is linear as a mapping from c_{00}(X, V) into V.

In particular, we can take V = k, considered as a one-dimensional vector space over itself. If y ∈ X, then let δ_{y}(x) be the k-valued function defined on X by

(1.3)  δ_{y}(x) = 1 when y = x
= 0 when y ≠ x,

where 0 and 1 are the additive and multiplicative identity elements in k. Thus δ_{y} ∈ c_{00}(X, k) for every y ∈ X, and the collection of δ_{y} with y ∈ X is a basis for c_{00}(X, k), as a vector space over k.

To say that A is an (associative) algebra over k means that A is a vector space over k equipped with a binary operation of multiplication which is bilinear over k and satisfies the associative law. If multiplication on A also satisfies the commutative law, then A is said to be a commutative algebra over k.

Let A be an algebra over k, and suppose that A has a multiplicative identity element e, so that e a = a e (1.4)

for every a ∈ A. Of course, e is unique when it exists, and A = {0} when e = 0. If e ≠ 0, then a ∈ A is said to be invertible in A when there is a b ∈ A such that a b = b a = e. (1.5)

In this case, b is unique, and is denoted a^{−1}. If a_{1}, a_{2} are invertible elements of A, then a_{1} a_{2} is invertible in A as well, with inverse a_{2}^{−1} a_{1}^{−1}.

If X is a nonempty set again, then the space c(X, k) of k-valued functions on X is a commutative algebra over k, with respect to pointwise multiplication of functions. Let 1_{X} be the k-valued function on X equal to the multiplicative identity element 1 in k at every x ∈ X. This is the multiplicative identity element in c(X, k).

If V and W are vector spaces over k, then the space L(V, W) of all linear mappings from V into W is a vector space over k with respect to pointwise addition and multiplication. The space L(V) = L(V, V) of all linear mappings from V into itself is an algebra over k, with composition of mappings as multiplication. The identity mapping I = I_{V} on V is the multiplicative identity element in L(V).

If A_{1} and A_{2} are algebras over k, then an (algebra) homomorphism from A_{1} into A_{2} is a linear mapping that preserves multiplication, as usual.

2 Polynomials and power series

Let k be a field, and let T be an indeterminate. As in [2, 5], we normally try to use upper-case letters like T for indeterminates, and lower-case letters for elements of k, or an algebra over k. A formal polynomial in T with coefficients in k can be expressed as

f(T) = ∑_{j=0}^{n} f_{j} T^{j}, (2.1)

where n is a nonnegative integer and f_{j} ∈ k for j = 0, 1, …, n. Similarly, a formal power series in T with coefficients in k can be expressed as

f(T) = ∑_{j=0}^{∞} f_{j} T^{j}, (2.2)

where f_{j} ∈ k for each nonnegative integer j. Thus a formal polynomial may be considered as a formal power series for which all but finitely many coefficients are equal to 0.

The spaces of formal polynomials and power series in T with coefficients in k are denoted k[T] and k[[T]], respectively. Let Z_{+} be the set of positive integers, so that Z_{+} ∪{0} is the set of nonnegative integers. To be more precise, k[[T]] can be identified with the space c(Z_{+} ∪ {0}, k) of all k-valued functions on the set of nonnegative integers, by identifying f(T) ∈ k[[T]] with j ↦ f_{j} as a k-valued function on Z_{+} ∪ {0}. Using this identification, k[T] corresponds to the space c_{00}(Z_{+} ∪{0}, k) of k-valued functions on Z_{+} ∪{0} with finite support.

We can also consider k[[T]] as a vector space over k with respect to termwise addition and scalar multiplication, which corresponds to pointwise addition and scalar multiplication of k-valued functions on Z_{+}∪{0}. Note that k[T] is a linear subspace of k[[T]], just as c_{00}(Z_{+} ∪ {0}, k) is a linear subspace of c(Z_{+} ∪ {0}, k).

Let (2.2) and

g(T) = ∑_{l=0}^{∞} g_{l} T^{l} (2.3)

be formal power series in T with coefficients in k. Their product

f(T) g(T) = h(T) = ∑_{n=0}^{∞} h_{n} T^{n} (2.4)

is defined as a formal power series in T with coefficients in k by putting

h_{n} = ∑_{j=0}^{n} f_{j} g_{n−j} (2.5)

for each nonnegative integer n. Of course, this corresponds exactly to multiplying f(T) and g(T) formally, and collecting terms with the same power of T. It is well known and not difficult to check that k[[T]] is a commutative algebra over k with respect to this definition of multiplication. It is easy to see that k[T] is a subalgebra of k[[T]], because h_{n} = 0 for all but finitely many n ≥ 0 when f_{j} = 0 for all but finitely many j ≥ 0 and g_{l} = 0 for all but finitely many l ≥ 0.

We can also identify each element of k with the “constant polynomial” such that the coefficient of T^{j} is the given element of k when j = 0, and is equal to 0 when j ≥ 1. It is easy to see that k corresponds to a subalgebra of k[T] in this way. The constant polynomial corresponding to the multiplicative identity element 1 in k is the multiplicative identity element in k[[T]].

If a(T) ∈ k[[T]], then

∑_{l=0}^{∞} a(T)^{l} T^{l} (2.6)

can be defined as an element of k[[T]], as follows. If l ∈ Z_{+}, then a(T)^{l} is defined as an element of k[[T]] using multiplication on k[[T]], as before. If l = 0, then a(T)^{l} is interpreted as being the constant polynomial corresponding to 1, as usual. Thus

∑_{l=0}^{n} a(T)^{l} T^{l} (2.7)

is defined as an element of k[[T]] for every nonnegative integer n. If j is any nonnegative integer, then the coefficient of T^{j} in a(T)^{l} T^{l} is equal to 0 when l > j. This implies that the coefficient of T^{j} in (2.7) does not depend on n when n ≥ j. In order to define (2.6) as a formal power series in T, we take the coefficient of T^{j} in (2.6) to be the same as the coefficient of T^{j} in (2.7) when n ≥ j.

Observe that

(1 − a(T) T) ∑_{l=0}^{n} a(T)^{l} T^{l} = 1 − a(T)^{n+1} T^{n+1} (2.8)

for each nonnegative integer n, by a standard computation. Using this, one can check that

(1 − a(T) T) ∑_{l=0}^{∞} a(T)^{l} T^{l} = 1, (2.9)

because the coefficient of T^{j} in the left side of (2.9) is the same as the coefficient of j in the left side of (2.8) when n ≥ j. This shows that 1 − a(T) T has a mutliplicative inverse in k[[T]], namely

(1 − a(T) T)^{−1} = ∑_{l=0}^{∞} a(T)^{l} T^{l}. (2.10)

If f(T) ∈ k[[T]] is as in (2.2) and f_{0} ≠ 0, then it is easy to see that f(T) has a multiplicative inverse in k[[T]], by reducing to the case where f_{0} = 1 and applying the remarks in the preceding paragraph. Conversely, if f(T) has a multiplicative inverse in k[[T]], then f_{0} ≠ 0. This follows from the fact that

f(T) ↦ f_{0} (2.11)

is an algebra homomorphism from k[[T]] onto k.

Let A be an algebra over k, and suppose that A has a multiplicative identity element e. If a ∈ A and f(T) ∈ k[T] is as in (2.1), then f(a) can be defined as an element of A by

f(a) = ∑_{j=0}^{n} f_{j} a^{j}, (2.12)

where a^{j} is interpreted as being equal to e when j = 0, as usual. One can verify that the mapping f(T) ↦ f(a) defines an algebra homomorphism from k[T] into A.

3 Laurent polynomials and series

Let k be a field again, and let T be an indeterminate. A formal Laurent polynomial in T with coefficients in k can be expressed as

f(T) = ∑_{j=−n}^{n} f_{j} T^{j}, (3.1)

where n is a nonnegative integer, and f_{j} ∈ k for each integer j with |j| ≤ n. A formal Laurent series in T with coefficients in k can be expressed as

f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j}, (3.2)

where f_{j} ∈ k for each integer j. Thus a formal Laurent polynomial may be considered as a formal Laurent series for which all but finitely many coefficients are equal to 0. Formal polynomials and power series in T may be considered as formal Laurent polynomials and series, respectively, for which the coefficients of T^{j} are equal to 0 when j < 0.

The space of formal Laurent series in T with coefficients in k can be identified with the space c(Z, k) of all k-valued functions on the set Z of integers, by identifying (3.2) with j ↦ f_{j} as a k-valued function on Z. Similarly, the space of formal Laurent polynomials in T with coefficients in k can be identified with the space c_{00}(Z, k) of all k-valued functions on Z with finite support. The space of formal Laurent series in T with coefficients in k is a vector space over k with respect to termwise addition and scalar multiplication, which corresponds to pointwise addition and scalar multiplication of k-valued functions on Z. The space of formal Laurent polynomials in T with coefficients in k is a linear subspace of the space of formal Laurent series in T with coefficients in k, just as c_{00}(Z, k) is a linear subspace of c(Z, k).

Put c^{+}(Z, k) = {f ∈ c(Z, k) : supp f ⊆ Z_{+} ∪ {0}}, (3.3) which is a linear subspace of c(Z, k). We can identify c(Z_{+} ∪ {0}, k) with (3.3), by identifying a k-valued function on Z_{+} ∪ {0} with the element of (3.3) whose restriction to Z_{+} ∪ {0} is the given function. This corresponds to identifying formal power series in T with coefficients in k with formal Laurent series in T with coefficients in k for which the coefficients of T^{j} are equal to 0 when j < 0, as before. In particular, k[[T]] corresponds to a linear subspace of the space of formal Laurent series in T with coefficients in k.

Similarly, put

(3.4) c_{00}^{+}(Z, k) = {f ∈ c_{00}(Z, k) : supp f ⊆ Z_{+} ∪ {0}} = c_{00}(Z, k) ∩ c^{+}(Z, k),

which is a linear subspace of c_{00}(Z, k). Using the identification of c(Z_{+} ∪{0}, k) with (3.3) mentioned in the preceding paragraph, we get an identification of c_{00}(Z_{+} ∪ {0}, k) with (3.4). This corresponds to identifying formal polynomials in T with coefficients in k with formal Laurent polynomials in T with coefficients in k for which the coefficients of T^{j} are equal to 0 when j < 0. Thus k[T] corresponds to a linear subspace of the space of formal Laurent polynomials in T with coefficients in k, which is the intersection of k[[T]] with the space of formal Laurent polynomials in T with coefficients in k.

Let f(T) be a formal Laurent series in T with coefficients in k, as in (3.2). If f_{j} = 0 for all but finitely many j < 0, then we may express f(T) as

f(T) = ∑_{j>>−∞} f_{j} T^{j}, (3.5)

as in [2]. The space of formal Laurent series in T with coefficients in k of this type is denoted k((T)). This is a linear subspace of the space of formal Laurent series in T with coefficients in k, which corresponds to the linear subspace of c(Z, k) consisting of k-valued functions on Z that are equal to 0 at all but finitely many negative integers. Note that k[[T]] and the space of formal Laurent polynomials in T with coefficients in k are linear subspaces of k((T)).

Let (3.5) and

g(T) = ∑_{l>>−∞} g_{l} T^{l} (3.6)

be elements of k((T)). Their product

f(T) g(T) = h(T) = ∑_{n=−∞}^{∞} h_{n} T^{n} (3.7)

is defined as a formal Laurent series in T with coefficients in k by putting

h_{n} = ∑_{j=−∞}^{∞} f_{j} g_{n−j} (3.8)

for each n ∈ Z. More precisely, one can check that all but finitely many terms in the sum on the right side of (3.8) are equal to 0 in this situation, so that the sum defines an element of k. One can also verify that h_{n} = 0 for all but finitely many n < 0, so that (3.7) is an element of k((T)) too. Similarly, if f(T) and g(T) are formal Laurent polynomials in T, then (3.7) is a formal Laurent polynomial in T.

As before, one can check that k((T)) is a commutative algebra over k with respect to this definition of multiplication. The space of formal Laurent polynomials in T with coefficients in k is a subalgebra of k((T)). The product (3.7) reduces to the one defined in the previous section when f(T) and g(T) are formal power series in T, so that k[[T]] corresponds to a subalgebra of k((T)) as well. Each element of k can be identified with the “constant” formal Laurent polynomial for which the coefficient of T^{j} is equal to the given element of k when j = 0, and is equal to 0 when j ≠ 0. This identifies k with a subalgebra of k((T)), which corresponds to the subalgebra of k[T] mentioned in the previous section.

In particular, the constant formal Laurent series corresponding to the multiplicative identity element 1 in k is the multiplicative identity element in k((T)). It is easy to see that every nonzero element f(T) of k((T)) has a multiplicative inverse in k((T)), using the description of multiplicative inverses in k[[T]] in the previous section. Thus k((T)) is a field, which contains a copy of k.

Let A be an algebra over k with a multiplicative identity element e, and let a be an invertible element of A. If f(T) is a formal Laurent polynomial in T with coefficients in k, as in (3.1), then

f(a) = ∑_{j=−n}^{n} f_{j} a^{j} (3.9)

defines an element of A. Of course, this reduces to (2.12) when f(T) ∈ k[T]. One can check that the mapping f(T) ↦ f(a) defines a homomorphism from the algebra of formal Laurent polynomials in T with coefficients in k into A, as before.

4 Some inequalities

Let X be a nonempty set, and let f be a nonnegative real-valued function on X with finite support. Put

∥f∥_{r} = ( ∑_{x∈X} f(x)^{r})^{1/r} (4.1)

for every positive real number r, and

∥f∥_{∞} = max_{x∈X} f(x). (4.2)

Observe that ∥f∥_{∞} ≤ ∥f∥_{r} (4.3) for every 0 < r < ∞. If 0 < r_{1} ≤ r_{2} < ∞, then

∥f∥_{r_{2}}^{r_{2}} = ∑_{x∈X} f(x)^{r_{2}} ≤ ∥f∥_{∞}^{r_{2}−r_{1}} ∑_{x∈X} f(x)^{r_{1}} = ∥f∥_{∞}^{r_{2}−r_{1}} ∥f∥_{r_{1}}^{r_{1}}. (4.4)

This implies that ∥f∥_{r_{2}} ≤ ∥f∥_{∞}^{1−(r_{1}/r_{2})} ∥f∥_{r_{1}}^{r_{1}/r_{2}} ≤ ∥f∥_{r_{1}}, (4.5) using (4.3) in the second step. If 0 < r < ∞, then it is easy to see that ∥f∥_{r} ≤ (# supp f)^{1/r} ∥f∥_{∞}, (4.6) where # supp f is the number of elements of the support of f. It follows that

lim_{r→∞} ∥f∥_{r} = ∥f∥_{∞}, (4.7)

using (4.2) and (4.6).

If a and b are nonnegative real numbers, then max(a, b) ≤ (a^{r} + b^{r})^{1/r} ≤ 2^{1/r} max(a, b). (4.8) This corresponds to (4.3) and (4.6), where the support of f has at most two elements. Hence

lim_{r→∞}(a^{r} + b^{r})^{1/r} = max(a, b), (4.9)

as in (4.7). Similarly, if 0 < r_{1} ≤ r_{2} < ∞, then (a^{r_{2}} + b^{r_{2}})^{1/r_{2}} ≤ (a^{r_{1}} + b^{r_{1}})^{1/r_{1}}, (4.10) by (4.5). This implies that (a + b)^{r} ≤ a^{r} + b^{r} (4.11) when 0 < r ≤ 1, by taking r_{1} = r and r_{2} = 1 in (4.10).

Let f, g be nonnegative real-valued functions on X with finite support. If 0 < r ≤ 1, then

∥f + g∥_{r}^{r} = ∑_{x∈X}(f(x) + g(x))^{r} ≤ ∑_{x∈X} f(x)^{r} + ∑_{x∈X} g(x)^{r} = ∥f∥_{r}^{r} + ∥g∥_{r}^{r}. (4.12)

This uses (4.11) in the second step. If 1 ≤ r ≤ ∞, then it is well known that ∥f + g∥_{r} ≤ ∥f∥_{r} + ∥g∥_{r}. (4.13) This is Minkowski’s inequality for finite sums. Of course, (4.12) and (4.13) are the same when r = 1, in which case the inequalities are clearly equalities. It is also easy to verify (4.13) directly when r = ∞.

5 q-Metrics and q-semimetrics

Let X be a set, and let q be a positive real number. A nonnegative real-valued function d(x, y) defined for x, y ∈ X is said to be a q-semimetric on X if it satisfies the following three conditions. First, (5.1) d(x, x) = 0 for every x ∈ X. Second, (5.2) d(x, y) = d(y, x) for every x, y ∈ X. Third, (5.3) d(x, z)^{q} ≤ d(x, y)^{q} + d(y, z)^{q} for every x, y, z ∈ X. If we also have that (5.4) d(x, y) > 0 for every x, y ∈ X with x ≠ y, then d(x, y) is said to be a q-metric on X. A q-metric or q-semimetric with q = 1 is also known simply as a metric or semimetric, as appropriate.

A nonnegative real-valued function d(x, y) defined for x, y ∈ X is said to be a semi-ultrametric on X if it satisfies (5.1), (5.2), and (5.5) d(x, z) ≤ max(d(x, y), d(y, z)) for every x, y, z ∈ X. If (5.4) also holds, then d(x, y) is said to be an ultrametric on X.

Observe that (5.3) is equivalent to saying that (5.6) d(x, z) ≤ (d(x, y)^{q} + d(y, z)^{q})^{1/q} for every x, y ∈ X.

The right side of of the inequality in (5.5) is the same as the limit of the right side of the inequality in (5.6) as q → ∞, as in (4.9). Thus we may consider ultrametrics and semi-ultrametrics as q-metrics and q-semimetrics with q = ∞, respectively.

The right side of the inequality in (5.6) decreases monotonically in q, as in (4.10). This includes the right side of the inequality in (5.5) as the analogue of the right side of the inequality in (5.6) with q = ∞, by the first inequality in (4.8). If 0 < q_{1} ≤ q_{2} ≤ ∞ and d(x, y) is a q_{2}-semimetric on X, then it follows that d(x, y) is a q_{1}-semimetric on X as well. Of course, this implies the analogous statement for q-metrics.

The discrete metric is defined on X by putting d(x, y) equal to 1 when x ≠ y, and to 0 when x = y, as usual. It is easy to see that the discrete metric is an ultrametric on X.

Let d(x, y) be a nonnegative real-valued function defined for x, y ∈ X again, and let a be a positive real number. It is easy to see that d(x, y) is a q-semimetric on X for some q > 0 if and only if

d(x, y)^{a} (5.7)

is a (q/a)-semimetric on X. Similarly, d(x, y) is a q-metric on X if and only if (5.7) is a (q/a)-metric on X. This includes the q = ∞ case, with q/a interpreted as being ∞ too.

Let d(x, y) be a q-semimetric on a set X for some q > 0. The open ball in X centered at a point x ∈ X and with radius r > 0 is defined as usual by

B(x, r) = B_{d}(x, r) = {y ∈ X : d(x, y) < r}. (5.8)

Similarly, the closed ball in X centered at x ∈ X with radius r ≥ 0 is defined by \overline{B}(x, r) = \overline{B}_{d}(x, r) = \{y ∈ X : d(x, y) ≤ r\}. (5.9)

If a is a positive real number, then

B_{d^{a}}(x, r^{a}) = B_{d}(x, r) (5.10)

for every x ∈ X and r > 0, and

\overline{B}_{d^{a}}(x, r^{a}) = \overline{B}_{d}(x, r) (5.11)

for every x ∈ X and r ≥ 0.

The topology determined on X by d(·, ·) is defined as usual by saying that U ⊆ X is an open set if for every x ∈ U there is an r > 0 such that B_{d}(x, r) ⊆ U. It is easy to see that an open set with respect to d(·, ·) is the same as an open set with respect to (5.7) for any a > 0, because of (5.11). In particular, one can use this to reduce to the case of ordinary semimetrics, by taking a = q when q < 1. One can check that this collection of open sets defines a topology on X, for which open balls are open sets, and closed balls are closed sets. If d(·, ·) is a q-metric on X, then X is Hausdorff with respect to this topology.

If d(·, ·) is a semi-ultrametric on X, then one can check that open balls are closed sets, and that closed balls in X of positive radius are open sets.

One can define uniform continuity of mappings with respect to q-semimetrics and q-metrics in the same way as for ordinary semimetrics and metrics, and with possibly different q’s for the domain and range. Note that uniform continuity is preserved when the q-semimetrics or q-metrics on the domain or range are replaced with positive powers of themselves, as in (5.7).

Similarly, Cauchy sequences and completeness can be defined with respect to a q-metric, in the same way as for an ordinary metric. These are also preserved when the q-metric is replaced with a positive power of itself, as in (5.7).

6 q-Absolute value functions

Let k be a field, and let q be a positive real number. A nonnegative real-valued function | · | defined on k is said to be a q-absolute value function if it satisfies the following three conditions. First,

(6.1) |x| = 0 if and only if x = 0.

Second, (6.2) |x y| = |x| |y| for every x, y ∈ k.

Third, (6.3) |x + y|^{q} ≤ |x|^{q} + |y|^{q} for every x, y ∈ k.

If these conditions hold with q = 1, then | · | is said to be an absolute value function on k. The standard absolute value functions on the fields R and C of real and complex numbers are absolute value functions in this sense.

If a nonnegative real-valued function | · | on a field satisfies (6.1) and (6.2), then it is easy to see that |1| = 1, where the first 1 is the multiplicative identity element in k, and the second 1 is the multiplicative identity element in R. This uses the fact that 1^{2} = 1 in k, so that 0 < |1| = |1|^{2}. Similarly, if x ∈ k satisfies x^{n} = 1 for some positive integer n, then |x| = 1, because |x|^{n} = |x^{n}| = 1. In particular, | − 1| = 1, where −1 is the additive inverse of 1 in k, because (−1)^{2} = 1.

If | · | is a q-absolute value function on k for some q > 0, then

d(x, y) = |x − y| (6.4)

defines a q-metric on k. This uses the fact that | − 1| = 1 to get that (6.4) is symmetric in x and y.

A nonnegative real-valued function | · | defined on a field k is said to be an ultrametric absolute value function on k if it satisfies (6.1), (6.2), and

(6.5) |x + y| ≤ max(|x|, |y|) for every x, y ∈ k.

In this case, (6.4) defines an ultrametric on k. The trivial absolute value function is defined on any field k by putting |0| = 0 and |x| = 1 for every x ∈ k with x ≠ 0.

This is an ultrametric absolute value function on k, for which the corresponding ultrametric is the discrete metric.

Note that (6.3) is equivalent to asking that

(6.6) |x + y| ≤ (|x|^{q} + |y|^{q})^{1/q} for every x, y ∈ k.

The right side of this inequality tends to the right side of the inequality in (6.5) as q → ∞, as in (4.9). An ultrametric absolute value function on k may also be considered as a q-absolute value function with q = ∞. If 0 < q_{1} ≤ q_{2} ≤ ∞ and | · | is a q_{2}-absolute value function on k, then one can check that | · | is a q_{1}-absolute value function on k, using (4.8) and (4.10).

If p is a prime number, then the p-adic absolute value |x|_{p} of a rational number x is defined as follows. Of course, we put |0|_{p} = 0. Otherwise, if x ≠ 0, then x can be expressed as p^{j} (a/b), where a, b, and j are integers, a, b ≠ 0, and neither a nor b is an integer multiple of p, and we put

|x|_{p} = p^{−j}. (6.7)

One can check that this defines an ultrametric absolute value function on the field Q of rational numbers. The corresponding ultrametric

d_{p}(x, y) = |x − y|_{p} (6.8)

is known as the p-adic metric on Q.

Let | · | be a q-absolute value function on a field k for some q > 0. If k is not already complete with respect to the associated q-metric (6.4), then one can pass to a completion, using standard arguments. The completion is also a field with a q-absolute value function, which contains k as a dense subfield. The completion is unique up to suitable isomorphic equivalence.

If p is a prime number, then the field Q_{p} of p-adic numbers is obtained by completing Q with respect to the p-adic metric, as in the preceding paragraph. The natural extensions of the p-adic absolute value and metric to Q_{p} are denoted in the same way as on Q.

Let | · | be a nonnegative real-valued function on a field k again, and let a be a positive real number. One can check that | · | is a q-absolute value function on k for some q > 0 if and only if

| · |^{a} (6.9)

is a (q/a)-absolute value function. In this case, the (q/a)-metric associated to (6.9) is the same as the ath power of the q-metric (6.4) associated to | · |. In particular, these (q/a) and q-metrics determine the same topology on k, as in the previous section.

Let | · |_{1} and | · |_{2} be q_{1} and q_{2}-absolute value functions on k, for some q_{1}, q_{2} > 0. If there is a positive real number a such that

|x|_{2} = |x|_{1}^{a} (6.10) for every x ∈ k, then | · |_{1} and | · |_{2} are said to be equivalent on k. This implies that the topologies determined on k by the associated q_{1} and q_{2}-metrics are the same, as in the preceding paragraph. Conversely, it is well known that | · |_{1} and | · |_{2} are equivalent on k when the topologies determined on k by the associated q_{1} and q_{2}-metrics are the same. Note that one can reduce to the case where q_{1} = q_{2} = 1, by replacing | · |_{1} with | · |_{1}^{q_{1}} when q_{1} ≤ 1, and similarly for | · |_{2}.

7 q-Norms and q-seminorms

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k. A nonnegative real-valued function N on V is said to be a q-seminorm on V with respect to | · | on k for some positive real number q if it satisfies the following two conditions. First,

(7.1) N(t v) = |t| N(v) for every t ∈ k and v ∈ V.

Second, (7.2) N(v + w)^{q} ≤ N(v)^{q} + N(w)^{q} for every v, w ∈ V.

Note that (7.1) implies that N(0) = 0, by taking t = 0. If N also satisfies

(7.3) N(v) > 0 for every v ∈ V with v ≠ 0,

then N is said to be a q-norm on V with respect to | · | on k. As usual, we may simply use the terms norm and seminorm when q = 1.

If a nonnegative real-valued function N on V satisfies (7.1) and

(7.4) N(v + w) ≤ max(N(v), N(w)) for every v, w ∈ V,

then N is said to be a semi-ultranorm on V with respect to | · | on k. If N also satisfies (7.3), then N is said to be an ultranorm on V with respect to | · | on k.

As before, (7.2) is equivalent to saying that

(7.5) N(v + w) ≤ (N(v)^{q} + N(w)^{q})^{1/q} for every v, w ∈ V.

An ultranorm or semi-ultranorm on V may be considered as a q-norm or qseminorm, as appropriate, with q = ∞, because the right side of the inequality in (7.5) tends to the right side of the inequality in (7.4) as q → ∞, by (4.9). If 0 < q_{1} ≤ q_{2} ≤ ∞ and N is a q_{2}-norm or q_{2}-seminorm on V with respect to | · | on k, then N is a q_{1}-norm or q_{1}-seminorm on V too, as appropriate, by (4.8) and (4.10).

If N is a q-norm or q-seminorm on V with respect to |·| on k for some q > 0, then d(v, w) = d_{N}(v, w) = N(v − w) (7.6)

defines a q-metric or q-semimetric on V, as appropriate. Note that | · | may be considered as a q_{k}-norm on k with respect to itself, and where k is considered as a one-dimensional vector space over itself.

Let a be a positive real number, and remember that | · |^{a} defines a (q_{k}/a)- absolute value function on k, as in the previous section. One can check that a nonnegative real-valued function N on V is a q-norm or q-seminorm with respect to | · | on k if and only if

N(v)^{a} (7.7)

is a (q/a)-norm or (q/a)-seminorm on V with respect to |·|^{a} on k, as appropriate. Of course, d_{N^{a}}(v, w) = N(v − w)^{a} = d_{N}(v, w)^{a} (7.8)

for every v, w ∈ V in this situation.

Suppose for the moment that | · | is the trivial absolute value function on k. In this case, the trivial ultranorm is defined on V by putting N(0) = 0 and N(v) = 1 for every v ∈ V with v ≠ 0. It is easy to see that this defines an ultranorm on V, for which the associated ultrametric as in (7.6) is the discrete metric on V.

Let | · | be any q_{k}-absolute value function on k again, and suppose that N is a q-seminorm on V with respect to | · | on k for some q > 0. If N(v) > 0 for some v ∈ V, then one can check that | · | is a q-absolute value function on k.

8 Absolute values on k((T))

Let k be a field, let T be an indeterminate, and remember that k((T)) denotes the space of formal Laurent series in T with coefficients in k such that the coefficient of T^{j} is equal to 0 for all but finitely many j < 0, as in Section 3. If

f(T) = ∑_{j>>−∞} f_{j} T^{j} (8.1)

is a nonzero element of k((T)), then let j_{0}(f(T)) be the unique integer such that f_{j_{0}(f(T))} ≠ 0 and f_{j} = 0 when j < j_{0}(f(T)). Otherwise, let us take j_{0}(f(T)) = ∞ when f(T) = 0. If

g(T) = ∑_{j>>−∞} g_{j} T^{j} (8.2)

is another element of k((T)), then it is easy to see that

j_{0}(f(T) + g(T)) ≥ min(j_{0}(f(T)), j_{0}(g(T))) (8.3)

and j_{0}(f(T) g(T)) = j_{0}(f(T)) + j_{0}(g(T)), (8.4)

with suitable interpretations when any of these terms is ∞.

Let r be a positive real number with r ≤ 1. If f(T) ∈ k((T)), then put

|f(T)|_{r} = r^{−j_{0}(f(T))} (8.5) when f(T) ≠ 0, and |0|_{r} = 0. One can check that | · |_{r} defines an ultrametric absolute value function on the field k((T)), using (8.3) and (8.4). If r = 1, then |·|_{r} is the same as the trivial absolute value function on k((T)). If a is a positive real number, then 0 < r^{a} ≤ 1, and

|f(T)|_{r^{a}} = |f(T)|_{r}^{a} (8.6)

for every f(T) ∈ k((T)).

Let us suppose from now on in this section that r < 1. If l ∈ Z, then f(T) ∈ k((T)) satisfies |f(T)|_{r} ≤ r^{−l} (8.7)

if and only if j_{0}(f(T)) ≥ l, which is the same as saying that

f_{j} = 0 when j < l. (8.8)

Let T^{l} k[[T]] be the collection of formal Laurent series in T with coefficients in k that can be expressed as T^{l} times a formal power series in T with coefficients in k. This is the same as the collection of f(T) ∈ k((T)) that satisfy (8.7), or equivalently (8.8).

A subset of k((T)) is bounded with respect to |·|_{r} when there is a finite upper bound for |f(T)|_{r} for elements f(T) of the set. This is the same as saying that the set is contained in T^{l_{0}} k[[T]] for some l_{0} ∈ Z, by the remarks in the preceding paragraph. A sequence of elements of k((T)) is said to be bounded with respect to |·|_{r} when the set of terms of the sequence is bounded with respect to |·|_{r}. If a sequence of elements of k((T)) converges to an element of k((T)) with respect to the ultrametric associated to | · |_{r}, then that sequence is bounded with respect to | · |_{r}, by standard arguments. Similarly, Cauchy sequences in k((T)) with respect to the ultrametric associated to | · |_{r} are bounded with respect to | · |_{r}.

As before, f(T), g(T) ∈ k((T)) satisfy

|f(T) − g(T)|_{r} ≤ r^{−l} (8.9)

for some l ∈ Z if and only if

f_{j} = g_{j} when j < l. (8.10)

Given a sequence of elements of k((T)), we get for each j ∈ Z a corresponding sequence of coefficients of T^{j} in k. Let us say that the sequence is termwise eventually constant if for each j ∈ Z, the corresponding sequence of coefficients of T^{j} in k is eventually constant. Similarly, let us say that the sequence converges termwise to f(T) ∈ k((T)) if for each j ∈ Z, the terms of the corresponding sequence of coefficients of T^{j} in k are eventually equal to the coefficient f_{j} or T^{j} in f(T).

If a sequence in k((T)) is a Cauchy sequence with respect to the ultrametric associated to | · |_{r}, then the sequence is termwise eventually constant. If a sequence in k((T)) converges to f(T) ∈ k((T)) with respect to the ultrametric associated to |·|_{r}, then the sequence converges termwise to f(T). In both cases, the converse holds when the sequence is bounded with respect to | · |_{r}.

If a sequence of elements of k((T)) is termwise eventually constant, then the eventual constant values of the coefficients of T^{j} for each j ∈ Z can be used to define a formal Laurent series f(T) in T with coefficients in k. If the sequence in k((T)) is also bounded with respect to | · |_{r}, then f(T) ∈ k((T)). It follows that k((T)) is complete with respect to the ultrametric associated to | · |_{r}.

9 q-Banach spaces and infinite series

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k with a q-norm N with respect to |·| on k for some q > 0. If V is complete with respect to the q-metric associated to N, then V is said to be a q-Banach space with respect to N. If q = 1, then we may simply say that V is a Banach space. If V is not complete, then one can pass to a completion to get a q-Banach space, using standard arguments. The completion is unique up to suitable isometric linear equivalence.

As usual, an infinite series

∑_{j=1}^{∞} v_{j} (9.1)

with terms in V is said to converge with respect to N if the corresponding sequence of partial sums

∑_{j=1}^{n} v_{j} (9.2)

converges to an element of V with respect to the q-metric associated to N. In this case, the value of the sum (9.1) is defined to be the limit of the partial sums (9.2). If (9.1) converges and t ∈ k, then ∑_{j=1}^{∞} t v_{j} converges in V too, with

∑_{j=1}^{∞} t v_{j} = t ∑_{j=1}^{∞} v_{j}, (9.3)

by standard arguments. Similarly, if ∑_{j=1}^{∞} w_{j} is another convergent series with terms in V, then ∑_{j=1}^{∞}(v_{j} + w_{j}) converges in V as well, with

∑_{j=1}^{∞}(v_{j} + w_{j}) = ∑_{j=1}^{∞} v_{j} + ∑_{j=1}^{∞} w_{j}. (9.4)

The sequence (9.2) of partial sums is a Cauchy sequence in V with respect to the q-metric associated to N if and only if for every ϵ > 0 there is a positive integer L such that

N( ∑_{j=l}^{n} v_{j}) < ϵ (9.5)

for every n ≥ l ≥ L. In particular, this implies that

lim_{j→∞} N(v_{j}) = 0, (9.6)

by taking l = n in (9.5). Of course, if (9.1) converges in V, then the sequence of partial sums (9.2) is a Cauchy sequence with respect to the q-metric associated to N, because convergent sequences are Cauchy sequences.

Suppose for the moment that q < ∞. An infinite series (9.1) with terms in V is said to converge q-absolutely with respect to N if

∑_{j=1}^{∞} N(v_{j})^{q} (9.7)

converges as an infinite series of nonnegative real numbers. We may refer to this as absolute convergence with respect to N when q = 1. Observe that

N( ∑_{j=l}^{n} v_{j})^{q} ≤ ∑_{j=l}^{n} N(v_{j})^{q} (9.8)

for every n ≥ l ≥ 1, by the q-norm version of the triangle inequality. If (9.1) converges q-absolutely with respect to N, then one can use (9.8) to verify that the Cauchy condition (9.5) holds. If V is a q-Banach space with respect to N, then it follows that (9.1) converges in V. One can also check that

N( ∑_{j=1}^{∞} v_{j})^{q} ≤ ∑_{j=1}^{∞} N(v_{j})^{q} (9.9)

under these conditions, using (9.8).

Suppose now that q = ∞, so that N is an ultranorm on V. Thus

N( ∑_{j=l}^{n} v_{j}) ≤ max_{l≤j≤n} N(v_{j}) (9.10)

for every n ≥ l ≥ 1, by the ultranorm version of the triangle inequality. If (9.6) holds, then it is easy to see that the Cauchy condition (9.5) holds too, using (9.10). This implies that (9.1) converges in V when V is a Banach space with respect to N. In this situation, one can verify that

N( ∑_{j=1}^{∞} v_{j}) ≤ max_{j≥1} N(v_{j}), (9.11)

using (9.10). Note that the maximum on the right side of (9.11) is automatically attained, because of (9.6). This is trivial when v_{j} = 0 for every j ≥ 1, and otherwise the maximum can be reduced to finitely many terms.

10 Bounded linear mappings

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. Also let N_{V}, N_{W} be q_{V}, q_{W} -seminorms on V, W, respectively, for some q_{V} , q_{W} > 0, and with respect to | · | on k. 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) (10.1)

for every v ∈ V. This implies that

N_{W} (T (u) − T (v)) = N_{W} (T (u − v)) ≤ C N_{V} (u − v) (10.2)

for every u, v ∈ V, so that T is uniformly continuous with respect to the q_{V}, q_{W} -semimetrics associated to N_{V}, N_{W} on V, W, respectively.

Let BL(V, W) be the space of bounded linear mappings from V into W with respect to N_{V}, N_{W}. If T ∈ BL(V, W), then put

(10.3) ∥T ∥_{op} = ∥T ∥_{op,V W} = inf{C ≥ 0 : (10.1) holds},

where more precisely the infimum is taken over all nonnegative real numbers C for which (10.1) holds. It is easy to see that (10.1) holds with C = ∥T ∥_{op}, so that the infimum is automatically attained. One can check that BL(V, W) is a vector space over k with respect to pointwise addition and scalar multiplication, and that (10.3) is a q_{W} -seminorm on BL(V, W) with respect to | · | on k.

Let Z be another vector space over k, and let N_{Z} be a q_{Z}-seminorm on Z with respect to | · | on k, for some q_{Z} > 0. If T_{1} is a bounded linear mapping from V into W with respect to N_{V}, N_{W}, and T_{2} is a bounded linear mapping from W into Z with respect to N_{W}, N_{Z}, then their composition T_{2} ∘ T_{1} is a bounded linear mapping from V into Z with respect to N_{V}, N_{Z}. More precisely, if v ∈ V, then

N_{Z}((T_{2} ∘ T_{1})(v)) = N_{Z}(T_{2}(T_{1}(v))) ≤ ∥T_{2}∥_{op,W Z} N_{W} (T_{1}(v))
(10.4)  ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z} N_{V} (v).

This implies that T_{2} ∘ T_{1} is bounded, with

∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}. (10.5)

Let us suppose from now on in this section that N_{W} is a q_{W} -norm on W. This implies that (10.3) is a q_{W} -norm on BL(V, W). If W is also complete with respect to the q_{W} -metric associated to N_{W}, then one can check that BL(V, W) is complete with respect to the q_{W} -metric associated to (10.3), using standard arguments.

Suppose that N_{V} is a q_{V} -norm on V, and let V_{0} be a linear subspace of V that is dense in V with respect to the q_{V} -metric associated to N_{V}. Note that the restriction of N_{V} to V_{0} defines a q_{V} -norm on V_{0} with respect to | · | on k. Let T_{0} be a bounded linear mapping from V_{0} into W, with respect to the restriction of N_{V} to V_{0}, and N_{W} on W. In particular, T_{0} is uniformly continuous with respect to the restriction of the q_{V} -metric associated to N_{V} to V_{0}, and the q_{W} -metric associated to N_{W} on W. If W is complete with respect to the q_{W} -metric associated to N_{W}, then there is a unique extension of T_{0} to a uniformly continuous mapping from V into W. This follows from a well-known extension theorem for uniformly continuous mappings between metric spaces, which is easily extended to q-metric spaces. In this situation, the extension is a bounded linear mapping from V into W, with the same operator q_{W} -norm as T_{0} has on V_{0}.

11 Submultiplicative q-seminorms

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let A be an algebra over k. Also let N be a q-seminorm on A with respect to | · | on k for some q > 0. If N(x y) ≤ N(x) N(y) (11.1)

for every x, y ∈ A, then N is said to be submultiplicative on A. Similarly, N is said to be multiplicative on A if

N(x y) = N(x) N(y) (11.2)

for every x, y ∈ A.

Suppose that A has a multiplicative identity element e. If N is submultiplicative on A, then N(x) ≤ N(e) N(x) for every x ∈ A. If N(x) > 0 for some x ∈ A, then it follows that N(e) ≥ 1. (11.3)

Similarly, if N is multiplicative on A, then N(x) = N(e) N(x) for every x ∈ A. If N(x) > 0 for some x ∈ A, then we get that

N(e) = 1. (11.4)

Suppose that N is a submultiplicative q-norm on A with respect to | · | on k for some q > 0. If A is complete with respect to the q-metric associated to N, then A is said to be a q-Banach algebra with respect to N. As usual, one may simply say that A is a Banach algebra when q = 1. If A is not complete, then one can pass to a completion, by standard arguments, which is unique up to a suitable isometric isomorphic equivalence. Sometimes the requirement that A have a multiplicative identity element e with N(e) = 1 is included in the definition of a q-Banach algebra.

Let V be a vector space over k, and let N_{V} be a q_{V} -seminorm on V with respect to | · | on k. Consider the space BL(V) = BL(V, V) of bounded linear mappings from V into itself, using N_{V} on the domain and range. This is an algebra over k, with composition of mappings as multiplication. The corresponding operator q_{V} -seminorm ∥ · ∥_{op} = ∥ · ∥_{op,V V} defined in the previous section is submultiplicative on BL(V), as in (10.5). The identity mapping I = I_{V} on V is bounded with respect to N_{V}, with

∥I∥_{op} = 1 (11.5) when N(v) > 0 for some v ∈ V, and ∥I∥_{op} = 0 otherwise. Of course, I is the multiplicative identity element in BL(V). If N_{V} is a q_{V} -norm on V, and V is complete with respect to the q_{V} -metric associated to N_{V}, then ∥ · ∥_{op} is a q_{V} -norm on BL(V), and BL(V) is complete with respect to the associated q_{V} -metric, as in the previous section. This means that BL(V) is a q_{V} -Banach algebra with respect to ∥ · ∥_{op}, at least if V ≠ {0}, so that (11.5) holds, if that is included in the definition of a q-Banach algebra.

12 Multiplication operators

Let k be a field, and let A be an algebra over k. If a ∈ A, then put

M_{a}(x) = a x (12.1)

for every x ∈ A. Thus M_{a} is a linear mapping from A into itself, where A is considered simply as a vector space over k. If b ∈ A too, then

(M_{a} ∘ M_{b})(x) = M_{a}(M_{b}(x)) = M_{a}(b x) = a b x = M_{a b}(x) (12.2)

for every x ∈ A. This means that

M_{a} ∘ M_{b} = M_{a b} (12.3)

as linear mappings from A into itself. It is easy to see that

a ↦ M_{a} (12.4)

is linear as a mapping from A into the algebra L(A) of linear mappings from A into itself, where A is considered as a vector space over k again. More precisely, (12.4) is an algebra homomorphism from A into L(A), because of (12.3). If A has a multiplicative identity element e, then M_{e} is the identity operator on A. In this case, (12.4) is injective, because

M_{a}(e) = a e = a (12.5)

for every a ∈ A.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let N be a submultiplicative q-seminorm on A with respect to | · | on k for some q > 0. Let a ∈ A be given, and observe that

N(M_{a}(x)) = N(a x) ≤ N(a) N(x) (12.6)

for every x ∈ A. This means that M_{a} is bounded as a linear mapping from A into itself, with respect to N on the domain and range, and that

∥M_{a}∥_{op} ≤ N(a). (12.7)

It follows that (12.4) is bounded as a linear mapping from A into BL(A), with respect to N on A and the corresponding operator q-seminorm ∥·∥_{op} on BL(A).

More precisely, the corresponding operator q-seminorm of (12.4) is less than or equal to 1, by (12.7).

If A has a multiplicative identity element e, then

N(a) = N(M_{a}(e)) ≤ ∥M_{a}∥_{op} N(e) (12.8)

for every a ∈ A. If N(e) = 1, then we get that

∥M_{a}∥_{op} = N(a) (12.9)

for every a ∈ A.

13 Supremum 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 V be a vector space over k, and let N_{V} be a q_{V} - seminorm on V with respect to | · | on k for some q_{V} > 0. A V -valued function f on X is said to be bounded with respect to N_{V} if N_{V} (f(x)) is bounded as a nonnegative real-valued function on X. Let ℓ^{∞}(X, V) be the space of V -valued functions on X that are bounded with respect to N_{V}, and put

∥f∥_{∞} = ∥f∥_{ℓ^{∞}(X,V)} = sup_{x∈X} N_{V} (f(x)) (13.1)

for every such function. One can check that ℓ^{∞}(X, V) is a linear subspace of the space c(X, V) of all V -valued functions on X, and that (13.1) defines a q_{V} -seminorm on ℓ^{∞}(X, V) with respect to | · | on k.

Suppose for the moment that N_{V} is a q_{V} -norm on V, which implies that (13.1) is a q_{V} -norm on ℓ^{∞}(X, V). If V is complete with respect to the q_{V} - metric associated to N_{V}, then ℓ^{∞}(X, V) is complete with respect to the q_{V} norm associated to (13.1), by standard arguments.

Let A be an algebra over k, and consider the space c(X, A) of all A-valued functions on X. This is an algebra over k too, with respect to pointwise multiplication of functions. If A has a multiplicative identity element e, then the constant function on X equal to e at every point is the multiplicative identity element in c(X, A).

Let N_{A} be a submultiplicative q_{A}-seminorm on A with respect to | · | on k for some q_{A} > 0. Using N_{A}, we can define the space ℓ^{∞}(X, A) of A-valued functions on X that are bounded with respect to N_{A}, as before. It is easy to see that ℓ^{∞}(X, A) is a subalgebra of c(X, A), and that the corresponding supremum q_{A}-seminorm ∥f∥_{∞} is submultiplicative on ℓ^{∞}(X, A).

Now let X be a nonempty topological space, and let V be a vector space over k with a q_{V} -seminorm N_{V} with respect to | · | on k for some q_{V} > 0 again. The q_{V} -semimetric on V associated to N_{V} determines a topology on V, as usual. Let C(X, V) be the space of V -valued functions on X that are continuous with respect to this topology on V. This is a linear subspace of the space c(X, V) of all V -valued functions on X, by standard arguments. Of course, if X is equipped with the discrete topology, then every function on X is continuous, so that C(X, V) is the same as c(X, V).

Let E be a nonempty compact subset of X. If f ∈ C(X, V), then f(E) is a compact subset of V. This implies that f(E) is bounded with respect to N_{V}, so that N_{V} (f(x)) is bounded as a nonnegative real-valued function on E. Put

∥f∥_{sup,E} = sup_{x∈E} N_{V} (f(x)). (13.2)

One can check that this defines a q_{V} -seminorm on C(X, V) with respect to | · | on k, which is the supremum q_{V} -seminorm associated to E and N_{V}.

Let C_{b}(X, V) = C(X, V) ∩ ℓ^{∞}(X, V) (13.3)

be the space of V -valued functions on X that are both bounded and continuous with respect to N_{V} on V. This is a linear subspace of C(X, V) and ℓ^{∞}(X, V). If f ∈ C_{b}(X, V), then we may also use ∥f∥_{sup} for the supremum q_{V} -seminorm (13.1). Note that C_{b}(X, V) is a closed set in ℓ^{∞}(X, V) with respect to the supremum q_{V} -semimetric, which is the q_{V} -semimetric associated to the supremum q_{V} -seminorm, by standard arguments. If X is compact, then every continuous V -valued function on X is bounded. If N_{V} is a q_{V} -norm on V, and V is complete with respect to the associated q_{V} -metric, then ℓ^{∞}(X, V) is complete with respect to the supremum q_{V} -metric, as before. In this case, it follows that C_{b}(X, V) is complete with respect to the restriction of the supremum q_{V} -metric to C_{b}(X, V).

Let A be an algebra over k with a submultiplicative q_{A}-seminorm N_{A} with respect to |·| on k for some q_{A} > 0 again. Observe that C(X, A) is a subalgebra of c(X, A), by standard arguments. If E is a nonempty compact subset of X, then (13.2) is submultiplicative on C(X, A). Of course, C_{b}(X, A) is a subalgebra of both C(X, A) and ℓ^{∞}(X, A).

14 More on absolute value functions

Let k be a field, and let |·| be a q_{k}-absolute value function on k for some q_{k} > 0. If x ∈ k and n ∈ Z_{+}, then let n · x be the sum of n x’s in k. If there are positive integers n such that |n · 1| can be arbitrarily large, then | · | is said to be archimedean on k. Here 1 refers to the multiplicative identity element in k. Otherwise, | · | is said to be non-archimedean if there is a finite upper bound for |n · 1|, n ∈ Z_{+}. If | · | is an ultrametric absolute value function on k, then it is easy to see that |n · 1| ≤ 1 (14.1)

for every n ∈ Z_{+}, 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.

Let | · | be a q-absolute value function on the field Q of rational numbers for some q > 0. A famous theorem of Ostrowski says that | · | is either equivalent to the standard absolute value function on Q, or | · | is the trivial absolute value function on Q, or | · | is equivalent to the p-adic absolute value function on Q for some prime number p.

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and suppose that k is complete with respect to the associated q_{k}-metric. If | · | is archimedean on k, then another famous theorem of Ostrowski says that k is isomorphic to R or C, in such a way that | · | corresponds to a q_{k}-absolute value function on R or C that is equivalent to the standard absolute value function.

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0 again. Thus {|x| : x ∈ k \ {0}} (14.2)

is a subgroup of the group R_{+} of positive real numbers with respect to multiplication. If 1 is not a limit point of (14.2) with respect to the standard Euclidean metric on R, then | · | is said to be discrete on k. In this case, one can show that (14.2) is the same as the set of integer powers of a positive real number r, with r = 1 when | · | is the trivial absolute value function on k. Otherwise, if | · | is not discrete on k, then (14.2) is dense in R_{+} with respect to the standard Euclidean metric.

Suppose that | · | is archimedean on k, and let us check that | · | is not discrete on k. The archimedean property implies that k has characteristic 0, since otherwise there are only finitely many elements of k of the form n · 1, with n ∈ Z_{+}. Hence there is a natural embedding of Q into k. Using this embedding, we get a q_{k}-absolute value function on Q, which is archimedean on Q. The first theorem of Ostrowski mentioned earlier implies that this q_{k}-absolute value function on Q is equivalent to the standard absolute value function on Q. In particular, this q_{k}-absolute value function on Q is not discrete on Q. It follows that | · | is not discrete on k, as desired. Equivalently, if | · | is discrete on k, then | · | is non-archimedean on k, which implies that | · | is an ultrametric absolute value function on k, as before.

Part II

Sums and norms

15 ℓ^{r} q-Seminorms

Let X be a nonempty set, and let f be a nonnegative real-valued function on X. The sum

∑_{x∈X} f(x) (15.1)

can be defined as a nonnegative extended real number as the supremum of the sums

∑_{x∈A} f(x) (15.2)

over all nonempty finite subsets A of X. If (15.1) is finite, then f is said to be summable on X. It is easy to see that

∑_{x∈X} t f(x) = t ∑_{x∈X} f(x) (15.3)

for every positive real number t, where t (+∞) is interpreted as +∞, as usual. Similarly, 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). (15.4)

Of course, the right side of this equation is interpreted as being +∞ when either of the sums is +∞. It is sometimes convenient to permit f to be a nonnegative extended-real valued function on X in (15.1), where the sum is interpreted as being +∞ when f(x) = +∞ for any x ∈ X.

If X is the set Z_{+} of positive integers, then (15.1) can also be obtained by taking the supremum of the partial sums ∑_{j=1}^{n} f(j) over n ∈ Z_{+}. In this case, the finiteness of the sum (15.1) corresponds to the convergence of the infinite series ∑_{j=1}^{∞} f(j) in R. Similarly, if X is the set Z_{+} ∪ {0} of nonnegative integers, then (15.1) can be obtained by taking the supremum of the partial sums ∑_{j=0}^{n} f(j) over n ≥ 0.

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k with a q_{V} -seminorm N_{V} with respect to | · | on k for some q_{V} > 0. Also let r be a positive real number, and let f be a V -valued function on a nonempty set X. If N_{V} (f(x))^{r} is summable as a nonnegative real-valued function on X, then f is said to be r-summable on X with respect to N_{V}. In this case, we put

∥f∥_{r} = ∥f∥_{ℓ^{r}(X,V)} = ( ∑_{x∈X} N_{V} (f(x))^{r})^{1/r}, (15.5)

which is defined as a nonnegative real number. Let ℓ^{r}(X, V) be the space of V -valued functions on X that are r-summable with respect to N_{V}.

If f ∈ ℓ^{r}(X, V) and t ∈ k, then it is easy to see that t f ∈ ℓ^{r}(X, V), with

∥t f∥_{r} = |t| ∥f∥_{r}. (15.6)

If g ∈ ℓ^{r}(X, V) too, then one can check that f + g ∈ ℓ^{r}(X, V). More precisely, we have that ∥f + g∥_{r}^{r} ≤ ∥f∥_{r}^{r} + ∥g∥_{r}^{r} (15.7)

when r ≤ q_{V}, and ∥f + g∥_{r}^{q_{V}} ≤ ∥f∥_{r}^{q_{V}} + ∥g∥_{r}^{q_{V}} (15.8)

when q_{V} ≤ r. To get (15.7), one can use the fact that N_{V} may be considered as an r-seminorm on V with respect to | · | on k when r ≤ q_{k}, as in Section 7. If q_{V} ≤ r, then (15.8) can be obtained from Minkowski’s inequality (4.13), with exponent r/q_{V} ≥ 1. Although Minkowski’s inequality was previously stated for finite sums, it can be extended to arbitrary sums in a standard way. Thus ℓ^{r}(X, V) is a linear subspace of the space c(X, V) of all V -valued functions on X, (15.5) is an r-seminorm on ℓ^{r}(X, V) with respect to | · | on k when r ≤ q_{V}, and (15.5) is a q_{V} -seminorm on ℓ^{r}(X, V) with respect to | · | on k when q_{V} ≤ r.

Suppose for the moment that N_{V} is a q_{V} -norm on V with respect to | · | on k. In this case, (15.5) is an r-norm on ℓ^{r}(X, V) with respect to | · | on k when r ≤ q_{V}, and (15.5) is a q_{V} -norm on ℓ^{r}(X, V) with respect to |·| on k when q_{V} ≤ r. Suppose that V is also complete with respect to the q_{V} -metric associated to N_{V}. Under these conditions, one can show that ℓ^{r}(X, V) is complete with respect to the r or q_{V} -metric associated to (15.5), as appropriate, for every r > 0, using standard arguments.

16 Comparing ℓ^{r} spaces

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 V be a vector space over k with a q_{V} -seminorm N_{V} with respect to | · | on k for some q_{V} > 0. If a V -valued function f on X is r-summable with respect to N_{V} for some positive real number r, then it is easy to see that f is bounded with respect to N_{V}, with

∥f∥_{∞} ≤ ∥f∥_{r}. (16.1)

Note that (16.1) corresponds to (4.3), applied to N_{V} (f(x)) as a nonnegative realvalued function on X. In Section 4 we considered nonnegative real-valued functions on X with finite support, but the same argument works for r-summable functions.

Suppose now that 0 < r_{1} ≤ r_{2} < ∞, and that f is a V -valued r_{1}-summable function on X with respect to N_{V}. Under these conditions, we have that

∑_{x∈X} N_{V} (f(x))^{r_{2}} ≤ ∥f∥_{∞}^{r_{2}−r_{1}} ∑_{x∈X} N_{V} (f(x))^{r_{1}} = ∥f∥_{∞}^{r_{2}−r_{1}} ∥f∥_{r_{1}}^{r_{1}}, (16.2)

as in (4.4). This implies that f is r_{2}-summable on X with respect to N_{V}, with

∥f∥_{r_{2}} ≤ ∥f∥_{∞}^{1−(r_{1}/r_{2})} ∥f∥_{r_{1}}^{r_{1}/r_{2}} ≤ ∥f∥_{r_{1}}, (16.3)

as in (4.5). In particular,

ℓ^{r_{1}}(X, V) ⊆ ℓ^{r_{2}}(X, V). (16.4)

This inclusion also holds when r_{2} = ∞, as in the preceding paragraph.

A V -valued function f on X is said to vanish at infinity with respect to N_{V} if for every ϵ > 0 we have that

N_{V} (f(x)) < ϵ (16.5)

for all but finitely many x ∈ X. Let c_{0}(X, V) be the space of all V -valued functions on X that vanish at infinity with respect to N_{V}. It is easy to see that c_{0}(X, V) is a linear subspace of the space c(X, V) of all V -valued functions on X. More precisely, c_{0}(X, V) ⊆ ℓ^{∞}(X, V). (16.6)

One can also check that ℓ^{r}(X, V) ⊆ c_{0}(X, V) (16.7)

for every positive real number r.

One can verify that c_{0}(X, V) is a closed set in ℓ^{∞}(X, V) with respect to the corresponding supremum q_{V} -semimetric, using standard arguments. If N_{V} is a q_{V} -norm on V, and V is complete with respect to the associated q_{V} -metric, then ℓ^{∞}(X, V) is complete with respect to the supremum q_{V} -metric, as before. In this case, we get that c_{0}(X, V) is complete with respect to the restriction of the supremum q_{V} -metric to c_{0}(X, V).

If f is a V -valued function on X such that N_{V} (f(x)) = 0 for all but finitely many x ∈ X, then f vanishes at infinity on X with respect to N_{V}, and f is r-summable on X with respect to N_{V} for every r > 0. In particular, this condition holds when f has finite support in X. If N_{V} is a q_{V} -norm on V, then this condition implies that f has finite support in X.

It is not difficult to check that c_{00}(X, V) is dense in c_{0}(X, V) with respect to the supremum q_{V} -semimetric. Thus c_{0}(X, V) is the same as the closure of c_{00}(X, V) in ℓ^{∞}(X, V), with respect to the supremum q_{V} -semimetric.

If r is a positive real number, then one can verify that c_{00}(X, V) is dense in ℓ^{r}(X, V), with respect to the r or q_{V} -semimetric associated to ∥ · ∥_{r}, as appropriate. More precisely, if f is an r-summable V -valued function on X with respect to N_{V}, then ∥f∥_{r}^{r} can be approximated by the sum of N_{V} (f(x))^{r} over suitable nonempty finite subsets of X. One can use these finite subsets of X to approximate f by V -valued functions on X with finite support with respect to the r or q_{V} -semimetric associated to ∥ · ∥_{r}, as appropriate.

If f is a V -valued function on X that vanishes at infinity with respect to N_{V}, then there are only finitely or countably many x ∈ X such that N_{V} (f(x)) > 0. More precisely, for each positive integer j, there are only finitely many x ∈ X such that N_{V} (f(x)) ≥ 1/j, and the previous statement follows by taking the union of these finite sets. If N_{V} is a q_{V} -norm on V, and f vanishes at infinity on X with respect to N_{V}, then the support of f has only finitely or countably many elements.

17 Doubly-infinite series

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k with a q_{V} -norm N_{V} with respect to | · | on k for some q_{V} > 0. Also let

∑_{j=−∞}^{∞} v_{j} (17.1)

be a doubly-infinite series with terms in V, so that v_{j} ∈ V for every j ∈ Z. Let us say that (17.1) converges in V with respect to N_{V} if

∑_{j=0}^{∞} v_{j} and ∑_{j=1}^{∞} v_{−j} (17.2)

converge as ordinary infinite series in V with respect to N_{V}. In this case, the value of the sum (17.1) is defined by

∑_{j=−∞}^{∞} v_{j} = ∑_{j=0}^{∞} v_{j} + ∑_{j=1}^{∞} v_{−j}. (17.3)

One could look at the convergence of (17.1) in terms of the convergence of the partial sums

∑_{j=−l}^{n} v_{j} (17.4)

as l, n → ∞. If l ≥ 1 and n ≥ 0, then (17.4) can be expressed as

∑_{j=0}^{n} v_{j} + ∑_{j=1}^{l} v_{−j}. (17.5)

Of course, the convergence of the series in (17.2) is the same as the convergence of these two sequences of partial sums.

Alternatively, one can consider

∑_{j=1}^{∞}(v_{j} + v_{−j}) (17.6)

as an infinite series with terms in V. If this series converges, then one might interpret (17.1) as being

v_{0} + ∑_{j=1}^{∞}(v_{j} + v_{−j}). (17.7)

Note that

∑_{j=−n}^{n} v_{j} = v_{0} + ∑_{j=1}^{n}(v_{j} + v_{−j}) (17.8)

for every positive integer n. If the series in (17.2) converge in V with respect to N_{V}, then (17.6) converges in V with respect to N_{V} as well. Under these conditions, (17.3) is equal to (17.7).

Suppose for the moment that k = R with the standard absolute value function, and that V = R, with N_{V} = |·|. If v_{j} is a nonnegative real number for each j ∈ Z, then the convergence of each of the infinite series in (17.2) and (17.6) is equivalent to the boundedness of the corresponding sequences of partial sums.

In this situation, the partial sums of (17.6) are bounded if and only if the partial sums of both series in (17.2) are bounded. Thus (17.6) converges if and only if both series in (17.2) converge, in which case (17.3) is equal to (17.7), as before. One can also define

∑_{j∈Z} v_{j} (17.9)

as a nonnegative extended real number as in Section 15. This is equivalent to taking the supremum of the partial sums in (17.4) over l, n ≥ 0, or with l = n, as in (17.8). This means that (17.9) is finite exactly when the series in (17.2) or (17.6) converge, with the same value of the sum as in (17.3) and (17.7).

Let k be any field with a q_{k}-absolute value function | · | for some q_{k} > 0 again, let V be a vector space over k with a q_{V} -norm N_{V} with respect to | · | on k for some q_{V} > 0, and let (17.1) be a doubly-infinite series with terms in V. Suppose for the moment that q_{V} < ∞. Let us say that (17.1) converges q_{V} -absolutely with respect to N_{V} if

∑_{j=−∞}^{∞} N_{V} (v_{j})^{q_{V}} (17.10)

converges as a doubly-infinite series of nonnegative real numbers. This is the same as saying that the series in (17.2) converge q_{V} -absolutely with respect to N_{V}. If V is complete with respect to the q_{V} -metric associated to N_{V}, then it follows that the series in (17.2) converge in V, as in Section 9. This means that (17.1) converges in V, and it is easy to see that

N_{V} ( ∑_{j=−∞}^{∞} v_{j})^{q_{V}} ≤ ∑_{j=−∞}^{∞} N_{V} (v_{j})^{q_{V}} . (17.11)

This uses the analogous statements for the series in (17.2), as in (9.9).

Suppose now that q_{V} = ∞, and that V is complete with respect to the ultrametric associated to N_{V}. If

N_{V} (v_{j}) → 0 as j → ±∞, (17.12)

then the series in (17.2) converge in V, as in Section 9. This means that (17.1) converges in V, and we have that

N_{V} ( ∑_{j=−∞}^{∞} v_{j}) ≤ max_{j∈Z} N_{V} (v_{j}). (17.13)

This follows from the analogous statements for the series in (17.2), as in (9.11). In particular, (17.12) implies that the maximum on the right side of (17.13) is attained, as before.

18 Sums over sets

Let X be a nonempty set, let k be a field, and let V be a vector space over k. Remember that

f ↦ ∑_{x∈X} f(x) (18.1)

defines a linear mapping from c_{00}(X, V) into V, as in Section 1. Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let N_{V} be a q_{V} -seminorm on V with respect to | · | on k for some q_{V} > 0. It is easy to see that

N_{V} ( ∑_{x∈X} f(x)) ≤ ∥f∥_{q_{V}} (18.2)

for every f ∈ c_{00}(X, V), where ∥f∥_{q_{V}} is as in Sections 13 and 15. Note that the restriction of ∥f∥_{q_{V}} to c_{00}(X, V) defines a q_{V} -seminorm on c_{00}(X, V) with respect to | · | on k. Thus (18.2) says that (18.1) is bounded as a linear mapping from c_{00}(X, V) into V, with respect to ∥ · ∥_{q_{V}} on c_{00}(X, V) and N_{V} on V, with operator q_{V} -seminorm less than or equal to 1. More precisely, the operator q_{V} -seminorm is equal to 1 when N_{V} (v) > 0 for some v ∈ V.

Let us suppose from now on in this section that N_{V} is a q_{V} -norm on V, and that V is complete with respect to the associated q_{V} -metric. If q_{V} < ∞, then there is a unique extension of (18.1) to a bounded linear mapping from ℓ^{q_{V}} (X, V) into V, as in Section 10. This uses the fact that c_{00}(X, V) is dense in ℓ^{q_{V}} (X, V) with respect to the q_{V} -metric associated to ∥ · ∥_{q_{V}} when q_{V} < ∞, as in Section 16. Similarly, if q_{V} = ∞, then there is a unique extension of (18.1) to a bounded linear mapping from c_{0}(X, V) into V, with respect to the supremum ultranorm on c_{0}(X, V) and N_{V} on V, because c_{00}(X, V) is dense in c_{0}(X, V) with respect to the supremum ultrametric on c_{0}(X, V). We can use these extensions to define

∑_{x∈X} f(x) (18.3)

as an element of V when f ∈ ℓ^{q_{V}} (X, V) and q_{V} < ∞, and when f ∈ c_{0}(X, V) and q_{V} = ∞. Observe that (18.2) holds for every f ∈ ℓ^{q_{V}} (X, V) when q_{V} < ∞, and for every f ∈ c_{0}(X, V) when q_{V} = ∞, because the operator q_{V} -norm of the extension is the same as on c_{00}(X, V). Of course, if X has only finitely many elements, then every V -valued function on X has finite support, and no extension is necessary.

If X = Z_{+}, then we can use the infinite series

∑_{j=1}^{∞} f(j) (18.4)

to define these extensions. More precisely, this series converges in V when f is an element of ℓ^{q_{V}} (Z_{+}, V) and q_{V} < ∞, and when f ∈ c_{0}(Z_{+}, V) and q_{V} = ∞, as in Section 9. This definition of the sum defines a bounded linear mapping into V, and (18.4) reduces to the usual finite sum when f has finite support in Z_{+}. Similarly, if X = Z_{+} ∪ {0}, then these extensions can be defined using the infinite series

∑_{j=0}^{∞} f(j). (18.5)

If X = Z, then these extensions can be defined by considering

∑_{j=−∞}^{∞} f(j) (18.6)

as a doubly-infinite series with terms in V, as in the previous section.

Let X be any countably-infinite set, and let {x_{j}}_{j=1}^{∞} be a sequence of elements of X in which every element of X occurs exactly once. Suppose that f ∈ ℓ^{q_{V}} (X, V) when q_{V} < ∞, and that f ∈ c_{0}(X, V) when q_{V} = ∞, as before. In both cases, the infinite series

∑_{j=1}^{∞} f(x_{j}) (18.7)

converges in V, as in Section 9 again. This can be used as another way to define (18.3) in this situation, as in the preceding paragraph. If X is any infinite set, and X_{0} is a countably-infinite subset of X, then one can deal with V -valued functions f on X whose support is contained in X_{0} in the same way.

19 Iterated sums

Let I and X be nonempty sets, and let E_{j} be a nonempty subset of X for every j ∈ I. Suppose that the E_{j}’s are pairwise disjoint, so that

E_{j} ∩ E_{l} = ∅ (19.1)

for every j, l ∈ I with j ≠ l. If f is a nonnegative real-valued function on X, then

∑_{x∈E_{j}} f(x) (19.2)

can be defined as a nonnegative extended real number for every j ∈ I. Hence

∑_{j∈I} ( ∑_{x∈E_{j}} f(x)) (19.3)

can be defined as an extended real number as well. Put

E = ⋃_{j∈I} E_{j}, (19.4)

so that

∑_{x∈E} f(x) (19.5)

can be defined as an extended real number too. One can check that (19.3) and (19.5) are the same in this situation. More precisely, one can verify that (19.3) and (19.5) are each less than or equal to the other.

Let k be a field, and let V be a vector space over k. If f is a V -valued function on X with finite support, then (19.2) can be defined as an element of V for every j ∈ I. We also have that (19.2) is equal to 0 for all but finitely many j ∈ I, so that (19.3) can be defined as an element of V too. Similarly, (19.5) can be defined as an element of V, and is equal to (19.3).

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let N_{V} be a q_{V} -norm on V with respect to | · | on k for some q_{V} > 0. Suppose for the rest of the section that V is complete with respect to the q_{V} -metric associated to N_{V}. Suppose for the moment that q_{V} < ∞, and let f ∈ ℓ^{q_{V}} (X, V) be given. It is easy to see that the restriction of f to E_{j} is q_{V} -summable with respect to N_{V} for every j ∈ I, so that (19.2) can be defined as an element of V as in the previous section. Note that

N_{V} ( ∑_{x∈E_{j}} f(x))^{q_{V}} ≤ ∑_{x∈E_{j}} N_{V} (f(x))^{q_{V}} (19.6)

for every j ∈ I, as in (18.2). This implies that

∑_{j∈I} N_{V} ( ∑_{x∈E_{j}} f(x))^{q_{V}} ≤ ∑_{j∈I} ( ∑_{x∈E_{j}} N_{V} (f(x))^{q_{V}} ) = ∑_{x∈E} N_{V} (f(x))^{q_{V}} , (19.7)

using the earlier remarks for nonnegative real-valued functions in the second step. Thus (19.2) is q_{V} -summable as a V -valued function of j ∈ I with respect to N_{V}, so that (19.3) can be defined as an element of V as in the previous section as well. The sum (19.5) can be defined as an element of V as in the previous section too, because the restriction of f to E is q_{V} -summable with respect to N_{V}. One can check that (19.3) is equal to (19.5), by approximating f by V -valued functions on X with finite support.

Suppose now that q_{V} = ∞, and let f ∈ c_{0}(X, V) be given. Observe that the restriction of f to E_{j} vanishes at infinity on E_{j} with respect to N_{V} for every j ∈ I, so that (19.2) can be defined as an element of V for every j ∈ I, as in the previous section. In this situation, we have that

N_{V} ( ∑_{x∈E_{j}} f(x)) ≤ max_{x∈E_{j}} N_{V} (f(x)) (19.8)

for every j ∈ I, as in (18.2). Using this, one can verify that (19.2) vanishes at infinity as a V -valued function of j ∈ I with respect to N_{V}, so that (19.3) can be defined as an element of V as in the previous section. The restriction of f to E vanishes at infinity with respect to N_{V}, so that (19.5) can be defined as an element of V as in the previous section as well. One can verify that (19.3) is equal to (19.5), by approximating f by V -valued functions on X with finite support again. Remember that the norm of (19.3) with respect to N_{V} is less than or equal to the maximum of the left side of (19.8) over j ∈ I, as in (18.2).

This is less than or equal to the maximum of the right side of (19.8) over j ∈ I, which is the same as the maximum of N_{V} (f(x)) over x ∈ E.

20 Cauchy products on Z_{+} ∪ {0}

Let k be a field, and let A be an algebra over k. Also let ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} be infinite series with terms in A, considered formally for the moment. Put

c_{n} = ∑_{j=0}^{n} a_{j} b_{n−j} (20.1)

for each nonnegative integer n. The infinite series ∑_{n=0}^{∞} c_{n} is called the Cauchy product of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l}. It is easy to see that

∑_{n=0}^{∞} c_{n} = ( ∑_{j=0}^{∞} a_{j}) ( ∑_{l=0}^{∞} b_{l}) (20.2)

formally. More precisely, consider

∑_{(j,l)∈(Z_{+}∪{0})^{2}} a_{j} b_{l} (20.3)

formally, where (Z_{+} ∪ {0})^{2} = (Z_{+} ∪ {0}) × (Z_{+} ∪ {0}). This can be identified formally with the iterated sums

∑_{j=0}^{∞} ( ∑_{l=0}^{∞} a_{j} b_{l}) (20.4)

and

∑_{l=0}^{∞} ( ∑_{j=0}^{∞} a_{j} b_{l}), (20.5)

each of which reduces formally to the product of the sums on the right side of (20.2). Put E_{n} = {(j, l) ∈ (Z_{+} ∪ {0})^{2} : j + l = n} (20.6)

for each nonnegative integer n, and observe that the E_{n}’s are pairwise-disjoint finite subsets of (Z_{+} ∪ {0})^{2} such that

⋃_{n=0}^{∞} E_{n} = (Z_{+} ∪ {0})^{2}. (20.7)

By construction,

c_{n} = ∑_{(j,l)∈E_{n}} a_{j} b_{l} (20.8)

for every n ≥ 0, so that the left side of (20.2) is the same as

∑_{n=0}^{∞} ( ∑_{(j,l)∈E_{n}} a_{j} b_{l}), (20.9)

which corresponds formally to (20.3) as well.

Suppose for the moment that a_{j} = 0 for all but finitely many j ≥ 0, and that b_{l} = 0 for all but finitely many l ≥ 0, so that ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} reduce to finite sums. In this case, one can check that c_{n} = 0 for all but finitely many n ≥ 0, so that ∑_{n=0}^{∞} c_{n} reduces to a finite sum too. In fact,

a_{j} b_{l} = 0 (20.10)

for all but finitely many (j, l) ∈ (Z_{+} ∪ {0})^{2}, so that (20.3) reduces to a finite sum as well. The iterated sums (20.4) and (20.5) also reduce to finite sums of finite sums in this situation. As in the previous section, (20.3) is equal to the iterated sums (20.4), (20.5), and (20.9), which implies that (20.2) holds.

Suppose for the moment again that k = A = R, and that a_{j}, b_{l} ≥ 0 for all j, l ≥ 0, so that c_{n} ≥ 0 for all n ≥ 0. The sums ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l}, and ∑_{n=0}^{∞} c_{n} may be considered as nonnegative extended real numbers, by considering them as sums over Z_{+} ∪ {0} as in Section 15, or by interpreting an infinite series with nonnegative terms as being +∞ when the series does not converge in R in the usual sense. The product of sums on the right side of (20.2) can be defined as a nonnegative extended real number in the usual way, except when one of the factors is equal to 0 and the other is equal to +∞. This can only happen when a_{j} = 0 for every j ≥ 0 or b_{l} = 0 for every l ≥ 0, in which case c_{n} = 0 for every n = 0, so that the left side of (20.2) is equal to 0. The sum (20.3) and the iterated sums (20.4) and (20.5) can also be defined as nonnegative extended real numbers, and we have that (20.3) is equal to the iterated sums (20.4), (20.5), and (20.9), as in the previous section. If ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} are finite, then the iterated sums (20.4) and (20.5) can be reduced to the product on the right side of (20.2). This implies that (20.2) holds, and in particular that ∑_{n=0}^{∞} c_{n} is finite. If either of ∑_{j=0}^{∞} a_{j} or ∑_{l=0}^{∞} b_{l} is +∞, and the other is positive, then the iterated sums (20.4), (20.5) are +∞. This implies that (20.3) is +∞, so that (20.9) is +∞, which is the same as saying that ∑_{n=0}^{∞} c_{n} = +∞.

Let k be any field again, and let | · | be a q_{k}-absolute value function on k for some q_{k} > 0. Also let A be an algebra over k with a submultiplicative q-norm with respect to | · | on k for some q > 0, and suppose for the rest of the section that A is complete with respect to the q-metric associated to N. Suppose for the moment that q < ∞, and that ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} are q-absolutely convergent infinite series with terms in A, with respect to N. If ∑_{n=0}^{∞} c_{n} is their Cauchy product, as before, then

N(c_{n})^{q} ≤ ∑_{j=0}^{n} N(a_{j} b_{n−j})^{q} ≤ ∑_{j=0}^{n} N(a_{j})^{q} N(b_{n−j})^{q} (20.11)

for each n ≥ 0. This uses the q-norm version of the triangle inequality in the first step, and submultiplicativity of N on A in the second set. The right side of (20.11) is the nth term of the Cauchy product of ∑_{j=0}^{∞} N(a_{j})^{q} and ∑_{l=0}^{∞} N(b_{l})^{q}, as infinite series of nonnegative real numbers. It follows that

(20.12)  ∑_{n=0}^{∞} N(c_{n})^{q} ≤ ∑_{n=0}^{∞} ( ∑_{j=0}^{n} N(a_{j})^{q} N(b_{n−j})^{q})

= ( ∑_{j=0}^{∞} N(a_{j})^{q}) ( ∑_{l=0}^{∞} N(b_{l})^{q}),

using the remarks in the preceding paragraph in the second step. In particular, this means that ∑_{n=0}^{∞} c_{n} converges q-absolutely with respect to N. Similarly,

∑_{(j,l)∈(Z_{+}∪{0})^{2}} N(a_{j} b_{l})^{q} ≤ ∑_{(j,l)∈(Z_{+}∪{0})^{2}} N(a_{j})^{q} N(b_{l})^{q}, (20.13)

because of the submultiplicativity of N on A. The right side of (20.13) is the same as the iterated sums

∑_{j=0}^{∞} ( ∑_{l=0}^{∞} N(a_{j})^{q} N(b_{l})^{q}) (20.14)

and

∑_{l=0}^{∞} ( ∑_{j=0}^{∞} N(a_{j})^{q} N(b_{l})^{q}), (20.15)

as before. These iterated sums are equal to the right side of (20.12), so that the right side of (20.13) is equal to the right side of (20.12). Hence the left side of (20.13) is finite, which means that f(j, l) = a_{j} b_{l} is q-summable as an A-valued function on (Z_{+} ∪{0})^{2}, with respect to N. The sum (20.3) can be defined as an element of A as in Section 18, and this sum is equal to the iterated sums (20.4) and (20.5), as in the previous section. Of course, ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge in A, as in Section 9. The product of these two sums is the same as each of the iterated sums (20.4) and (20.5). The sum (20.3) is equal to the iterated sum (20.9) too, which is the same as ∑_{n=0}^{∞} c_{n}. This shows that (20.2) holds in this situation.

Suppose now that q = ∞, and that

lim_{j→∞} N(a_{j}) = lim_{l→∞} N(b_{l}) = 0. (20.16)

This implies that ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge in A, as in Section 9. If c_{n} is as in (20.1), then

N(c_{n}) ≤ max_{0≤j≤n} N(a_{j} b_{n−j}) ≤ max_{0≤j≤n}(N(a_{j}) N(b_{n−j})) (20.17)

for every n ≥ 0, using the ultranorm version of the triangle inequality in the first step, and the submultiplicativity of N on A in the second step. One can check that

lim_{n→∞} N(c_{n}) = 0, (20.18)

using (20.16) and (20.17). We also have that

(20.19)  max_{n≥0} N(c_{n}) ≤ ( max_{j≥0} N(a_{j})) ( max_{l≥0} N(b_{l})),

by (20.17). Of course, N(a_{j} b_{l}) ≤ N(a_{j}) N(b_{l}) (20.20)

for all j, l ≥ 0, because of the submultiplicativity of N on A. One can verify that f(j, l) = a_{j} b_{l} vanishes at infinity as an A-valued function on (Z_{+} ∪ {0})^{2} with respect to N, using (20.16) and (20.20). More precisely, N(a_{j}) N(b_{l}) vanishes at infinity as a nonnegative real-valued function on (Z_{+} ∪ {0})^{2}. This implies that the sum (20.3) can be defined as an element of A, as in Section 18. This sum is equal to the iterated sums (20.4) and (20.5), and these iterated sums are both equal to the product of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l}. It follows that (20.2) holds in this situation as well, because (20.3) is also equal to the iterated sum (20.9), which is the same as ∑_{n=0}^{∞} c_{n}.

21 Cauchy products on Z

Let k be a field again, and let A be an algebra over k. Let us now consider doubly-infinite series ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l} with terms in A, at least formally at first. Consider

c_{n} = ∑_{j=−∞}^{∞} a_{j} b_{n−j} (21.1)

for each n ∈ Z, at least formally. The doubly-infinite series ∑_{n=−∞}^{∞} c_{n} is the formal Cauchy product of ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l}. As before,

∑_{n=−∞}^{∞} c_{n} = ( ∑_{j=−∞}^{∞} a_{j}) ( ∑_{l=−∞}^{∞} b_{l}) (21.2)

formally. To see this, consider

∑_{(j,l)∈Z^{2}} a_{j} b_{l} (21.3)

formally, where Z^{2} = Z×Z. As before, this corresponds formally to the iterated sums

∑_{j=−∞}^{∞} ( ∑_{l=−∞}^{∞} a_{j} b_{l}) (21.4)

and

∑_{l=−∞}^{∞} ( ∑_{j=−∞}^{∞} a_{j} b_{l}), (21.5)

which reduce formally to the product on the right side of (21.2). In this situation, we put E_{n} = {(j, l) ∈ Z^{2} : j + l = n} (21.6)

for every n ∈ Z, which are pairwise-disjoint subsets of Z^{2} such that

⋃_{n=−∞}^{∞} E_{n} = Z^{2}. (21.7)

Thus (21.1) is formally the same as

c_{n} = ∑_{(j,l)∈E_{n}} a_{j} b_{l}, (21.8)

so that the left side of (21.2) corresponds formally to

∑_{n=−∞}^{∞} ( ∑_{(j,l)∈E_{n}} a_{j} b_{l}). (21.9)

This can be identified formally with (21.3), because of (21.7).

If a_{j} = 0 for all but finitely many j ∈ Z, and b_{l} = 0 for all but finitely many l ∈ Z, then ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l} reduce to finite sums in A. Under these conditions, the right side of (21.1) also reduces to a finite sum, so that c_{n} is defined as an element of A for every n ∈ Z. It is easy to see that c_{n} = 0 for all but finitely many n ∈ Z, so that ∑_{n=−∞}^{∞} c_{n} reduces to a finite sum in A. Similarly, a_{j} b_{l} = 0 (21.10)

for all but finitely many (j, l) ∈ Z^{2}, so that (21.3) reduces to a finite sum in A as well. The iterated sums (21.4) and (21.5) reduce to finite sums of finite sums, and (21.3) is equal to the iterated sums (21.4), (21.5), and (21.9), so that (21.2) holds.

Suppose that k = A = R, and that a_{j}, b_{l} ≥ 0 for every j, l ∈ Z. The sums ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l} may be considered as nonnegative extended real numbers, by considering them as sums over Z, as in Section 15, or by interpreting a doubly-infinite series with nonnegative terms as being +∞ when the series does not converge in R in the usual sense. Similarly, c_{n} is defined as a nonnegative extended real number for each n ∈ Z, so that ∑_{n=−∞}^{∞} c_{n} is defined as a nonnegative extended real number. The sum (21.3) and the iterated sums (21.4) and (21.5) are defined as extended real numbers as well, and (21.3) is equal to the iterated sums (21.4), (21.5), and (21.9), as in Section 19. If ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l} are finite, then the iterated sums (21.4) and (21.5) are the same as the product of these two sums. It follows that (21.2) holds, and in particular that ∑_{n=−∞}^{∞} c_{n} is finite. If either ∑_{j=−∞}^{∞} a_{j} or ∑_{l=−∞}^{∞} b_{l} is equal to 0, then either a_{j} = 0 for every j ∈ Z or b_{l} = 0 for every l ∈ Z. In both cases, c_{n} = 0 for every n ∈ Z, and the sum (21.3) and the iterated sums (21.4), (21.5), and (21.9) are equal to 0. If ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l} are both positive, and at least one of these sums is +∞, then the iterated sums (21.4) and (21.5) are +∞. This implies that (21.3) is +∞, and hence that (21.9) is +∞, which means that ∑_{n=−∞}^{∞} c_{n} = +∞.

Let k be a field with a q_{k}-absolute value function |·| for some q_{k} > 0, and let A be an algebra over k with a submultiplicative q-norm N with respect to | · | on k for some q > 0. Suppose for the rest of the section that A is complete with respect to the q-metric associated to N. Suppose for the moment that q < ∞, and that

∑_{j=−∞}^{∞} N(a_{j})^{q}, ∑_{l=−∞}^{∞} N(b_{l})^{q} < ∞. (21.11)

This implies that ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l} can be defined as elements of A, as q-absolutely convergent doubly-infinite series, as in Section 17, or as sums over Z, as in Section 18. Observe that

∑_{j=−∞}^{∞} N(a_{j} b_{n−j})^{q} ≤ ∑_{j=−∞}^{∞} N(a_{j})^{q} N(b_{n−j})^{q} < ∞ (21.12)

for every n ∈ Z. This uses the fact that the terms in the sums in (21.11) are bounded, because the sums are finite. Thus c_{n} can be defined as an element of A as in (21.1) for each n ∈ Z, with

N(c_{n})^{q} ≤ ∑_{j=−∞}^{∞} N(a_{j} b_{n−j})^{q}. (21.13)

Of course,

∑_{(j,l)∈Z^{2}} N(a_{j} b_{l})^{q} ≤ ∑_{(j,l)∈Z^{2}} N(a_{j})^{q} N(b_{l})^{q}, (21.14)

by the submultiplicativity of N on A. The right side of (21.14) is equal to the iterated sums

∑_{j=−∞}^{∞} ( ∑_{l=−∞}^{∞} N(a_{j})^{q} N(b_{l})^{q}) (21.15)

and

∑_{l=−∞}^{∞} ( ∑_{j=−∞}^{∞} N(a_{j})^{q} N(b_{l})^{q}), (21.16)

as in the preceding paragraph. These iterated sums reduce to the product

( ∑_{j=−∞}^{∞} N(a_{j})^{q}) ( ∑_{l=−∞}^{∞} N(b_{l})^{q}), (21.17)

which is finite, by (21.11). The right side of (21.14) is also equal to the iterated sum

∑_{n=−∞}^{∞} ( ∑_{(j,l)∈E_{n}} N(a_{j})^{q} N(b_{l})^{q}) = ∑_{n=−∞}^{∞} ( ∑_{j=−∞}^{∞} N(a_{j})^{q} N(b_{n−j})^{q}), (21.18)

as before. The finiteness of the right side of (21.14) implies that f(j, l) = a_{j} b_{l} is q-summable as an A-valued function on Z^{2} with respect to N, so that (21.3) can be defined as an element of A, as in Section 18. The iterated sums (21.4) and (21.5) can also be defined as elements of A, and are equal to (21.3), as in Section 19. Of course, these iterated sums reduce to the product on the right side of (21.2). Similarly, the iterated sum (21.9) can be defined as an element of A, and is equal to (21.3). This includes the fact that

∑_{n=−∞}^{∞} N(c_{n})^{q} < ∞, (21.19)

which follows from the finiteness of (21.18). Thus ∑_{n=−∞}^{∞} c_{n} can be defined as an element of A, and satisfies (21.2).

Suppose now that q = ∞, and that

lim_{j→±∞} N(a_{j}) = lim_{l→±∞} N(b_{l}) = 0. (21.20)

This implies that ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l} can be defined as elements of A, as convergent doubly-infinite series, as in Section 17, or as sums over Z, as in Section 18. Remember that

N(a_{j} b_{l}) ≤ N(a_{j}) N(b_{l}) (21.21)

for every j, l ∈ Z, by the submultiplicativity of N on A. It is easy to see that

lim_{j→±∞} N(a_{j} b_{n−j}) = 0 (21.22)

for each n ∈ Z, because of (21.20), which implies in particular that N(a_{j}) and N(b_{l}) are bounded. Hence c_{n} can be defined as an element of A as in (21.1) for each n ∈ Z, and satisfies

N(c_{n}) ≤ max_{j∈Z} N(a_{j} b_{n−j}). (21.23)

One can check that N(a_{j}) N(b_{l}) vanishes at infinity as a nonnegative real-valued function on Z^{2}, using (21.20). This means that f(j, l) = a_{j} b_{l} vanishes at infinity as an A-valued function on Z^{2} with respect to N, by (21.21). Thus the sum (21.3) can be defined as an element of A, as in Section 18. This sum is equal to the iterated sums (21.4) and (21.5), as in Section 19, and these iterated sums reduce to the product of ∑_{j=−∞}^{∞} a_{j} and ∑_{l=−∞}^{∞} b_{l}. The iterated sum (21.9) can also be defined as an element of A, and is equal to (21.3), as in Section 19. This includes the fact that

lim_{n→±∞} N(c_{n}) = 0, (21.24)

which can be obtained from the vanishing at infinity of f(j, l) = a_{j} b_{l} on Z^{2}. This means that ∑_{n=−∞}^{∞} c_{n} can be defined as an element of A, and satisfies (21.2).

22 Hilbert spaces

Let V and W be vector spaces over the complex numbers. Thus V and W may also be considered as vector spaces over the real numbers. A mapping T from V into W may be called complex-linear to indicate that T is a linear mapping from V into W as vector spaces over C. Similarly, T may be called real-linear when T is linear as a mapping from V into W as vector spaces over R. A real-linear mapping T from V into W is complex-linear when

T (i v) = i T (v) (22.1)

for every v ∈ V. A real-linear mapping T from V into W is said to be conjugatelinear if T (a v) = \overline{a} T (v) (22.2)

for every a ∈ C and v ∈ V, where \overline{a} is the complex-conjugate of a. This holds when T (i v) = −i T (v) (22.3)

for every v ∈ V, because T is real-linear.

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⟩, as appropriate, defined for v, w ∈ V, that satisfies the following properties. First, for each w ∈ V, ⟨v, w⟩ should be real-linear as a function of v into R, or complex-linear as a function of v into C, as appropriate. Second, ⟨v, w⟩ should be symmetric in the real case, so that ⟨v, w⟩ = ⟨w, v⟩ (22.4)

for every v, w ∈ V. Combining this with the first property, we get that ⟨v, w⟩ is linear in w for each v ∈ V. In the complex case, ⟨v, w⟩ should be Hermitiansymmetric, which means that

⟨w, v⟩ = \overline{⟨v, w⟩} (22.5)

for every v, w ∈ V. It follows that ⟨v, w⟩ is conjugate-linear in w for every v ∈ V. Note that ⟨v, v⟩ ∈ R (22.6) for every v ∈ V in the complex case, by (22.5). The third condition in both cases is that ⟨v, v⟩ > 0 (22.7)

for every v ∈ V with v ≠ 0. Of course, ⟨v, w⟩ = 0 when v = 0 or w = 0, by the first two conditions.

Let ⟨v, w⟩ be an inner product on V, and put

∥v∥ = ⟨v, v⟩^{1/2} (22.8)

for every v ∈ V, using the nonnegative square root on the right side. It is well known that |⟨v, w⟩| ≤ ∥v∥ ∥w∥ (22.9)

for every v, w ∈ V, which is the Cauchy–Schwarz inequality. Using this, one can show that ∥v + w∥ ≤ ∥v∥ + ∥w∥ (22.10)

for every v, w ∈ V. This implies that (22.8) defines a norm on V with respect to the standard absolute value function on R or C, as appropriate, because of the other properties of the inner product. If V is complete with respect to the metric associated to this norm, then V is said to be a Hilbert space.

Let X be a nonempty set, and let f, g be real or complex-valued functions on X that are square-summable, which is to say that |f|^{2} and |g|^{2} are summable on X. Remember that a b ≤ (1/2) (a^{2} + b^{2}) (22.11)

for all nonnegative real numbers a and b, because (a − b)^{2} ≥ 0. This implies that

∑_{x∈X} |f(x)| |g(x)| ≤ (1/2) ∑_{x∈X} |f(x)|^{2} + (1/2) ∑_{x∈X} |g(x)|^{2}, (22.12)

so that |f| |g| is summable on X. Put

⟨f, g⟩ = ⟨f, g⟩_{ℓ^{2}(X,R)} = ∑_{x∈X} f(x) g(x) (22.13)

in the real case, and

⟨f, g⟩ = ⟨f, g⟩_{ℓ^{2}(X,C)} = ∑_{x∈X} f(x) \overline{g(x)} (22.14)

in the complex case. It is easy to see that these define inner products on ℓ^{2}(X, R) and ℓ^{2}(X, C), respectively. In both cases we have that

⟨f, f⟩ = ∑_{x∈X} |f(x)|^{2}, (22.15)

so that the norms corresponding to (22.13) and (22.14) are the same as the usual ℓ^{2} norms. Thus ℓ^{2}(X, R) and ℓ^{2}(X, C) are Hilbert spaces, because they are complete with respect to the corresponding ℓ^{2} metrics.

23 Hilbert space adjoints

Let (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) be Hilbert spaces, both real or both complex, and let ∥ · ∥_{V} and ∥ · ∥_{W} be the corresponding norms on V and W, respectively. If T is a bounded linear mapping from V into W, then it is well known that there is a unique bounded linear mapping T^{∗} from W into V such that

⟨T (v), w⟩_{W} = ⟨v, T^{∗}(w)⟩_{V} (23.1)

for every v ∈ V and w ∈ W. This defines a mapping T ↦ T^{∗} from BL(V, W) into BL(W, V), which is real-linear in the real case, and conjugate-linear in the complex case. One can check that

(T^{∗})^{∗} = T (23.2)

for every T ∈ BL(V, W), directly from this characterization of the adjoint. If V = W with the same inner product, and I is the identity operator on V, then I is a bounded linear mapping from V into itself, and I^{∗} = I.

If T ∈ BL(V, W), then

|⟨v, T^{∗}(w)⟩_{V} | = |⟨T (v), w⟩_{W} | ≤ ∥T ∥_{op,V W} ∥v∥_{V} ∥w∥_{W} (23.3)

for every v ∈ V and w ∈ W, using the Cauchy–Schwarz inequality and the definition of the operator norm ∥T ∥_{op,V W} in the second step. This implies that

∥T^{∗}(w)∥_{V} ≤ ∥T ∥_{op,V W} ∥w∥_{W} (23.4)

for every w ∈ W, by taking v = T^{∗}(w)/∥T^{∗}(w)∥_{V} when T^{∗}(w) ≠ 0. It follows that ∥T^{∗}∥_{op,W V} ≤ ∥T ∥_{op,V W} . (23.5)

The opposite inequality can be obtained similarly, or by applying this inequality to T^{∗} in place of T, and using (23.2). It follows that

∥T^{∗}∥_{op,W V} = ∥T ∥_{op,V W} . (23.6)

Let (Z, ⟨·, ·⟩_{Z}) be another Hilbert space, which is real or complex depending on whether V, W are real or complex. Also let T_{1} be a bounded linear mapping from V into W, and let T_{2} be a bounded linear mapping from W into Z, so that their composition T_{2} ∘T_{1} is a bounded linear mapping from V into Z. Note that the adjoints T_{1}^{∗} and T_{2}^{∗} of T_{1} and T_{2} map W into V and Z into W, respectively. If v ∈ V and z ∈ Z, then

(23.7)  ⟨(T_{2} ∘ T_{1})(v), z⟩_{Z} = ⟨T_{2}(T_{1}(v)), z⟩_{Z} = ⟨T_{1}(v), T_{2}^{∗}(z)⟩_{W}
= ⟨v, T_{1}^{∗}(T_{2}^{∗}(z))⟩_{V} .

This implies that (T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗}. (23.8)

Let T be a bounded linear mapping from V into W again, so that T^{∗} is a bounded linear mapping from W into V. Thus T^{∗} ∘ T is a bounded linear mapping from V into itself, with

∥T^{∗} ∘ T ∥_{op,V V} ≤ ∥T ∥_{op,V W} ∥T^{∗}∥_{op,W V} = ∥T ∥_{op,V W}^{2} , (23.9)

using (23.6) in the second step. If v ∈ V, then

⟨(T^{∗} ∘ T)(v), v⟩_{V} = ⟨T^{∗}(T (v)), v⟩_{V} = ⟨T (v), T (v)⟩_{W} = ∥T (v)∥_{W}^{2} . (23.10)

This implies that ∥T (v)∥_{W}^{2} ≤ ∥T^{∗} ∘ T ∥_{op,V V} ∥v∥_{V}^{2} , (23.11)

using the Cauchy–Schwarz inequality. It follows that

∥T ∥_{op,V W}^{2} ≤ ∥T^{∗} ∘ T ∥_{op,V V} , (23.12)

and hence ∥T^{∗} ∘ T ∥_{op,V V} = ∥T ∥_{op,V W}^{2} , (23.13)

because of (23.9).

24 Isometric linear mappings

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. Also let N_{V}, N_{W} be q_{V}, q_{W} -seminorms on V, W with respect to | · | on k, respectively, for some q_{V} , q_{W} > 0. A linear mapping T from V into W is said to be an isometry if

N_{W} (T (v)) = N_{V} (v) (24.1)

for every v ∈ V. This implies that T is bounded with respect to N_{V} and N_{W}, with the corresponding operator q_{W} -seminorm ∥T ∥_{op,V W} equal to 1 when N_{V} (v) > 0 for some v ∈ V. If N_{V} is a q_{V} -norm on V, then (24.1) implies that T is injective.

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} is a isometric linear mapping from V into W, and T_{2} is an isometric linear mapping from W into Z, then their composition T_{2} ∘ T_{1} is an isometric linear mapping from V into Z.

Let T be a one-to-one linear mapping from V onto W. If T^{−1} is a bounded linear mapping from W into V, then

N_{V} (T^{−1}(w)) ≤ ∥T^{−1}∥_{op,W V} N_{W} (w) (24.2)

for every w ∈ W. This is the same as saying that

N_{V} (v) ≤ ∥T^{−1}∥_{op,W V} N_{W} (T (v)) (24.3) for every v ∈ V, by taking w = T (v). If T and T^{−1} are both bounded linear mappings, with ∥T ∥_{op,V W} , ∥T^{−1}∥_{op,W V} ≤ 1, (24.4)

then it is easy to see that T is an isometric linear mapping. Of course, if T is an isometric linear mapping, then T^{−1} is an isometric linear mapping from W onto V.

Let k = R or C with the standard absolute value function, and suppose that (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) are inner product spaces over k. If a linear mapping T from V into W is an isometry with respect to the corresponding norms ∥ · ∥_{V} and ∥ · ∥_{W}, respectively, then one can check that

⟨T (v), T (v′)⟩_{W} = ⟨v, v′⟩_{V} (24.5)

for every v, v′ ∈ V, using polarization identities. Conversely, (24.5) implies that T is an isometry, by taking v = v′. Suppose now that V and W are Hilbert spaces, and let T be a bounded linear mapping from V into W. Observe that

⟨(T^{∗} ∘ T)(v), v′⟩_{V} = ⟨T^{∗}(T (v)), v′⟩_{V} = ⟨T (v), T (v′)⟩_{W} (24.6)

for every v, v′ ∈ V, where T^{∗} is the adjoint of T, as in the previous section. Thus T is an isometry if and only if

⟨(T^{∗} ∘ T)(v), v′⟩_{V} = ⟨v, v′⟩_{V} (24.7)

for every v, v′ ∈ V. It follows that T is an isometry if and only if

T^{∗} ∘ T = I_{V} , (24.8)

where I is the identity mapping on V.

A one-to-one linear mapping T from V onto W is said to be unitary if (24.5) holds for every v, v′ ∈ V, which is equivalent to asking that T be an isometry, as before. In this case, T^{∗} is a one-to-one linear mapping from W onto V, and

T^{∗} = T^{−1}, (24.9)

by (24.8).

25 C^{∗} Algebras

Let A be an algebra over the complex numbers. A conjugate-linear mapping

x ↦ x^{∗} (25.1)

from A into itself is said to be an involution on A if

(x y)^{∗} = y^{∗} x^{∗} (25.2)

and (x^{∗})^{∗} = x (25.3) for every x, y ∈ A. Let us suppose from now on in this section that A has a nonzero multiplicative identity element e. It is easy to see that

e^{∗} = e, (25.4)

using (25.2).

Let ∥ · ∥ be a submultiplicative norm on A with respect to the standard absolute value function on C, and suppose that A is complete with respect to the metric associated to ∥ · ∥. If we also have that

∥x^{∗} x∥ = ∥x∥^{2} (25.5)

for every x ∈ A, then A is said to be a C^{∗} algebra. Using (25.5) and submultiplicativity, we get that ∥x∥^{2} ≤ ∥x^{∗}∥ ∥x∥ (25.6)

for every x ∈ A, and hence ∥x∥ ≤ ∥x^{∗}∥. (25.7)

Applying this to x^{∗} instead of x, we get that ∥x^{∗}∥ ≤ ∥x∥, so that

∥x^{∗}∥ = ∥x∥ (25.8)

for every x ∈ A. Note that ∥e∥ = 1, (25.9)

by (25.4) and (25.5).

Let X be a nonempty topological space, and consider the space C_{b}(X, C) of bounded continuous complex-valued functions on X, using the standard absolute value function on C. This is a commutative algebra over C with respect to pointwise addition and multiplication of functions, and the constant function equal to 1 on X is the multiplicative identity element in C_{b}(X, C). The supremum norm ∥f∥_{sup} on C_{b}(X, C) associated to the standard absolute value function on C is a submultiplicative norm on C_{b}(X, C), and C_{b}(X, C) is complete with respect to the corresponding supremum metric. If f ∈ C_{b}(X, C), then the complex-conjugate \overline{f} of f is an element of C_{b}(X, C) too, and f ↦ \overline{f} defines an involution on C_{b}(X, C). It is easy to see that

∥f \overline{f}∥_{sup} = ∥|f|^{2}∥_{sup} = ∥f∥_{sup}^{2} (25.10)

for every f ∈ C_{b}(X, C), so that C_{b}(X, C) is a C^{∗} algebra with respect to the supremum norm.

Let (V, ⟨·, ·⟩) be a complex Hilbert space, let ∥ · ∥ be the norm on V corresponding to the inner product, and consider the algebra BL(V) of bounded linear mappings from V into itself with respect to ∥ · ∥. The identity operator I on V is the multiplicative identity element in BL(V), and let us suppose that V ≠ {0}, so that I ≠ 0. The mapping from T ∈ BL(V) to its adjoint T^{∗} defines an involution on V, as in Section 23. The operator norm ∥T ∥_{op} on BL(V) corresponding to ∥ · ∥ is submultiplicative on BL(V), and BL(V) is complete with respect to the associated metric, because V is complete with respect to the metric associated to ∥ · ∥, by hypothesis. Remember that ∥T ∥_{op} satisfies (25.5), as in (23.13), so that BL(V) is a C^{∗} algebra with respect to the operator norm.

A famous theorem implies that every commutative C^{∗} algebra can be realized as the algebra C(X, C) of continuous complex-valued functions on a nonempty compact Hausdorff topological space X, using the supremum norm and complexconjugation as the involution, as before.

26 Some weighted conditions

Let X be a nonempty set, and let w be a positive real-valued function on X. Also let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let V be a vector space over k with a q_{V} -seminorm N_{V} with respect to | · | on k for some q_{V} > 0. A V -valued function f on X is said to be bounded with respect to w on X and N_{V} on V if N_{V} (f(x)) w(x) is bounded as a nonnegative real-valued function on X. In this case, we put

∥f∥_{∞,w} = ∥f∥_{ℓ_{w}^{∞}(X,V)} = sup_{x∈X}(N_{V} (f(x)) w(x)). (26.1)

One can check that the space ℓ_{w}^{∞}(X, V) of these functions is a linear subspace of the space c(X, V) of all V -valued functions on X, and that (26.1) defines a q_{V} -seminorm on ℓ_{w}^{∞}(X, V) with respect to | · | on k. If N_{V} is a q_{V} -norm on V, then (26.1) is a q_{V} -norm on ℓ_{w}^{∞}(X, V). If V is also complete with respect to the q_{V} -metric associated to N_{V}, then ℓ_{w}^{∞}(X, V) is complete with respect to the q_{V} -metric associated to (26.1), by standard arguments.

Let r be a positive real number. A V -valued function f on X is said to be r-summable with respect to w on X and N_{V} on V if N_{V} (f(x))^{r} w(x)^{r} is summable as a nonnegative real-valued function on X. Let ℓ_{w}^{r}(X, V) be the space of these functions, and put

∥f∥_{r,w} = ∥f∥_{ℓ_{w}^{r}(X,V)} = ( ∑_{x∈X} N_{V} (f(x))^{r} w(x)^{r})^{1/r} (26.2)

for every f ∈ ℓ_{w}^{r}(X, V). One can verify that ℓ_{w}^{r}(X, V) is a linear subspace of c(X, V), and that (26.2) is an r-seminorm on ℓ_{w}^{r}(X, V) with respect to | · | on k when r ≤ q_{V}, and a q_{V} -seminorm on ℓ_{w}^{r}(X, V) when q_{V} ≤ r, as in Section 15. If N_{V} is a q_{V} -norm on V, then (26.2) is an r-norm on ℓ_{w}^{r}(X, V) when r ≤ q_{V}, and a q_{V} -norm on ℓ_{w}^{r}(X, V) when r ≥ q_{V}. If V is also complete with respect to the q_{V} -metric associated to N_{V}, then ℓ_{w}^{r}(X, V) is complete with respect to the r or q_{V} -metric associated to (26.2), as appropriate. If 0 < r_{1} ≤ r_{2} ≤ ∞, then

ℓ_{w}^{r_{1}}(X, V) ⊆ ℓ_{w}^{r_{2}}(X, V), (26.3)

and ∥f∥_{r_{2},w} ≤ ∥f∥_{r_{1},w} (26.4) for every f ∈ ℓ_{w}^{r_{1}}(X, V). This is basically the same as in Section 16. More precisely, if f is a V -valued function on X, then one can reduce to the earlier statements applied to N_{V} (f(x)) w(x) as a nonnegative real-valued function on X.

Similarly, a V -valued function f on X is said to vanish at infinity with respect to w on X and N_{V} on V if N_{V} (f(x)) w(x) vanishes at infinity as a nonnegative real-valued function on X. It is easy to see that the space c_{0,w}(X, V) of these functions is a linear subspace of c(X, V). Observe that

c_{0,w}(X, V) ⊆ ℓ_{w}^{∞}(X, V), (26.5)

and that ℓ_{w}^{r}(X, V) ⊆ c_{0,w}(X, V) (26.6)

when 0 < r < ∞, as before. One can check that c_{0,w}(X, V) is a closed set in ℓ_{w}^{∞}(X, V) with respect to the q_{V} -semimetric associated to (26.1).

Remember that c_{00}(X, V) is the space of V -valued functions on X with finite support. Clearly c_{00}(X, V) ⊆ ℓ_{w}^{r}(X, V) (26.7)

for every r > 0, and thus

c_{00}(X, V) ⊆ c_{0,w}(X, V) (26.8)

in particular. As before, one can verify that c_{00}(X, V) is dense in c_{0,w}(X, V) with respect to the q_{V} -semimetric associated to (26.1). If 0 < r < ∞, then c_{00}(X, V) is dense in ℓ_{w}^{r}(X, V) with respect to the r or q_{V} -semimetric associated to (26.2), as appropriate.

Let us now take k = R or C, with the standard absolute value function, and V = k. Let f, g be real or complex-valued functions on X that are 2-summable or equivalently square-summable with respect to w, so that |f|^{2} w^{2} and |g|^{2} w^{2} are summable as nonnegative real-valued functions on X. This implies that |f| |g| w^{2} is also summable as a nonnegative real-valued function on X, as in Section 22. As before,

⟨f, g⟩ = ⟨f, g⟩_{ℓ_{w}^{2}(X,R)} = ∑_{x∈X} f(x) g(x) w(x)^{2} (26.9)

and

⟨f, g⟩ = ⟨f, g⟩_{ℓ_{w}^{2}(X,C)} = ∑_{x∈X} f(x) \overline{g(x)} w(x)^{2} (26.10)

define inner products on ℓ_{w}^{2}(X, R) and ℓ^{2}(X, C), respectively. In both cases,

⟨f, f⟩ = ∑_{x∈X} |f(x)|^{2} w(x)^{2} = ∥f∥_{2,w}^{2}, (26.11)

so that ∥f∥_{2,w} is the norm associated to the inner product.

27 Lebesgue spaces

Let (X, A, μ) be a measure space, so that X is a set, A is a σ-algebra of measurable subsets of X, and μ is a nonnegative countably-additive measure defined on elements of A. If f is a nonnegative real-valued function on X that is measurable with respect to A, then the Lebesgue integral ∫_{X} f dμ can be defined as a nonnegative extended real number in the usual way. Remember that the Lebesgue spaces L^{q}(X, R) and L^{q}(X, C) are defined for 0 < q ≤ ∞, and consist of (equivalence classes of) real and complex-valued functions f on X that are measurable with respect to A (and equal almost everywhere with respect to μ). If q < ∞, then |f(x)|^{q} is integrable on X with respect to μ, so that

∥f∥_{q} = ( ∫_{X} |f|^{q} dμ)^{1/q} (27.1)

is defined as a nonnegative real number. If q = ∞, then f is essentially bounded on X with respect to μ, and ∥f∥_{∞} is defined to be the essential supremum of |f| on X with respect to μ. It is well known that ∥f∥_{q} defines a norm on L^{q}(X, R) and L^{q}(X, C) with respect to the standard absolute value functions on R and C, respectively, when 1 ≤ q ≤ ∞. If 0 < q ≤ 1, then it is easy to see that (27.1) defines a q-norm on L^{q}(X, R) and L^{q}(X, C). It is also well known that L^{q}(X, R) and L^{q}(X, C) are complete with respect to the metric or q-metric associated to ∥f∥_{q}, as appropriate, for every q > 0.

Suppose that 0 < q_{1}, q_{2}, q_{3} ≤ ∞ satisfy

1/q_{3} = 1/q_{1} + 1/q_{2}, (27.2)

where 1/∞ is interpreted as being equal to 0, as usual. If f ∈ L^{q_{1}}(X, R) or L^{q_{1}}(X, C) and g ∈ L^{q_{2}}(X, R) or L^{q_{2}}(X, C), then Hölder’s inequality implies that f g ∈ L^{q_{3}}(X, R) or L^{q_{3}}(X, C), as appropriate, with

∥f g∥_{q_{3}} ≤ ∥f∥_{q_{1}} ∥g∥_{q_{2}}. (27.3)

This is often stated with q_{3} = 1, and it is easy to reduce to this case. Let us now take q_{1} = q_{2} = 2, so that q_{3} = 1, and (27.3) is an integral version of the Cauchy–Schwarz inequality. It is easy to see that

⟨f, g⟩ = ⟨f, g⟩_{L^{2}(X,R)} = ∫_{X} f g dμ (27.4)

defines an inner product on L^{2}(X, R), and that

⟨f, g⟩ = ⟨f, g⟩_{L^{2}(X,C)} = ∫_{X} f \overline{g} dμ (27.5)

defines an inner product on L^{2}(X, C). In both cases, the norm associated to the inner product is the L^{2} norm. Thus L^{2}(X, R) and L^{2}(X, C) are Hilbert spaces over the real and complex numbers, respectively, with respect to these inner products.

Let X be a nonempty set, and let w_{0}, w_{1} be positive real-valued functions on X. If t is a positive real number with t < 1, then put

w_{t}(x) = w_{0}(x)^{1−t} w_{1}(x)^{t} (27.6)

for every x ∈ X. This defines another positive real-valued function on X, which reduces to w_{0}, w_{1} when t = 0, 1, respectively. 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_{V} be a q_{V} -seminorm on V with respect to | · | on k for some q_{V} > 0. Suppose that f is a V -valued function on X that is bounded with respect to N_{V} on V and both w_{0} and w_{1} on X, as in the previous section. If x ∈ X and 0 < t < 1, then

(27.7) N_{V} (f(x)) w_{t}(x) = (N_{V} (f(x)) w_{0}(x))^{1−t} (N_{V} (f(x)) w_{1}(x))^{t}
≤ ∥f∥_{∞,w_{0}}^{1−t} ∥f∥_{∞,w_{1}}^{t},

where ∥f∥_{∞,w} is as in (26.1). This implies that f is bounded with respect to w_{t} on X and N_{V} on V, with

∥f∥_{∞,w_{t}} ≤ ∥f∥_{∞,w_{0}}^{1−t} ∥f∥_{∞,w_{1}}^{t}. (27.8)

If we also have that f vanishes at infinity with respect to N_{V} on V and either w_{0} or w_{1} on X, then one can check that f vanishes at infinity with respect to N_{V} on V and w_{t} on X when 0 < t < 1, using the first step in (27.7).

Let r be a positive real number, and suppose now that f is a V -valued function on X that is r-summable with respect to N_{V} on V and both w_{0} and w_{1} on X, as in the previous section. If 0 < t < 1, then f is r-summable with respect to w_{t} on X and N_{V} on V, with

∥f∥_{r,w_{t}} ≤ ∥f∥_{r,w_{0}}^{1−t} ∥f∥_{r,w_{1}}^{t}, (27.9)

and where ∥f∥_{r,w} is as in (26.2). To see this, put q_{1} = r/(1 − t), q_{2} = r/t, and q_{3} = r, which automatically satisfy (27.2). The r-summability of f with respect to w_{0} on X implies that (N_{V} (f(x)) w_{0}(x))^{1−t} is q_{1}-summable on X in the usual sense, with

∥f∥_{r,w_{0}}^{1−t} = ( ∑_{x∈X}(N_{V} (f(x)) w_{0}(x))^{(1−t) q_{1}})^{1/q_{1}}. (27.10)

Similarly, the r-summability of f with respect to w_{1} on X is the same as saying that (N_{V} (f(x)) w_{1}(x))^{t} is q_{2}-summable on X in the usual sense, with

∥f∥_{r,w_{1}}^{t} = ( ∑_{x∈X}(N_{V} (f(x)) w_{1}(x))^{t q_{2}})^{1/q_{2}}. (27.11)

The r-summability of f with respect to w_{t} is the same as the q_{3}-summability of N_{V} (f(x)) w_{t}(x) on X in the usual sense, with

∥f∥_{r,w_{t}} = ( ∑_{x∈X}(N_{V} (f(x)) w_{t}(x))^{q_{3}})^{1/q_{3}}. (27.12)

Using the first step in (27.7) and Hölder’s inequality for sums, we get that the first two summability conditions just mentioned imply the third one, and that (27.12) is less than or equal to the product of (27.10) and (27.11). This means that f is r-summable on X with respect to w_{t}, and that (27.9) holds, as desired.

Part III

Power series

28 Some spaces of power series

Let k be a field, and let T be an indeterminate. The space of formal power series

f(T) = ∑_{j=0}^{∞} f_{j} T^{j} (28.1)

with coefficients in k is denoted k[[T]], as in Section 2. This can be identified with the space c(Z_{+} ∪ {0}, k) of all k-valued functions on the set Z_{+} ∪ {0} of nonnegative integers, as before. Similarly, the space k[[T]] of formal polynomials in T with coefficients in k corresponds to the space c_{00}(Z_{+} ∪ {0}, k) of k-valued functions on Z_{+} ∪ {0} with finite support.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let r be a positive real number. Put w_{r}(j) = r^{j} (28.2)

for every nonnegative integer j, which defines a positive real-valued function on Z_{+} ∪ {0}. If q is a positive real number, then put

k_{r}^{q}[[T]] = {f(T) ∈ k[[T]] : ∑_{j=0}^{∞} |f_{j}|^{q} r^{q j} < ∞}, (28.3)

and put

∥f(T)∥_{q,r} = ∥f(T)∥_{k_{r}^{q}[[T]]} = ( ∑_{j=0}^{∞} |f_{j}|^{q} r^{q j})^{1/q} (28.4)

for every f(T) ∈ k_{r}^{q}[[T]]. Equivalently, (28.3) can be identified with the space ℓ_{w_{r}}^{q}(Z_{+} ∪ {0}, k) of k-valued functions on Z_{+} ∪ {0} that are q-summable with respect to w_{r} on Z_{+} ∪ {0} and | · | on k, as a linear subspace of c(Z_{+} ∪ {0}, k), as in Section 26. Similarly, (28.4) corresponds to ∥ · ∥_{ℓ_{wr}^{q} (Z_{+}∪{0},k)} applied to j ↦ f_{j} as a k-valued function on Z_{+} ∪ {0}. Thus (28.4) defines a q-norm on (28.3) with respect to | · | on k when q ≤ q_{k}, and a q_{k}-norm on (28.3) when q ≥ q_{k}, as before. Note that

k[T] ⊆ k_{r}^{q}[[T]], (28.5) and that k[T] is dense in k_{r}^{q}[[T]] with respect to the q or q_{k}-metric associated to (28.4).

Put

(28.6) k_{r}^{∞}[[T]] = {f(T) ∈ k[[T]] : |f_{j}| r^{j} is bounded on Z_{+} ∪ {0}},

and

∥f(T)∥_{∞,r} = ∥f(T)∥_{k_{r}^{∞}[[T]]} = sup_{j≥0}(|f_{j}| r^{j}) (28.7)

for every f(T) ∈ k_{r}^{∞}[[T]]. As before, (28.6) can be identified with the space ℓ_{w_{r}}^{∞}(Z_{+}∪{0}, k) of k-valued functions on Z_{+}∪{0} that are bounded with respect to w_{r} on Z_{+} ∪ {0} and | · | on k, and (28.7) corresponds to ∥ · ∥_{ℓ_{wr}^{∞} (Z_{+}∪{0},k)} applied to j ↦ f_{j} as a k-valued function on Z_{+} ∪ {0}. In particular, (28.6) is a linear subspace of k[[T]], and (28.7) is a q_{k}-norm on (28.6) with respect to | · | on k. If 0 < q ≤ q̃ ≤ ∞, then

k_{r}^{q}[[T]] ⊆ k_{q} ; r[[T]], (28.8)

and

∥f(T)∥_{_{q},r} ≤ ∥f(T)∥_{q,r} (28.9)

for every f(T) ∈ k_{r}^{q}[[T]], as in (26.3) and (26.4).

Put

k_{0,r}[[T]] = {f(T) ∈ k[[T]] : lim_{j→∞} |f_{j}| r^{j} = 0}, (28.10)

which can be identified with the space c_{0,w_{r}}(Z_{+} ∪ {0}, k) of k-valued functions on Z_{+} ∪ {0} that vanish at infinity with respect to w_{r} on Z_{+} ∪ {0} and | · | on k. Thus (28.10) is a closed linear subspace of (28.6) with respect to the q_{k}-metric associated to (28.7). We also have that

k[T] ⊆ k_{0,r}[[T]], (28.11)

and that k[T] is dense in k_{0,r}[[T]] with respect to the q_{k}-metric associated to (28.7). If 0 < q < ∞, then

k_{r}^{q}[[T]] ⊆ k_{0,r}[[T]], (28.12)

as in (26.6).

Suppose for the moment that | · | is the trivial absolute value function on k. In this case, k_{r}^{q}[[T]] = k[[T]] (28.13)

when 0 < q < ∞ and 0 < r < 1, and when q = ∞ and 0 < r ≤ 1. Otherwise,

k_{r}^{q}[[T]] = k[T] (28.14)

when 0 < q < ∞ and r ≥ 1, and when q = ∞ and r > 1. Similarly,

k_{0,r}[[T]] = k[[T]] (28.15)

when 0 < r < 1, and k_{0,r}[[T]] = k[T] (28.16)

when r ≥ 1.

29 Submultiplicativity

Let k be a field, and let T be an indeterminate. Also let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} and g(T) = ∑_{l=0}^{∞} g_{l} T^{l} be formal power series in T with coefficients in k. Remember that their product f(T) g(T) = h(T) = ∑_{n=0}^{∞} h_{n} T^{n} is defined by

h_{n} = ∑_{j=0}^{n} f_{j} g_{n−j} (29.1)

for every n ≥ 0, as in Section 2.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let 0 < q ≤ q_{k} be given. Thus | · | may be considered as a q-absolute value function on k as well. If q < ∞, then

|h_{n}|^{q} ≤ ∑_{j=0}^{n} |f_{j}|^{q} |g_{n−j}|^{q} (29.2)

for every n ≥ 0. If q = ∞, then

|h_{n}| ≤ max_{0≤j≤n}(|f_{j}| |g_{n−j}|) (29.3)

for every n ≥ 0.

Let r be a positive real number. If q < ∞, then

|h_{n}|^{q} r^{q n} ≤ ∑_{j=0}^{n} |f_{j}|^{q} |g_{n−j}|^{q} r^{q n} = ∑_{j=0}^{n}(|f_{j}|^{q} r^{q j}) (|g_{n−j}|^{q} r^{q (n−j)}) (29.4)

for every n ≥ 0, using (29.2) in the first step. If q = ∞, then

|h_{n}| r^{n} ≤ max_{0≤j≤n}(|f_{j}| |g_{n−j}|) r^{n} = max_{0≤j≤n}((|f_{j}| r^{j}) (|g_{n−j}| r^{n−j})) (29.5)

for every n ≥ 0, using (29.3) in the first step.

Suppose that f(T), g(T) ∈ k_{r}^{q}[[T]]. If q < ∞, then

∑_{n=0}^{∞} |h_{n}|^{q} r^{q n} ≤ ∑_{n=0}^{∞} ∑_{j=0}^{n}(|f_{j}|^{q} r^{q j}) (|g_{n−j}|^{q} r^{q (n−j)}), (29.6)

by (29.4). The right side is equal to

( ∑_{j=0}^{∞} |f_{j}|^{q} r^{q j}) ( ∑_{l=0}^{∞} |g_{l}|^{q} r^{q l}). (29.7)

More precisely, the right side of (29.4) corresponds to the nth term of the Cauchy product of these two series, as in Section 20. It follows that f(T) g(T) = h(T) is an element of k_{r}^{q}[[T]] too, with

∥f(T) g(T)∥_{q,r} ≤ ∥f(T)∥_{q,r} ∥g(T)∥_{q,r}. (29.8)

If q = ∞, then we get that |h_{n}| r^{n} ≤ ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r} (29.9) for every n ≥ 0, using (29.5). This implies that f(T) g(T) = h(T) ∈ k_{r}^{∞}[[T]], and satisfies (29.8). If f(T), g(T) ∈ k_{0,r}[[T]], then one can check that

lim_{n→∞} |h_{n}| r^{n} = 0, (29.10)

using (29.5) again. This means that f(T) g(T) = h(T) ∈ k_{0,r}[[T]] in this situation. Thus k_{r}^{q}[[T]] is a subalgebra of k[[T]] when q ≤ q_{k}, and k_{0,r}[[T]] is a subalgebra of k[[T]] when q_{k} = ∞.

30 Radius of convergence

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let T be an indeterminate. Suppose that f(T) = ∑_{j=0}^{∞} f_{j} T^{j} ∈ k_{r}^{q}[[T]] for some q > 0 and 0 < r < ∞. If 0 < r_{1} ≤ r, then it is easy to see that f(T) ∈ k_{r_{1}}^{q}[[T]], with ∥f(T)∥_{q,r_{1}} ≤ ∥f(T)∥_{q,r}. (30.1)

Suppose that f(T) ∈ k_{r}^{∞}[[T]] and 0 < r_{1} < r, and observe that |f_{j}| r_{1}^{j} = |f_{j}| r^{j} (r_{1}/r)^{j} ≤ ∥f(T)∥_{∞,r} (r_{1}/r)^{j} (30.2) for every j ≥ 0. This implies that

lim_{j→∞} |f_{j}| r_{1}^{j} = 0, (30.3)

so that f(T) ∈ k_{0,r_{1}}[[T]]. If 0 < q < ∞, then we get that

(30.4)  ∑_{j=0}^{∞} |f_{j}|^{q} r_{1}^{q j} ≤ ∥f(T)∥_{∞,r}^{q} ∑_{j=0}^{∞}(r_{1}/r)^{q j}
= ∥f(T)∥_{∞,r}^{q} (1 − (r_{1}/r)^{q})^{−1}.

It follows that f(T) ∈ k_{r_{1}}^{q}[[T]], with ∥f(T)∥_{q,r_{1}} ≤ ∥f(T)∥_{∞,r} (1 − (r_{1}/r)^{q})^{−1/q}. (30.5)

The radius of convergence of f(T) ∈ k[[T]] may be defined as a nonnegative extended real number as the supremum of the nonnegative real numbers r such that (30.6) |f_{j}| r^{j} is bounded on Z_{+} ∪ {0}. Of course, (30.6) holds trivially when r = 0, so that the set of r ≥ 0 with this property is nonempty. Equivalently, the radius of convergence of f(T) is the supremum of the set of r ≥ 0 such that

lim_{j→∞} |f_{j}| r^{j} = 0. (30.7)

If q is any positive real number, then the radius of convergence of f(T) can also be characterized as the supremum of the set of r ≥ 0 such that

∑_{j=0}^{∞} |f_{j}|^{q} r^{q j} < ∞. (30.8)

Indeed, (30.8) implies (30.7), and (30.7) implies (30.6). If (30.6) holds for some r > 0, then the analogues of (30.7) and (30.8) hold with r replaced by any r_{1} ∈ [0, r), as in the preceding paragraph. Hence the suprema of these sets are the same. If r ≥ 0 is strictly less than the radius of convergence of f(T), then f(T) satisfies each of these conditions.

If t ∈ k and t ≠ 0, then it is easy to see that the radius of convergence of t f(T) is equal to the radius of convergence of f(T). If g(T) ∈ k[[T]] too, then the radius of convergence of f(T) + g(T) is greater than or equal to the minimum of the radii of convergence of f(T) and g(T). Similarly, the radius of convergence of f(T) g(T) is greater than or equal to the minimum of the radii of convergence of f(T) and g(T). This follows from (29.8), where 0 < q ≤ q_{k}, as before.

If 0 < r ≤ ∞, then we let k_{r}[[T]] be the space of f(T) ∈ k[[T]] whose radius of convergence is greater than or equal to r. This is a subalgebra of k[[T]], by the remarks in the preceding paragraph.

31 A multiplicativity condition

Let k be a field with an ultrametric absolute value function | · |, and let T be an indeterminate. Suppose that f(T) = ∑_{j=0}^{∞} f_{j} T^{j} and g(T) = ∑_{l=0}^{∞} g_{l} T^{l} are elements of k_{0,r}[[T]] for some positive real number r. This implies that f(T) g(T) ∈ k_{0,r}[[T]], with

∥f(T) g(T)∥_{∞,r} ≤ ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r}, (31.1)

as in Section 29. In fact, it is well known that

∥f(T) g(T)∥_{∞,r} = ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r} (31.2)

under these conditions. To see this, it suffices to verify that

∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r} ≤ ∥f(T) g(T)∥_{∞,r}. (31.3)

Of course, this is trivial when f(T) = 0 or g(T) = 0. Thus we may suppose that f(T) ≠ 0 and g(T) ≠ 0.

By hypothesis, |f_{j}| r^{j} → 0 as j → ∞, and |g_{l}| r^{l} → 0 as l → ∞. These conditions imply that the suprema in the definitions of ∥f(T)∥_{∞,r} and ∥g(T)∥_{∞,r} are attained. Let j_{0} and l_{0} be the smallest nonnegative integers such that

|f_{j_{0}}| r^{j_{0}} = ∥f(T)∥_{∞,r} (31.4) and |g_{l_{0}}| r^{l_{0}} = ∥g(T)∥_{∞,r}. (31.5)

In particular, ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r} = |f_{j_{0}}| |g_{l_{0}}| r^{j_{0}+l_{0}}. (31.6)

Remember that f(T) g(T) = h(T) = ∑_{n=0}^{∞} h_{n} T^{n}, where h_{n} = ∑_{j=0}^{n} f_{j} g_{n−j} for each n ≥ 0. Note that

|h_{j_{0}+l_{0}}| r^{j_{0}+l_{0}} ≤ ∥h(T)∥_{∞,r} = ∥f(T) g(T)∥_{∞,r}. (31.7)

To get (31.3), it is enough to check that

|f_{j_{0}}| |g_{l_{0}}| r^{j_{0}+l_{0}} ≤ |h_{j_{0}+l_{0}}| r^{j_{0}+l_{0}}. (31.8)

Clearly |f_{j}| r^{j} ≤ ∥f(T)∥_{∞,r} = |f_{j_{0}}| r^{j_{0}} (31.9)

for every j ≥ 0, with a strict inequality when j < j_{0}, by the minimality of j_{0}. Similarly, |g_{l}| r^{l} ≤ ∥g(T)∥_{∞,r} = |g_{l_{0}}| r^{l_{0}} (31.10)

for every l ≥ 0, with a strict inequality when l < l_{0}, by the minimality of l_{0}. It follows that |f_{j}| |g_{l}| r^{j+l} ≤ |f_{j_{0}}| |g_{l_{0}}| r^{j_{0}+l_{0}} (31.11)

for every j, l ≥ 0, with a strict inequality when either j < j_{0} or l < l_{0}. If j + l = j_{0} + l_{0}, then we get that

|f_{j}| |g_{l}| ≤ |f_{j_{0}}| |g_{l_{0}}|, (31.12)

with a strict inequality when either j < j_{0} or l < l_{0}.

We would like to use these inequalities to obtain that

|h_{j_{0}+l_{0}} − f_{j_{0}} g_{l_{0}}| < |f_{j_{0}}| |g_{l_{0}}|. (31.13)

Remember that h_{j_{0}+l_{0}} is equal to the sum of f_{j} g_{l} over j, l ≥ 0 with j+l = j_{0}+l_{0}. This condition on j + l implies that j ≠ j_{0} if and only if l ≠ l_{0}, in which case either j < j_{0} or l < l_{0}. This means that h_{j_{0}+l_{0}} − f_{j_{0}} g_{l_{0}} can be expressed as the sum of finitely many terms, each of which corresponds to a strict inequality in (31.12). Thus (31.13) follows from the ultrametric version of the triangle inequality.

Of course, |f_{j_{0}}| |g_{l_{0}}| ≤ max(|h_{j_{0}+l_{0}}|, |h_{j_{0}+l_{0}} − f_{j_{0}} g_{l_{0}}|), (31.14)

by the ultrametric version of the triangle inequality. This implies that

|f_{j_{0}}| |g_{l_{0}}| ≤ |h_{j_{0}+l_{0}}|, (31.15)

because of (31.13). Hence (31.8) holds, as desired. We also have that |h_{j_{0}+l_{0}}| is less than or equal to |f_{j_{0}}| |g_{l_{0}}|, by (31.13) and the ultrametric version of the triangle inequality. Thus we have equality in (31.15), even if this is not needed here.

32 Some continuity properties

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, let T be an indeterminate, and let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} be a formal power series in T with coefficients in k. Thus

sup_{j≥0}(|f_{j}| r^{j}) (32.1)

is defined as a nonnegative extended real number for every positive real number r. If 0 < r_{1} ≤ r, then

sup_{j≥0}(|f_{j}| r_{1}^{j}) (32.2)

is automatically less than or equal to (32.1). In fact, (32.2) tends to (32.1) as r_{1} → r−, which is to say as r_{1} ∈ (0, r) approaches r, with suitable interpretations for extended real numbers. This can be obtained from the continuity of r^{j} on R for each j ≥ 0.

Similarly, if q is a positive real number, then

∑_{j=0}^{∞} |f_{j}|^{q} r^{j} (32.3)

can be defined as a nonnegative extended real number for every positive real number r. As before,

∑_{j=0}^{∞} |f_{j}|^{q} r_{1}^{j} (32.4)

is automatically less than or equal to (32.3) when 0 < r_{1} ≤ r. One can check that (32.4) tends to (32.3) as r_{1} → r−, with suitable interpretations for extended real numbers. This uses the continuity of the partial sums

∑_{j=0}^{n} |f_{j}|^{q} r^{j} (32.5)

in r for each nonnegative integer n. Of course, (32.3) is the same as the supremum of (32.5) over all nonnegative integers n.

Suppose now that | · | is an ultrametric absolute value function on k, and that f(T), g(T) are elements of k_{r}^{∞}[[T]] for some positive real number r. Thus f(T) g(T) ∈ k_{r}^{∞}[[T]], with

∥f(T) g(T)∥_{∞,r} ≤ ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r}, (32.6)

as in Section 29. If 0 < r_{1} < r, then f(T), g(T) ∈ k_{0,r_{1}}[[T]], as in Section 30, and ∥f(T) g(T)∥_{∞,r_{1}} = ∥f(T)∥_{∞,r_{1}} ∥g(T)∥_{∞,r_{1}}, (32.7)

as in the previous section. It follows that

∥f(T) g(T)∥_{∞,r} = ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r}, (32.8)

by taking r_{1} → r− in (32.7).

33 Functions on Banach algebras

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let A be an algebra over k with a submultiplicative q-norm N with respect to | · | on k for some q > 0. Suppose that A has a multiplicative identity element e with N(e) = 1, and that A is complete with respect to the q-metric associated to N. Let T be an indeterminate, and let r be a positive real number. Suppose for the moment that q < ∞, and that f(T) = ∑_{j=0}^{∞} f_{j} T^{j} is an element of k_{r}^{q}[[T]]. If a ∈ A satisfies N(a) ≤ r, (33.1)

then we would like to put

f(a) = ∑_{j=0}^{∞} f_{j} a^{j}, (33.2)

where a^{j} is interpreted as being equal to e when j = 0. Of course,

N(a^{j}) ≤ N(a)^{j} ≤ r^{j} (33.3)

for each j ≥ 0, so that

∑_{j=0}^{∞} N(f_{j} a^{j})^{q} = ∑_{j=0}^{∞} |f_{j}|^{q} N(a^{j})^{q} ≤ ∑_{j=0}^{∞} |f_{j}|^{q} r^{q j} = ∥f(T)∥_{q,r}^{q}, (33.4)

where ∥f(T)∥_{q,r} is as in (28.4). In particular, this means that the series on the right side of (33.2) converges q-absolutely with respect to N. It follows that this series converges in A, because A is complete with respect to the q-metric associated to N, as in Section 9. We also get that

N(f(a)) ≤ ( ∑_{j=0}^{∞} N(f_{j} a^{j})^{q})^{1/q} ≤ ∥f(T)∥_{q,r}, (33.5)

using (9.9) in the first step.

Suppose now that q = ∞, and that f(T) ∈ k_{0,r}[[T]]. If a ∈ A satisfies (33.1), then N(f_{j} a^{j}) = |f_{j}| N(a^{j}) ≤ |f_{j}| r^{j} (33.6)

for each j ≥ 0, so that

lim_{j→∞} N(f_{j} a^{j}) = 0. (33.7)

This implies that the series on the right side of (33.2) converges in A, because A is complete with respect to the ultrametric associated to N, as in Section 9 again. In this situation, we have that

N(f(a)) ≤ max_{j≥0} N(f_{j} a^{j}) ≤ max_{j≥0}(|f_{j}| r^{j}) = ∥f(T)∥_{∞,r}, (33.8)

using (9.11) in the first step.

We may as well ask that q ≤ q_{k}, as in Section 7. Hence k_{r}^{q}[[T]] is a subalgebra of k[[T]], and k_{0,r}[[T]] is a subalgebra of k[[T]] when q = ∞, as in Section 29. If q < ∞, then f(T) ↦ f(a) (33.9)

defines a homomorphism from k_{r}^{q}[[T]] into A, as algebras over k. More precisely, this mapping is clearly linear, and products of elements of k_{r}^{q}[[T]] correspond to Cauchy products of series in A. Similarly, if q = ∞, then (33.9) defines an algebra homomorphism from k_{0,r}[[T]] into A.

Let f(T) be an element of k_{r}^{q}[[T]] when q < ∞, or an element of k_{0,r}[[T]] when q = ∞. In both cases, (33.2) defines a mapping from the closed ball

{a ∈ A : N(a) ≤ r} (33.10)

in A centered at 0 with radius r into A. Observe that the partial sums of (33.2) converge uniformly on (33.10), by estimating the errors as before. It is easy to see that the partial sums are uniformly continuous on (33.10), with respect to the q-metric on A associated to N, and its restriction to (33.10). This implies that (33.2) is uniformly continuous on (33.10) too, by standard arguments.

Suppose now that 0 < r ≤ ∞, and that f(T) is an element of the space k_{r}[[T]] defined in Section 30. If 0 < r_{1} < r, then f(T) is an element of k_{r_{1}}^{q}[[T]] and k_{0,r_{1}}[[T]], as before. If a ∈ A satisfies N(a) < r, then we can use this to define f(a) as an element of A, as in (33.2). This defines f(a) as a mapping from {a ∈ A : N(a) < r} (33.11)

into A. It is easy to see that this mapping is continuous, because its restriction to any closed ball in A centered at 0 with radius r_{1} < r is continuous, as in the preceding paragraph.

34 Holomorphic functions

In this section, we take k to be the field C of complex numbers, with the standard absolute value function. If U is a nonempty open subset of the complex plane, then let H(U) be the space of holomorphic functions on U. This is a subalgebra of the algebra C(U, C) of all continuous complex-valued functions on U. If 0 < r ≤ ∞, then put U_{r} = {z ∈ C : |z| < r}. (34.1)

Let T be an indeterminate, and let f(T) = ∑_{j=0}^{∞} f_{j} T^{j} be an element of the space C_{r}[[T]] of formal power series in T with complex coefficients and radius of convergence greater than or equal to r, as in Section 30. If z ∈ U_{r}, then put

f(z) = ∑_{j=0}^{∞} f_{j} z^{j}, (34.2)

where the series on the right converges absolutely, as in the previous section. It is well known that this defines a holomorphic function on U_{r}, and that every holomorphic function on U_{r} can be represented by a unique power series in this way.

If U is a nonempty open subset of C again, then let H^{∞}(U) be the space of bounded holomorphic functions on U. This is a subalgebra of the algebra C_{b}(U, C) of all bounded continuous complex-valued functions on U. Note that H^{∞}(U) contains all constant complex functions on U. It is well known that H^{∞}(U) is a closed set in C_{b}(U, C), with respect to the supremum metric.

Let r be a positive real number, and let

\overline{U_{r}} = \{z ∈ C : |z| ≤ r\} (34.3)

be the closed disk in C centered at 0 with radius r, which is the same as the closure in C of the open disk U_{r} centered at 0 with radius r. Let A(U_{r}) be the space of continuous complex-valued functions on \overline{U_{r}} that are holomorphic on U_{r}. This is a subalgebra of the algebra C(\overline{U_{r}}, C) of continuous complex-valued functions on \overline{U_{r}}. It is well known that A(U_{r}) is a closed set in C(\overline{U_{r}}, C) with respect to the supremum metric, as in the previous paragraph.

Let f(T) be an element of the space C_{r}^{1}[[T]] defined in Section 28, so that

∥f(T)∥_{1,r} = ∑_{j=0}^{∞} |f_{j}| r^{j} (34.4)

is finite. If z ∈ \overline{U_{r}}, then the series on the right side of (34.2) converges absolutely, and (34.2) defines a continuous complex-valued function on \overline{U_{r}}, as in the previous section. We also have that (34.2) is holomorphic on U_{r}, as before, so that (34.2) defines an element of A(U_{r}). Note that

|f(z)| ≤ ∥f(T)∥_{1,r} (34.5)

for every z ∈ \overline{U_{r}}.

Now let f(z) by any continuous complex-valued function on \overline{U_{r}}. If t ∈ C and |t| ≤ 1, then f(t z) defines a continuous complex-valued function on \overline{U_{r}}. It is well known that \overline{U_{r}} is compact with respect to the standard Euclidean metric on C, and hence that f(z) is uniformly continuous on \overline{U_{r}} with respect to the standard Euclidean metric on C and its restriction to \overline{U_{r}}. This implies that f(t z) tends to f(z) uniformly as a function of z on \overline{U_{r}}, as t ∈ \overline{U_{1}} tends to 1. Of course, f(t z) is holomorphic as a function of z on U_{r/|t|}. If |t| < 1, then r/|t| > r, and it follows that f(t z) can be represented by an absolutely convergent power series in z on \overline{U_{r}}. In particular, one can use this to approximate f(z) by polynomials in z uniformly on \overline{U_{r}}. More precisely, one can first approximate f(z) uniformly on \overline{U_{r}} by f(t z), by taking t ∈ U_{1} sufficiently close to 1. One can approximate f(t z) uniformly as a function of z ∈ \overline{U_{r}} by polynomials when |t| < 1, using the absolute convergence of its power series expansion in z on \overline{U_{r}}.

35 Rescaling power series

Let k be a field, and let T be an indeterminate. If t ∈ k and f(T) = ∑_{j=0}^{∞} f_{j} T^{j} is a formal power series in T with coefficients in k, then define R_{t}(f(T)) ∈ k[[T]] by

R_{t}(f(T)) = ∑_{j=0}^{∞} f_{j} t^{j} T^{j}. (35.1)

This may also be denoted (R_{t}(f))(T), or simply R_{t}(f). One can check that R_{t} defines a homomorphism from k[[T]] into itself, as an algebra over k. Of course, R_{1} is the identity mapping on k[[T]], and R_{0} reduces to the mapping from f(T) to f_{0}. If u is another element of k, then it is easy to see that

R_{t} ∘ R_{u} = R_{t u}. (35.2)

If t ≠ 0, then R_{t} is a one-to-one mapping from k[[T]] onto itself, with R_{t}^{−1} = R_{1/t}. Note that R_{t} maps k[T] into itself, and onto itself when t ≠ 0.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let r be a positive real number. Also let t ∈ k \ {0}, f(T) ∈ k[[T]], and 0 < q ≤ ∞ be given. It is easy to see that R_{t}(f(T)) is in the space k_{r}^{q}[[T]] defined in Section 28 if and only if f(T) ∈ k_{r |t|}^{q}[[T]]. In this case, we have that

∥R_{t}(f(T))∥_{q,r} = ∥f(T)∥_{q,r |t|}. (35.3)

Similarly, R_{t}(f(T)) is an element of the space k_{0,r}[[T]] defined earlier if and only if f(T) ∈ k_{0,r |t|}[[T]]. If 0 < r ≤ ∞, then R_{t}(f(T)) is in the space k_{r}[[T]] defined in Section 30 if and only if f(T) ∈ k_{r |t|}[[T]]. This follows from the previous statements, applied to radii less than r.

Let A be an algebra over k with a submultiplicative q-norm N with respect to | · | on k for some 0 < q ≤ q_{k}. Suppose that A has a multiplicative identity element e, and that A is complete with respect to the q-metric associated to N. Let t ∈ k \ {0} and a positive real number r be given, and let a be an element of A with N(a) ≤ r. Suppose for the moment that q < ∞, and that f(T) ∈ k_{r |t|}^{q}[[T]], so that R_{t}(f(T)) ∈ k_{r}^{q}[[T]]. Note that

N(t a) = |t| N(a) ≤ r |t|, (35.4)

so that f(t a) can be defined as an element of A as in Section 33. Similarly, (R_{t}(f))(a) can be defined as an element of A, and in fact

(R_{t}(f))(a) = ∑_{j=0}^{∞} f_{j} t^{j} a^{j} = ∑_{j=0}^{∞} f_{j} (t a)^{j} = f(t a). (35.5)

If q = ∞ and f(T) ∈ k_{0,r |t|}[[T]], then R_{t}(f(T)) ∈ k_{0,r}[[T]], f(t a) and (R_{t}(f))(a) can be defined as elements in A, and they are equal, as in (35.5).

Suppose now that 0 < r ≤ ∞ and f(T) ∈ k_{r |t|}[[T]], so that R_{t}(f(T)) is an element of k_{r}[[T]]. If a ∈ A satisfies N(a) < r, then N(t a) < r |t|, f(t a) and (R_{t}(f))(a) can be defined as elements of A, and they are equal, as in (35.5) again.

36 Hardy spaces

Let r be a positive real number, and let f be a holomorphic function on the open disk U_{r} in the complex plane centered at 0 with radius r. If q and r_{1} are positive real numbers, with r_{1} < r, then put

M_{q}(f, r_{1}) = (\frac{1}{2 π r_{1}} ∫_{|z|=r_{1}} |f(z)|^{q} |dz|)^{1/q}. (36.1)

More precisely, the integral is taken over the set of z ∈ C with |z| = r_{1}, with respect to arclength. Similarly, put

M_{∞}(f, r_{1}) = sup{|f(z)| : z ∈ C, |z| = r_{1}}. (36.2)

Equivalently,

M_{q}(f, r_{1}) = (\frac{1}{2 π} ∫_{|z|=1} |f(r_{1} z)|^{q} |dz|)^{1/q} (36.3)

when 0 < q < ∞, and

M_{∞}(f, r_{1}) = sup{|f(r_{1} z)| : z ∈ C, |z| = 1}. (36.4)

In particular, M_{q}(f, r_{1}) can be defined for r_{1} = 0, using (36.3) when q = ∞. It is well known that M_{q}(f, r_{1}) is monotonically increasing in r_{1} for each q > 0, which follows from the maximum principle when q = ∞. Note that M_{q}(f, r_{1}) is monotonically increasing in q as well, by the inequalities of Hölder or Jensen.

The Hardy space H^{q}(U_{r}) is defined for 0 < q ≤ ∞ as the space of holomorphic functions f on U_{r} such that M_{q}(f, r_{1}) is bounded for 0 ≤ r_{1} < r, in which case we put

∥f∥_{H^{q}(U_{r})} = sup_{0≤r_{1}<r} M_{q}(f, r_{1}). (36.5)

One can check that H^{q}(U_{r}) is a linear subspace of the space H(U_{r}) of all holomorphic functions on U_{r}. If q ≥ 1, then (36.5) defines a norm on H^{q}(U_{r}), with respect to the standard absolute value function on C. If 0 < q ≤ 1, then (36.5) defines a q-norm on H^{q}(U_{r}). It is easy to see that f is bounded on U_{r} if and only if M_{∞}(f, r_{1}) is bounded for 0 ≤ r_{1} < ∞, so that this definition of H^{∞}(U_{r}) is the same as the space of bounded holomorphic functions on U_{r}, as in Section 34. If 0 < q_{1} ≤ q_{2} ≤ ∞, then

H^{q_{2}}(U_{r}) ⊆ H^{q_{1}}(U_{r}), (36.6)

with ∥f∥_{H^{q_{1}} (U_{r})} ≤ ∥f∥_{H^{q_{2}} (U_{r})} (36.7)

for every f ∈ H^{q_{2}}(U_{r}). This follows from the monotonicity of M_{q}(f, r_{1}) in q.

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. Suppose that f(T) has radius of convergence greater than or equal to r, so that f(z) = ∑_{j=0}^{∞} f_{j} z^{j} defines a holomorphic function on U_{r}. If 0 ≤ r_{1} < r, then

M_{2}(f, r_{1}) = ( ∑_{j=0}^{∞} |f_{j}|^{2} r_{1}^{2 j})^{1/2}, (36.8)

by a standard computation. If f ∈ H^{2}(U_{r}), then it follows that f(T) is an element of the space C_{r}^{2}[[T]] defined in Section 28, using the standard absolute value function on C, with

∥f∥_{H^{2}(U_{r})} = ∥f(T)∥_{C_{r}^{2}[[T]]}, (36.9)

where the right side is as in (28.4). Conversely, if f(T) ∈ C_{r}^{2}[[T]], then f(T) has radius of convergence greater than or equal to r, and the corresponding holomorphic function f on U_{r} is an element of H^{2}(U_{r}).

Suppose that f(T) is an element of the space C_{r}[[T]] defined in Section 30 again, so that f(z) defines a holomorphic function on U_{r}, as before. If j is a nonnegative integer and 0 < r_{1} < r, then

f_{j} r_{j}^{j} = \frac{1}{2 π} ∫_{|z|=1} f(r_{1} z) z^{−j} |dz|, (36.10)

by a standard computation. This implies that

|f_{j}| r_{1}^{j} ≤ \frac{1}{2 π} ∫_{|z|=1} |f(z)| |dz| = M_{1}(f, r_{1}) (36.11)

for each j ≥ 0. If f ∈ H^{1}(U_{r}), then it follows that

|f_{j}| r^{j} ≤ ∥f∥_{H^{1}(U_{r})} (36.12)

for every j ≥ 0. This means that f(T) ∈ C_{r}^{∞}[[T]], with

∥f(T)∥_{C_{r}^{∞}[[T]]} ≤ ∥f∥_{H^{1}(U_{r})}, (36.13)

where the left side is as in (28.7). In fact, it is well known that

lim_{j→∞} |f_{j}| r^{j} = 0 (36.14)

in this case, so that f(T) ∈ C_{0,r}[[T]]. This is typically stated with r = 1, and one can reduce to that situation by rescaling.

37 Geometric series

Let k be a field, and let T be an indeterminate. The usual geometric series

a(T) = ∑_{j=0}^{∞} T^{j} (37.1)

may be considered as a formal power series in T with coefficients in k, where the coefficient of T^{j} is the multiplicative identity element 1 in k for each j ≥ 0. Remember that (37.1) is the multiplicative inverse of 1−T in k[[T]], as in Section 2.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let q, r be positive real numbers. If r < 1, then (37.1) is an element of the space k_{r}^{q}[[T]] defined in Section 28, with

∥a(T)∥_{q,r} = ( ∑_{j=0}^{∞} r^{q j})^{1/q} = (1 − r^{q})^{−1/q}. (37.2)

In particular, (37.1) is in the space k_{0,r}[[T]] defined in Section 28 when r < 1. If 0 < r ≤ 1, then (37.1) is in the space k_{r}^{∞}[[T]] defined in Section 28, with

∥a(T)∥_{∞,r} = 1. (37.3)

Otherwise, (37.1) is not in k_{r}^{q}[[T]] for any q < ∞, nor in k_{0,r}[[T]], when r ≥ 1, and (37.1) is not in k_{r}^{∞}[[T]] when r > 1.

Let A be an algebra over k, and let N be a submultiplicative q-norm N on A with respect to | · | on k for some q > 0. Suppose that A has a multiplicative identity element e with N(e) = 1, and that A is complete with respect to the q-metric associated to N. Note that

N(x^{j}) ≤ N(x)^{j} (37.4)

for every x ∈ A and nonnegative integer j, where x^{j} is interpreted as being equal to e when j = 0. In particular, if N(x) < 1, then it follows that

N(x^{j}) → 0 as j → ∞. (37.5)

Let us check that

∑_{j=0}^{∞} x^{j} (37.6)

converges in A when N(x) < 1. If q < ∞, then

∑_{j=0}^{∞} N(x^{j})^{q} ≤ ∑_{j=0}^{∞} N(x)^{q j} = (1 − N(x)^{q})^{−1}, (37.7)

so that (37.6) converges q-absolutely with respect to N. If q = ∞, then the convergence of (37.6) follows from (37.5), as in Section 9.

If x is any element of A and n is a nonnegative integer, then

(e − x) ∑_{j=0}^{n} x^{j} = ( ∑_{j=0}^{n} x^{j}) (e − x) = e − x^{n+1}, (37.8)

by a standard computation. Suppose that N(x) < 1, so that

(e − x) ∑_{j=0}^{∞} x^{j} = ( ∑_{j=0}^{∞} x^{j}) (x − e) = e (37.9)

by (37.5). Thus (37.6) is the multiplicative inverse of e − x in A. If q < ∞, then it follows that

N((e − x)^{−1}) = N( ∑_{j=0}^{∞} x^{j}) ≤ ( ∑_{j=0}^{∞} N(x^{j})^{q})^{1/q} ≤ (1 − N(x)^{q})^{−1/q}, (37.10)

using (9.9) in the second step. If q = ∞, then

N((e − x)^{−1}) = N( ∑_{j=0}^{∞} x^{j}) ≤ max_{j≥0} N(x^{j}) ≤ 1, (37.11)

using (9.11) in the second step.

38 Boundary values

Let r be a positive real number, let U_{r} be the open disk in the complex plane centered at 0 with radius r again, and let f be a holomorphic function on U_{r}. If t ∈ C and t ≠ 0, then f_{t}(z) = f(t z) (38.1)

defines a holomorphic function on U_{r/|t|}. Let 0 < q ≤ ∞ and 0 ≤ r_{1} < r be given, and observe that

M_{q}(f_{t}, r_{1}/|t|) = M_{q}(f, r_{1}), (38.2)

where M_{q}(·, ·) is as in Section 36. In particular, if f is an element of the Hardy space H^{q}(U_{r}) defined earlier, then f_{t} ∈ H^{q}(U_{r/|t|}), and

∥f_{t}∥_{H^{q}(U_{r/|t|})} = ∥f∥_{H^{q}(U_{r})}. (38.3)

If |t| ≤ 1, then U_{r} ⊆ U_{r/|t|}, and so the restriction of f_{t} to U_{r} defines a holomorphic function on U_{r}. If f ∈ H^{q}(U_{r}), then the restriction of f_{t} to U_{r} is an element of H^{q}(U_{r}) too, with

∥f_{t}∥_{H^{q}(U_{r})} ≤ ∥f∥_{H^{q}(U_{r})}. (38.4)

Let us now restrict our attention to positive real numbers t less than 1. Let

∂U_{r} = {z ∈ C : |z| = r} (38.5)

be the boundary of U_{r} in C, which is the same as the circle centered at 0 with radius r. This may be considered as a measure space, using arclength measure normalized by dividing by 2 π r. If f ∈ H^{q}(U_{r}), then it is well known that

lim_{t→1−} f(t z) (38.6)

exists for almost every z ∈ ∂U_{r}. Let f(z) denote these radial boundary values of f for almost every z ∈ ∂U_{r}. It is well known that this defines f as an element of L^{q}(∂U_{r}) = L^{q}(∂U_{r}, C), with

∥f∥_{H^{q}(U_{r})} = ∥f∥_{L^{q}(∂U_{r})}. (38.7)

If q < ∞, then it is also well known that f(t z) converges to f(z) in L^{q}(∂U) as t → 1−.

It is well known that H^{q}(U_{r}) is complete for every q > 0, with respect to the metric or q-metric associated to (36.5), as appropriate. This implies that the radial boundary values of elements of H^{q}(U_{r}) form a closed linear subspace of L^{q}(∂U_{r}).

Suppose that 0 < q_{1}, q_{2}, q_{3} ≤ ∞ satisfy 1/q_{3} = 1/q_{1}+1/q_{2}. Let f ∈ H^{q_{1}}(U_{r}), g ∈ H^{q_{2}}(U_{r}) be given, and note that their product f g is holomorphic on U_{r} as well. If 0 ≤ r_{1} < r, then

M_{q_{3}}(f g, r_{1}) ≤ M_{q_{1}}(f, r_{1}) M_{q_{2}}(g, r_{1}), (38.8)

by Hölder’s inequality. It follows that f g ∈ H^{q_{3}}(U_{r}), with

∥f g∥_{H^{q_{3}} (U_{r})} ≤ ∥f∥_{H^{q_{1}} (U_{r})} ∥g∥_{H^{q_{2}} (U_{r})}. (38.9)

Of course, the radial boundary values of f g on ∂U_{r} are equal to the product of the radial boundary values of f and g, when they exist.

39 Some remarks about invertibility

Let k be a field, and let A be an algebra over k with a multiplicative identity element e. The collection of elements of A with a multiplicative inverse in A is a group with respect to multiplication, that we shall denote G(A). Let |·| be a q_{k}- absolute value function on k for some q_{k} > 0, and let N be a submultiplicative q-seminorm on A with respect to | · | on k for some q > 0. If a, b ∈ G(A), then

a^{−1} − b^{−1} = a^{−1} (b − a) b^{−1}, (39.1)

and hence N(a^{−1} − b^{−1}) ≤ N(a^{−1}) N(b^{−1}) N(a − b). (39.2)

Suppose for the moment that q < ∞, so that

(39.3)  N(b^{−1})^{q} ≤ N(a^{−1})^{q} + N(a^{−1} − b^{−1})^{q}
≤ N(a^{−1})^{q} + N(a^{−1})^{q} N(b^{−1})^{q} N(a − b)^{q}.

Thus (1 − N(a^{−1})^{q} N(a − b)^{q}) N(b^{−1})^{q} ≤ N(a^{−1})^{q}. (39.4)

If N(a^{−1}) N(a − b) < 1, (39.5) then it follows that N(b^{−1})^{q} ≤ (1 − N(a^{−1})^{q} N(a − b)^{q})^{−1} N(a^{−1})^{q}. (39.6) Equivalently, this means that N(b^{−1}) ≤ (1 − N(a^{−1})^{q} N(a − b)^{q})^{−1/q} N(a^{−1}) (39.7) when (39.5) holds. Combining this with (39.2), we obtain that N(a^{−1} − b^{−1}) ≤ (1 − N(a^{−1})^{q} N(a − b)^{q})^{−1/q} N(a^{−1})^{2} N(a − b) (39.8) when (39.5) holds.

If q = ∞, then

(39.9)  N(b^{−1}) ≤ max(N(a^{−1}), N(a^{−1} − b^{−1}))
≤ max(N(a^{−1}), N(a^{−1}) N(b^{−1}) N(a − b)).

One can use this to get that N(b^{−1}) ≤ N(a^{−1}) (39.10) when (39.5) holds. This implies that N(a^{−1} − b^{−1}) ≤ N(a^{−1})^{2} N(a − b) (39.11) when (39.5) holds, because of (39.2).

Suppose now that N is a submultiplicative q-norm on A, and that A is complete with respect to the q-metric associated to N. If x ∈ A and N(x) < 1, then e − x is invertible in A, as in Section 37. Let a ∈ G(A) and b ∈ A be given, and observe that b = a − (a − b) = a (e − a^{−1} (a − b)). (39.12) If (39.5) holds, then e − a^{−1} (a − b) is invertible in A, as before. This implies that b ∈ G(A) when (39.5) holds, by (39.12).

Let {a_{j}}_{j=1}^{∞} be a sequence of invertible elements of A that converges to an element a of A with respect to the q-metric associated to N. Suppose also that {a_{j}^{−1}}_{j=1}^{∞} is bounded with respect to N, so that N(a_{j}^{−1}) ≤ C for some C ≥ 0 and every j ≥ 1. It follows that N(a_{j}^{−1} − a_{l}^{−1}) ≤ C^{2} N(a_{j} − a_{l}) (39.13) for every j, l ≥ 1, as in (39.2). This implies that {a_{j}^{−1}}_{j=1}^{∞} is a Cauchy sequence in A with respect to the q-metric associated to N, because {a_{j}}_{j=1}^{∞} is a Cauchy sequence, since it converges. Thus {a_{j}^{−1}}_{j=1}^{∞} converges in A, by completeness, and the limit of this sequence is the multiplicative inverse of a.

Part IV

Laurent series

40 Some spaces of Laurent series

Let k be a field, and let T be an indeterminate. Let us now use LS(k) = LS(k, T) to denote the space of formal Laurent series

f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j} (40.1)

in T with coefficients in k. As in Section 3, LS(k) can be defined precisely as the space c(Z, k) of all k-valued functions on the set Z of integers. In particular, LS(k) is a vector space over k with respect to termwise addition and scalar multiplication, which corresponds to pointwise addition and scalar multiplication of functions on Z.

Let LP (k) = LP (k, T) be the space of formal Laurent polynomials in T with coefficients in k. This is a linear subspace of LS(k), as before, which corresponds to the space c_{00}(Z, k) of k-valued functions on Z with finite support. Remember that k((T)) is the linear subspace of LS(k) consisting of formal Laurent series in T with coefficients in k such that the coefficient of T^{j} is equal to 0 for all but finitely many j < 0. This is a field with respect to formal multiplication of these series, as before. Of course, LP (k) is a subalgebra of k((T)), as an algebra over k.

Let P S(k) = P S(k, T) be the space of formal Laurent series in T with coefficients in k such that the coefficient of T^{j} is equal to 0 when j < 0. This is a linear subspace of LS(k), whose elements can be identified with formal power series in T with coefficients in k. Similarly, let P (k) = P (k, T) be the space of formal Laurent polynomials in T with coefficients in k such that the coefficient of T^{j} is 0 when j < 0, which is the linear subspace of LP (k) corresponding to formal polynomials in T with coefficients in k. Equivalently,

P (k) = LP (k) ∩ P S(k). (40.2)

Note that P (k), P S(k) are subalgebras of LP (k), k((T)), respectively, as algebras over k.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, and let r be a positive real number. Put w_{r}(j) = r^{j} (40.3)

for every integer j, which defines a positive real-valued function on Z. Let q be a positive real number, and put

LS_{r}^{q}(k) = LS_{r}^{q}(k, T) = {f(T) ∈ LS(k) : ∑_{j=−∞}^{∞} |f_{j}|^{q} r^{q j} < ∞}. (40.4)

This sum may be considered as a sum over Z, as in Section 15, or as a doublyinfinite series of nonnegative real numbers, as in Section 17. If f(T) ∈ LS_{r}^{q}(k), then we put

∥f(T)∥_{q,r} = ∥f(T)∥_{LS_{r}^{q}(k)} = ( ∑_{j=−∞}^{∞} |f_{j}|^{q} r^{q j})^{1/q}. (40.5)

Note that (40.4) corresponds to the space ℓ_{w_{r}}^{q}(Z, k) of k-valued functions on Z that are q-summable with respect to w_{r} on Z and | · | on k, as in Section 26. Similarly, (40.5) corresponds to ∥ · ∥_{ℓ_{wr}^{q} (Z,k)} in Section 26, applied to j ↦ f_{j} as a k-valued function on Z. In particular, (40.5) defines a q-norm on (40.4) with respect to | · | on k when q ≤ q_{k}, and a q_{k}-norm on (40.4) when q ≥ q_{k}. As before, LS_{r}^{q}(k) is a linear subspace of LS(k),

LP (k) ⊆ LS_{r}^{q}(k), (40.6)

and LP (k) is dense in LS_{r}^{q}(k) with respect to the q or q_{k}-metric associated to (40.5). Put P S_{r}^{q}(k) = P S_{r}^{q}(k, T) = LS_{r}^{q}(k) ∩ P S(k), (40.7)

which corresponds to k_{r}^{q}[[T]] in Section 28.

Put

(40.8) LS_{r}^{∞}(k) = LS_{r}^{∞}(k, T) = {f(T) ∈ LS(k) : |f_{j}| r^{j} is bounded on Z},

and

∥f(T)∥_{∞,r} = ∥f(T)∥_{LS_{r}^{∞}(k)} = sup_{j∈Z}(|f_{j}| r^{j}) (40.9)

for each f(T) ∈ LS_{r}^{∞}(k). Thus (40.8) corresponds to the space ℓ_{w_{r}}^{∞}(Z, k) of k-valued functions on Z that are bounded with respect to w_{r} on Z and | · | on k, and (40.9) corresponds to ∥·∥_{ℓ_{wr}^{∞} (Z,k)} applied to j ↦ f_{j} as a k-valued function on Z. As before, (40.8) is a linear subspace of LS(k), and (40.9) defines a q_{k}-norm on (40.8) with respect to | · | on k. If 0 < q ≤ q̃ ≤ ∞, then

LS_{r}^{q}(k) ⊆ LS_{q} ; r(k), (40.10)

and

∥f(T)∥_{_{q},r} ≤ ∥f(T)∥_{q,r} (40.11)

for every f(T) ∈ LS_{r}^{q}(k), as in Section 26. Put

P S_{r}^{∞}(k) = P S_{r}^{∞}(k, T) = LS_{r}^{∞}(k) ∩ P S(k), (40.12)

which corresponds to k_{r}^{∞}[[T]] in Section 28.

Put

LS_{0,r}(k) = LS_{0,r}(k, T) = {f(T) ∈ LS(k) : lim_{j→±∞} |f_{j}| r^{j} = 0}, (40.13)

which corresponds to the space c_{0,w_{r}}(Z, k) of k-valued functions on Z that vanish at infinity with respect to w_{r} on Z and | · | on k. This is a closed linear subspace of (40.8) with respect to the q_{k}-metric associated to (40.9). As before,

LP (k) ⊆ LS_{0,r}(k), (40.14)

and LP (k) is dense in LS_{0,r}(k) with respect to the q_{k}-metric associated to (40.9). We also have that LS_{r}^{q}(k) ⊆ LS_{0,r}(k) (40.15)

when 0 < q < ∞, as in Section 26. Put

P S_{0,r}(k) = P S_{0,r}(k, T) = LS_{0,r}(k) ∩ P S(k), (40.16)

which corresponds to k_{0,r}[[T]] in Section 28.

If | · | is the trivial absolute value function on k, then LS_{1}^{∞}(k) = LS(k), LS_{1}^{q}(k) = LP (k) when 0 < q < ∞, and LS_{0,1}(k) = LP (k). If 0 < r < 1, then LS_{r}^{q}(k) = k((T)) for every q > 0, and LS_{0,r}(k) = k((T)) as well.

41 Multiplying Laurent series

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let T be an indeterminate. Suppose that k is complete with respect to the q_{k}-metric associated to | · |. Let a positive real number r and 0 < q ≤ q_{k} be given, so that | · | may be considered as a q-absolute value function on k too. Suppose for the moment that q < ∞, and that f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j}, g(T) = ∑_{l=−∞}^{∞} g_{l} T^{l} are elements of the space LS_{r}^{q}(k) defined in (40.4). We would like to define

f(T) g(T) = h(T) = ∑_{n=−∞}^{∞} h_{n} T^{n} (41.1)

as a formal Laurent series in T with coefficients in k, with

h_{n} = ∑_{j=−∞}^{∞} f_{j} g_{n−j} (41.2)

for every n ∈ Z.

Observe that

r^{q n} ∑_{j=−∞}^{∞} |f_{j} g_{n−j}|^{q} = ∑_{j=−∞}^{∞}(|f_{j}|^{q} r^{q j}) (|g_{n−j}|^{q} r^{q (n−j)}) (41.3)

for every n ∈ Z. The right side corresponds to the nth term of the Cauchy product of ∑_{j=−∞}^{∞} |f_{j}|^{q} r^{q j} and ∑_{l=−∞}^{∞} |g_{l}|^{q} r^{q l}, as in Section 21. It follows that

∑_{n=−∞}^{∞} (r^{q n} ∑_{j=−∞}^{∞} |f_{j} g_{n−j}|^{q}) ≤ ( ∑_{j=−∞}^{∞} |f_{j}|^{q} r^{q j}) ( ∑_{l=−∞}^{∞} |g_{l}|^{q} r^{q l})
(41.4)  = ∥f(T)∥_{q,r}^{q} ∥g(T)∥_{q,r}^{q},

as before. In particular, (41.3) is finite for each n ∈ Z. This permits us to define (41.2) as an element of k for every n ∈ Z, as a q-absolutely convergent doubly-infinite series, as in Section 17, or as a sum over Z, as in Section 18. We also get that

|h_{n}|^{q} r^{q n} ≤ r^{q n} ∑_{j=−∞}^{∞} |f_{j} g_{n−j}|^{q} (41.5)

for every n ∈ Z, so that

∑_{n=−∞}^{∞} |h_{n}|^{q} r^{q n} ≤ ∥f(T)∥_{q,r}^{q} ∥g(T)∥_{q,r}^{q}, (41.6)

by (41.4). This implies that h(T) ∈ LS_{r}^{q}(k) too, with

∥h(T)∥_{q,r} ≤ ∥f(T)∥_{q,r} ∥g(T)∥_{q,r}. (41.7)

Suppose now that q = ∞, so that q_{k} = ∞ in particular. In this case, we ask that f(T) and g(T) be elements of the space LS_{0,r}(k) defined in the previous section. Of course,

|f_{j} g_{n−j}| = r^{−n} (|f_{j}| r^{j}) (|g_{n−j}| r^{n−j}) (41.8)

for every j, n ∈ Z. It is easy to see that

lim_{j→±∞} |f_{j} g_{n−j}| = 0 (41.9)

for every n ∈ Z, using (41.8) and the hypothesis that f(T), g(T) ∈ LS_{0,r}(k). Thus we can define (41.2) as an element of k for every n ∈ Z, as a convergent doubly-infinite series, as in Section 17, or as a sum over Z, as in Section 18. Note that

(41.10) |h_{n}| r^{n} ≤ r^{n} max_{j∈Z} |f_{j} g_{n−j}| = max_{j∈Z}((|f_{j}| r^{j}) (|g_{n−j}| r^{n−j}))

for every n ∈ Z. One can use this to verify that

lim_{n→±∞} |h_{n}| r^{n} = 0, (41.11)

because of the analogous conditions for f(T) and g(T). This means that h(T) is an element of LS_{0,r}(k), and we can also use (41.10) to get that

∥h(T)∥_{∞,r} ≤ ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r}. (41.12)

One can check that LS_{r}^{q}(k) is a commutative algebra over k with respect to this definition of multiplication when 0 < q ≤ q_{k} and q < ∞, and similarly that LS_{0,r}(k) is a commutative algebra over k when q_{k} = ∞. Of course, this definition of (41.1) reduces to the one in Section 3 when f(T), g(T) ∈ LP (k). Thus LP (k) is a subalgebra of LS_{r}^{q}(k) when 0 < q ≤ q_{k} and q < ∞, and a subalgebra of LS_{0,r}(k) when q_{k} = ∞. Remember that k can be indentified with a subalgebra of LP (k), as in Section 3. The multiplicative identity element 1 in k corresponds to the multiplicative identity element in LS_{r}^{q}(k) when 0 < q ≤ q_{k} and q < ∞, and to the multiplicative identity element in LS_{0,r}(k) when q_{k} = ∞.

42 Another multiplicativity condition

Let k be a field with an ultrametric absolute value function | · |, and suppose that k is complete with respect to the ultrametric associated to | · |. Also let T be an indeterminate, and let r be a positive real number. If f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j}, g(T) = ∑_{l=−∞}^{∞} g_{l} T^{l} ∈ LS_{0,r}(k), then ∥f(T) g(T)∥_{∞,r} = ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r}. (42.1) It suffices to show that ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r} ≤ ∥f(T) g(T)∥_{∞,r}, (42.2) because of (41.12). We may as well suppose that f(T), g(T) ≠ 0.

As in Section 31, we would like to choose j_{0}, l_{0} ∈ Z such that |f_{j_{0}}|r^{j_{0}} = ∥f(T)∥_{∞,r} (42.3) and |g_{l_{0}}| r^{l_{0}} = ∥g(T)∥_{∞,r}. (42.4) This would imply that |f_{j}| r^{j} ≤ ∥f(T)∥_{∞,r} = |f_{j_{0}}| r^{j_{0}} (42.5) and |g_{l}| r^{l} ≤ ∥g(T)∥_{∞,r} = |g_{l_{0}}| r^{l_{0}} (42.6) for every j, l ∈ Z. Remember that |f_{j}| r^{j} → 0 as j → ±∞, and |g_{l}| r^{l} → ∞ as l → ±∞, by hypothesis. Hence the suprema in the definitions of ∥f(T)∥_{∞,r} and ∥g(T)∥_{∞,r} are attained. This means that there are j_{0}, l_{0} ∈ Z such that (42.3) and (42.4) hold. In fact, there are only finitely many such j_{0} and l_{0}, because f(T), g(T) ≠ 0. Thus we can choose j_{0} and l_{0} to be the first integers such that (42.3) and (42.4) hold, so that the inequalities in (42.5) and (42.6) are strict when j < j_{0} and l < l_{0}, respectively.

Using (42.3) and (42.4), we get that ∥f(T)∥_{∞,r} ∥g(T)∥_{∞,r} = |f_{j_{0}}| |g_{l_{0}}| r^{j_{0}+l_{0}}. (42.7) As in the previous section, f(T) g(T) = h(T) = ∑_{n=−∞}^{∞} h_{n} T^{n}, where h_{n} ∈ k is as in (41.2). In particular, |h_{j_{0}+l_{0}}| r^{j_{0}+l_{0}} ≤ ∥h(T)∥_{∞,r} = ∥f(T) g(T)∥_{∞,r}. (42.8) Thus we would like to check that |f_{j_{0}}| |g_{l_{0}}|r^{j_{0}+l_{0}} ≤ |h_{j_{0}+l_{0}}| r^{j_{0}+l_{0}} (42.9) to get (42.2). Of course, this is the same as saying that |f_{j_{0}}| |g_{l_{0}}| ≤ |h_{j_{0}+l_{0}}|. (42.10)

Note that |f_{j_{0}}| |g_{l_{0}}| ≤ max(|h_{j_{0}+l_{0}}|, |h_{j_{0}+l_{0}} − f_{j_{0}} g_{l_{0}}|), (42.11)

by the ultrametric version of the triangle inequality. In order to get (42.10), it is enough to verify that

|h_{j_{0}+l_{0}} − f_{j_{0}} g_{l_{0}}| < |f_{j_{0}}| |g_{l_{0}}|. (42.12)

Using (42.5) and (42.6), we get that

|f_{j}| |g_{l}| r^{j+l} ≤ |f_{j_{0}}| |g_{l_{0}}| r^{j_{0}+l_{0}} (42.13)

for every j, l ∈ Z. If j + l = j_{0} + l_{0}, then it follows that

|f_{j}| |g_{l}| ≤ |f_{j_{0}}| |g_{l_{0}}|. (42.14)

This inequality is strict when either j < j_{0} or l < l_{0}, because of the corresponding strict inequalities in (42.5) and (42.6), mentioned earlier. Observe that h_{j_{0}+l_{0}} is basically the sum of f_{j} g_{l} with j, l ∈ Z and j + l = j_{0} + l_{0}, as in (41.2). Thus h_{j_{0}+l_{0}} − f_{j_{0}} g_{l_{0}} reduces to the sum of f_{j} g_{l} over j, l ∈ Z with j +l = j_{0} +l_{0} and either j < j_{0} or l < l_{0}. Each of these terms has absolute value strictly less than |f_{j_{0}}| |g_{l_{0}}|, because of the strict inequality in (42.14) in these cases. This implies (42.12), because of the ultrametric version of the triangle inequality, and the convergence of the sum in this situation.

43 Some subsets of algebras

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let A be an algebra over k with a submultiplicative q-seminorm N with respect to | · | on k, for some q > 0. Suppose that A has a multiplicative identity element e with N(e) = 1, and let G(A) be the group of invertible elements of A, as before. Let r be a positive real number, and consider the set

C(r) = C_{A}(r) = {a ∈ G(A) : N(a) ≤ r, N(a^{−1}) ≤ 1/r}. (43.1)

Note that e ∈ C(1), and that C(1) is a subgroup of G(A). If t ∈ k \ {0}, then a ∈ C(r) if and only if t a ∈ C(r |t|).

If a ∈ C(r), then N(a^{j}) ≤ r^{j} (43.2)

for every j ∈ Z. Of course,

1 ≤ N(a^{j}) N(a^{−j}) (43.3)

for every a ∈ G(A) and j ∈ Z. It follows that

N(a^{j}) = r^{j} (43.4)

for every a ∈ C(r) and j ∈ Z.

Suppose that N is a q-norm on A, and that A is complete with respect to the q-metric associated to N. One can check that C(r) is a closed set in A with respect to the q-metric associated to N, using the remarks in Section 39. If q = ∞, then C(r) is an open set in A with respect to the q-metric associated to N.

If 0 < q ≤ q_{k} and q < ∞, then LS_{r}^{q}(k) is a commutative algebra over k, as in the previous section, and ∥ · ∥_{q,r} is a submultiplicative q-norm on LS_{r}^{q}(k). Similarly, if q_{k} = ∞, then LS_{0,r}(k) is a commutative algebra over k, and ∥·∥_{∞,r} is a multiplicative ultranorm on LS_{0,∞}(k). If j ∈ Z, then T^{j} may be considered as a formal Laurent polynomial in T with coefficient equal to the multiplicative identity element 1 in k, and ∥T^{j}∥_{q,r} = r^{j} for every q > 0. Note that LS_{r}^{q}(k) is complete with respect to the q or q_{k}-metric associated to ∥ · ∥_{q,r} when k is complete with respect to the q_{k}-metric associated to | · |, as in Section 26. This implies that LS_{0,r}(k) is complete with respect to the q_{k}-metric associated to ∥ · ∥_{∞,r}, because LS_{0,r}(k) is a closed set in LS_{r}^{∞}(k) with respect to this q_{k}-metric, as before.

Let V be a vector space over k, and let N_{V} be a q_{V} -seminorm on V with respect to |·| on k for some q_{V} > 0. Remember that the space BL(V) of bounded linear mappings from V into itself with respect to N_{V} is an algebra over k with respect to composition of mappings, and that the corresponding operator q_{V} - seminorm ∥ · ∥_{op} is submultiplicative on BL(V). Suppose that N_{V} (v) > 0 for some v ∈ V, which implies that the identity mapping on V satisfies ∥I∥_{op} = 1. If A is a one-to-one linear mapping from V onto itself that satisfies

N_{V} (A(v)) = r N_{V} (v) (43.5)

for every v ∈ V, then N_{V} (A^{j}(v)) = r^{j} N_{V} (v) (43.6)

for every v ∈ V and j ∈ Z. This implies that A^{j} is a bounded linear mapping from V into itself for every j ∈ Z, with

∥A^{j}∥_{op} = r^{j}. (43.7)

Conversely, if A is a bounded linear mapping from V into itself with a bounded inverse, and if ∥A∥_{op} ≤ r, ∥A^{−1}∥_{op} ≤ 1/r, then one can check that (43.5) holds. If N_{V} is a q_{V} -norm on V, then ∥ · ∥_{op} is a q_{V} -norm on BL(V), and BL(V) is complete with respect to the q_{V} -metric associated to ∥ · ∥_{op} when V is complete with respect to the q_{V} -metric associated to N_{V}.

44 Functions on these subsets

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let A be an algebra over k with a submultiplicative q-norm N with respect to | · | on k for some q > 0. Suppose that A has a multiplicative identity element e with N(e) = 1, and that A is complete with respect to the q-metric associated to N. Let T be an indeterminate, let r be a positive real number, and let a be an element of the set C(r) defined in (43.1). Suppose for the moment that q < ∞, and that f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j} is an element of LS_{r}^{q}(k). We would like to define f(a) as an element of A by

f(a) = ∑_{j=−∞}^{∞} f_{j} a^{j}, (44.1)

where a^{j} = e when j = 0, as usual. Note that

∑_{j=−∞}^{∞} N(f_{j} a^{j})^{q} = ∑_{j=−∞}^{∞} |f_{j}|^{q} N(a^{j})^{q} = ∑_{j=−∞}^{∞} |f_{j}|^{q} r^{q j} = ∥f(T)∥_{q,r}^{q}. (44.2)

Thus (44.1) can be defined as an element of A, as a q-absolutely convergent doubly-infinite series as in Section 17, or as a sum over Z, as in Section 18. We also have that

N(f(a)) ≤ ( ∑_{j=−∞}^{∞} N(f_{j} a^{j})^{q})^{1/q} = ∥f(T)∥_{q,r}. (44.3)

Suppose now that q = ∞, and that f(T) ∈ LS_{0,r}(k). If a ∈ C(r), then

N(f_{j} a^{j}) = |f_{j}| N(a^{j}) = |f_{j}| r^{j} → 0 as j → ±∞. (44.4)

This permits us to define (44.1) as an element of A, as a convergent doublyinfinite series, as in Section 17, or as a sum over Z, as in Section 18. In this case, we get that

N(f(a)) ≤ max_{j∈Z} N(f_{j} a^{j}) = max_{j∈Z}(|f_{j}| r^{j}) = ∥f(T)∥_{∞,r}. (44.5)

Let f(T) be an element of LS_{r}^{q}(k) when q < ∞, or an element of LS_{0,r}(k) when q = ∞. If a ∈ C(r), then

∑_{j=−n}^{n} f_{j} a^{j} (44.6)

converges to f(a) as n → ∞, with respect to the q-metric on A associated to N. It is easy to see that the convergence is uniform on C(r) in both cases, using the same type of estimates as before.

If j ∈ Z, then a ↦ a^{j} is uniformly continuous as a mapping from C(r) into A, with respect to the q-metric associated to N on A and its restriction to C(r). This is clear when j = 0, 1, and the case where j = −1 follows from (39.2). One can reduce to these cases when |j| ≥ 2, because products of bounded uniformly continuous functions are uniformly continuous as well. It follows that (44.6) is uniformly continuous on C(r) for every n ≥ 0. This implies that f(a) is uniformly continuous on C(r) under the conditions in the preceding paragraph, because of uniform convergence.

As in Section 7, we may as well suppose that q ≤ q_{k}. Let us also ask that k be complete with respect to the q_{k}-metric associated to | · |. Let f(T), g(T) be elements of LS_{r}^{q}(k) when q < ∞, or elements of LS_{0,r}(k) when q = ∞. Thus f(T) g(T) = h(T) is defined as an element of the same space, as in Section 41. If a ∈ C(r), then f(a), g(a), and h(a) are defined as elements of A, as before. Observe that h(a) = f(a) g(a), (44.7)

because h(a) is defined by the sum over Z corresponding to the Cauchy product of the sums over Z that define f(a) and g(a), as in Section 21. It follows that f(T) ↦ f(a) defines an algebra homomorphism from LS_{r}^{q}(k) into A when q < ∞, and from LS_{0,r}(k) into A when q = ∞.

45 Rescaling Laurent series

Let k be a field, let T be an indeterminate, and let t ∈ k \ {0} be given. If f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j} is a formal Laurent series in T with coefficients in k, then put

R_{t}(f(T)) = ∑_{j=−∞}^{∞} f_{j} t^{j} T^{j}. (45.1)

This defines R_{t}(f(T)) as an element of LS(k), and it may also be denoted (R_{t}(f))(T) or simply R_{t}(f). Note that R_{t} defines a one-to-one linear mapping from LS(k) onto itself, which is the identity mapping on LS(k) when t = 1. It is easy to see that R_{t} ∘ R_{u} = R_{t u} (45.2)

for every t, u ∈ k \ {0}. Of course, (45.1) reduces to the analogous definition for formal power series in Section 35 when f(T) ∈ P S(k). Note that R_{t} maps LP (k) onto itself for each t ∈ k \ {0}, and that R_{t} is a homomorphism from LP (k) into itself, as an algebra over k.

Let | · | be a q_{k}-absolute value function on k for some q_{k} > 0, let r be a positive real number, and let t ∈ k \ {0}, f(T) ∈ LS(k), and 0 < q ≤ ∞ be given. Observe that R_{t}(f(T)) ∈ LS_{r}^{q}(k) if and only if f(T) ∈ LS_{r |t|}^{q}(k), in which case ∥R_{t}(f(T))∥_{q,r} = ∥f(T)∥_{q,r |t|}. (45.3)

Similarly, R_{t}(f(T)) ∈ LS_{0,r}(k) if and only if f(T) ∈ LS_{0,r |t|}(k).

Let A be an algebra over k with a multiplicative identity element e and a submultiplicative q-norm N with respect to | · | on k for some 0 < q ≤ q_{k}. Suppose that N(e) = 1, and that A is complete with respect to the q-metric associated to N. Let t ∈ k \ {0} and a positive real number r be given, and let a be an element of the subset C(r) of A defined in (43.1). Thus t a ∈ C(r |t|), as before. Suppose for the moment that q < ∞ and f(T) ∈ LS_{r |t|}^{q}(k), so that R_{t}(f(T)) ∈ LS_{r}^{q}(k). Under these conditions, f(t a) and (R_{t}(f))(a) can be defined as elements of A, as in the previous section, with

(R_{t}(f))(a) = ∑_{j=−∞}^{∞} f_{j} t^{j} a^{j} = ∑_{j=−∞}^{∞} f_{j} (t a)^{j} = f(t a). (45.4)

If q = ∞ and f(T) ∈ LS_{0,r |t|}(k), then R_{t}(f(T)) ∈ LS_{0,r}(k), f(t a) and (R_{t}(f))(a) can be defined as elements of A again, and satisfy (45.4).

Suppose that k is complete with respect to the q_{k}-metric associated to |·|. If 0 < q ≤ q_{k} and q < ∞, then LS_{r}^{q}(k) is a commutative algebra over k for every positive real number r, as in Section 41. One can check that for every t ∈ k \{0} and 0 < r < ∞, R_{t} is a homomorphism from LS_{r |t|}^{q}(k) into LS_{r}^{q}(k), as algebras over k. Similarly, if q_{k} = ∞, then LS_{0,r}(k) is a commutative algebra over k for every 0 < r < ∞. One can verify that for every t ∈ k \ {0} and 0 < r < ∞, R_{t} is a homomorphism from LS_{0,r |t|}(k) into LS_{0,r}(k), as algebras over k.

46 Fourier series

Let r be a positive real number, and let

C(r) = {z ∈ C : |z| = r} (46.1)

be the circle in the complex plane centered at 0 with radius r, with respect to the standard absolute value function on C. This may be considered as a measure space, using arclength measure divided by 2 π, r, as before. Let L^{q}(C(r)) = L^{q}(C(r), C) be the corresponding L^{q} space of complex-valued functions on C(r) for each q > 0. If f ∈ L^{1}(C(r)) and j ∈ Z, then the jth Fourier coefficient of f is defined by

f̂(j) = \frac{1}{2 π r} ∫_{C(r)} f(w) w^{−j} |dw|, (46.2)

where |dw| refers to arclength measure on C(r). The corresponding Fourier series is defined formally by

∑_{j=−∞}^{∞} f̂(j) z^{j}. (46.3)

If T is an indeterminate, then

∑_{j=−∞}^{∞} f̂(j) T^{j} (46.4)

may be considered as a formal Laurent series in T with complex coefficients. Note that f ↦ f̂ is linear as a mapping from L^{1}(C(r)) into the space c(Z, C) of complex-valued functions on Z. One can also consider the mapping from f to (46.4) as a linear mapping from L^{1}(C(r)) into LS(C).

Observe that

|f̂(j)| ≤ \frac{1}{2 π r} ∫_{C(r)} |f(w)| |w|^{−j} |dw| = r^{−j} ∥f∥_{L^{1}(C(r))} (46.5)

for every j ∈ Z. Equivalently,

|f̂(j)| r^{j} ≤ ∥f∥_{L^{1}(C(r))} (46.6)

for every j ∈ Z. Put w_{r}(j) = r^{j} for every j ∈ Z, which defines a positive realvalued function on Z. Thus f̂ is an element of the space ℓ_{w_{r}}^{∞}(Z, C) of complexvalued functions on Z that are bounded with respect to w_{r}, as in Section 26, with ∥f̂∥_{ℓ_{wr}^{∞} (Z,C)} ≤ ∥f∥_{L^{1}(C(r))}. (46.7)

It is well known that

lim_{j→±∞} |f̂(j)| r^{j} = 0, (46.8)

by the Riemann–Lebesgue lemma. This means that f̂ vanishes at infinity on Z with respect to w_{r}. This is the same as saying that (46.4) is an element of the space LS_{0,r}(C) defined in Section 40.

If f ∈ L^{2}(C(r)), then it is well known that

∑_{j=−∞}^{∞} |f̂(j)|^{2} r^{2 j} = \frac{1}{2 π r} ∫_{C(r)} |f(w)|^{2} |dw| = ∥f∥_{L^{2}(C(r))}^{2}, (46.9)

by Parseval’s theorem. This means that f̂ is in the space ℓ_{w_{r}}^{2}(Z, C) defined in Section 26, with ∥f̂∥_{ℓ_{wr}^{2} (Z,C)} = ∥f∥_{L^{2}(C(r))}. (46.10)

In particular, (46.4) is in the space LS_{r}^{2}(C) defined in Section 40.

Suppose that f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j} is an element of the space LS_{r}^{1}(C) defined in Section 40, so that ∑_{j=−∞}^{∞} |f_{j}| r^{j} is finite. In this case,

f(w) = ∑_{j=−∞}^{∞} f_{j} w^{j} (46.11)

defines a continuous complex-valued function on C(r), as in Section 44. It is well known that f̂(l) = f_{l} (46.12)

for every l ∈ Z in this situation. If f(T) ∈ LS_{r}^{2}(C), then one can define an element of L^{2}(C(r)) as in (46.11), with convergence with respect to the L^{2} norm. This element of L^{2}(C(r)) also satisfies (46.12), and (46.10) is the same as ∥f(T)∥_{LS_{r}^{2}(C)}.

47 Pairs of radii

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, let T be an indeterminate, and let r_{1}, r_{2} be positive real numbers with r_{1} ≤ r_{2}. If 0 < q ≤ ∞, then put

LS_{r_{1},r_{2}}^{q}(k) = LS_{r_{1},r_{2}}^{q}(k, T) = LS_{r_{1}}^{q}(k) ∩ LS_{r_{2}}^{q}(k), (47.1) where LS_{r}^{q}(k) is as in Section 40. Let w_{r_{1},r_{2}} be the positive real-valued function defined on Z by

(47.2)  w_{r_{1},r_{2}}(j) = max(r_{1}^{j}, r_{2}^{j}) = r_{1}^{j} when j ≤ 0
= r_{2}^{j} when j ≥ 0.

If f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j} is an element of (47.1), then put

∥f(T)∥_{q,r_{1},r_{2}} = ∥f(T)∥_{LS_{r_{1},r_{2}}^{q} (k)} = ( ∑_{j=−∞}^{∞} |f_{j}|^{q} w_{r_{1},r_{2}}(j)^{q})^{1/q} (47.3)

when q < ∞, and

∥f(T)∥_{∞,r_{1},r_{2}} = ∥f(T)∥_{LS_{r_{1},r_{2}}^{∞} (k)} = sup_{j∈Z}(|f_{j}| w_{r_{1},r_{2}}(j)) (47.4)

when q = ∞. Note that (47.1) corresponds to the space ℓ_{w_{r_{1},r_{2}}}^{q} (Z, k) defined in Section 26. Similarly, (47.3) and (47.4) correspond to ∥ · ∥_{ℓ_{wr_{1},r_{2}}^{q} (Z,k)} applied to j ↦ f_{j} as a k-valued function on Z. In particular, this is a q-norm with respect to | · | on k when q ≤ q_{k}, and a q_{k}-norm when q_{k} ≤ q.

If f(T) is an element of (47.1), then it is easy to see that

max(∥f(T)∥_{q,r_{1}}, ∥f(T)∥_{q,r_{2}}) ≤ ∥f(T)∥_{q,r_{1},r_{2}}, (47.5)

where ∥f(T)∥_{q,r} is as in Section 40. In fact,

∥f(T)∥_{∞,r_{1},r_{2}} = max(∥f(T)∥_{∞,r_{1}}, ∥f(T)∥_{∞,r_{2}}) (47.6)

when q = ∞. If q < ∞, then we also have that

∥f(T)∥_{q,r_{1},r_{2}} ≤ (∥f(T)∥_{q,r_{1}}^{q} + ∥f(T)∥_{q,r_{2}}^{q})^{1/q}. (47.7)

In this case, LP (k) is dense in (47.1) with respect to the q or q_{k}-metric associated to (47.3), as appropriate.

Put LS_{0,r_{1},r_{2}}(k) = LS_{0,r_{1},r_{2}}(k, T) = LS_{0,r_{1}}(k) ∩ LS_{0,r_{2}}(k), (47.8)

where LS_{0,r}(k) is as in Section 40. Equivalently, f(T) ∈ LS(k) is an element of (47.8) if and only if

lim_{j→−∞} |f_{j}| r_{1}^{j} = lim_{j→+∞} |f_{j}| r_{2}^{j} = 0. (47.9)

Thus (47.8) corresponds to the space c_{0,w_{r_{1},r_{2}}} (Z, k) of k-valued functions on Z that vanish at infinity with respect to w_{r_{1},r_{2}} on Z and | · | on k, as in Section 26. As before, (47.8) is a closed linear subspace of LS_{r_{1},r_{2}}^{∞}(k) with respect to the q_{k}-metric associated to (47.4). Note that LP (k) is dense in (47.8) with respect to the q_{k}-metric associated to (47.4).

Let t ∈ k\{0} be given, and let R_{t} be the rescaling operator defined on LS(k) in Section 45. Also let f(T) ∈ LS(k) and 0 < q ≤ ∞ be given. Observe that R_{t}(f(T)) ∈ LS_{r_{1},r_{2}}^{q}(k) if and only if f(T) ∈ LS_{r_{1} |t|,r_{2} |t|}^{q}(k), by the analogous statement for LS_{r}^{q}(k) mentioned earlier. In this case, one can check that

∥R_{t}(f(T))∥_{q,r_{1},2_{2}} = ∥f(T)∥_{q,r_{1} |t|,r_{2} |t|}. (47.10)

Similarly, R_{t}(f(T)) ∈ LS_{0,r_{1},r_{2}}(k) if and only if f(T) ∈ LS_{0,r_{1} |t|,r_{2} |t|}(k).

48 Products with two radii

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and suppose that k is complete with respect to the q_{k}-metric associated to | · |. Also let T be an indeterminate, let r_{1}, r_{2} be positive real numbers with r_{1} ≤ r_{2}, and let 0 < q ≤ q_{k} be given. Suppose for the moment that q < ∞, and let f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j} and g(T) = ∑_{l=−∞}^{∞} g_{l} T^{l} be elements of the space LS_{r_{1},r_{2}}^{q}(k) defined in (47.1). Under these conditions, f(T) g(T) = h(T) = ∑_{n=−∞}^{∞} h_{n} T^{n} can be defined as an element of LS_{r_{1},r_{2}}^{q}(k) as well, as in Section 41. More precisely, if f(T), g(T) ∈ LS_{r}^{q}(k) for some r > 0, then we saw that

∑_{j=−∞}^{∞} |f_{j} g_{n−j}|^{q} < ∞ (48.1)

for every n ∈ Z. This was used to define

h_{n} = ∑_{j=−∞}^{∞} f_{j} g_{n−j} (48.2)

as an element of k for every n ∈ Z. If f(T), g(T) ∈ LS_{r_{1},r_{2}}^{q}(k), then this criterion for (48.1) holds for r = r_{1}, r_{2}, but of course (48.2) does not depend on which criterion for (48.1) was used.

The remarks in Section 41 imply that h(T) ∈ LS_{r}^{q}(k) for r = r_{1}, r_{2}, so that h(T) ∈ LS_{r_{1},r_{2}}^{q}(k). We would like to check that

∥h(T)∥_{q,r_{1},r_{2}} ≤ ∥f(T)∥_{q,r_{1},r_{2}} ∥g(T)∥_{q,r_{1},r_{2}}. (48.3)

Remember that

|h_{n}|^{q} ≤ ∑_{j=−∞}^{∞} |f_{j}|^{q} |g_{n−j}|^{q} (48.4)

for every n ∈ Z. This uses the fact that | · | may be considered as a q-absolute value function on k, because q ≤ q_{k}.

Let w_{r_{1},r_{2}} be defined on Z as in (47.2), and observe that

w_{r_{1},r_{2}}(j + l) = max(r_{1}^{j} r_{1}^{l}, r_{2}^{j} r_{2}^{l}) ≤ w_{r_{1},r_{2}}(j) w_{r_{1},r_{2}}(l) (48.5) for every j, l ∈ Z. This implies that

|h_{n}|^{q} w_{r_{1},r_{2}}(n)^{q} ≤ ∑_{j=−∞}^{∞}(|f_{j}|^{q} w_{r_{1},r_{2}}(j)^{q}) (|g_{n−j}|^{q} w_{r_{1},r_{2}}(n − j)^{q}) (48.6)

for every n ∈ Z, because of (48.4). The right side of (48.6) is the same as the nth term of the Cauchy product of the series ∑_{j=−∞}^{∞} |f_{j}|^{q} w_{r_{1},r_{2}}(j)^{q} and ∑_{l=−∞}^{∞} |g_{l}|^{q} w_{r_{1},r_{2}}(l)^{q}, as in Section 21. It follows that

∥h(T)∥_{q,r_{1},r_{2}}^{q} = ∑_{n=−∞}^{∞} |h_{n}|^{q} w_{r_{1},r_{2}}(n)^{q}

(48.7)  ≤ ( ∑_{j=−∞}^{∞} |f_{j}|^{q} w_{r_{1},r_{2}}(j)^{q}) ( ∑_{l=−∞}^{∞} |g_{l}|^{q} w_{r_{1},r_{2}}(l)^{q})
= ∥f(T)∥_{q,r_{1},r_{2}}^{q} ∥g(T)∥_{q,r_{1},r_{2}}^{q},

as desired.

Suppose now that q_{k} = ∞. If f(T), g(T) ∈ LS_{0,r}(k) for some r > 0, then we saw previously that

lim_{j→±∞} |f_{j} g_{n−j}| = 0 (48.8)

for every n ∈ Z, which was used to define (48.2) as an element of k for every n. If f(T), g(T) are elements of the space LS_{0,r_{1},r_{2}}(k) defined in (47.8), then this criterion holds for r = r_{1}, r_{2}, but (48.2) does not depend on which criterion for (48.8) was used. In this case, we get that h(T) ∈ LS_{0,r}(k) for r = r_{1}, r_{2}, as before, so that h(T) ∈ LS_{0,r_{1},r_{2}}(k). We also have that

∥h(T)∥_{∞,r_{1},r_{2}} ≤ ∥f(T)∥_{∞,r_{1},r_{2}} ∥g(T)∥_{∞,r_{1},r_{2}}, (48.9)

using (41.12) with r = r_{1}, r_{2}, and (47.6).

49 Related sets and functions

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, and let A be an algebra over k with a submultiplicative q-seminorm N with respect to | · | on k for some q > 0. Suppose that A has a multiplicative identity element e with N(e) = 1, and remember that G(A) denotes the group of invertible elements in A. Let r_{1}, r_{2} be positive real numbers with r_{1} ≤ r_{2}, and put

C[r_{1}, r_{2}] = C_{A}[r_{1}, r_{2}] = {a ∈ G(A) : N(a) ≤ r_{2}, N(a^{−1}) ≤ 1/r_{1}}. (49.1)

This reduces to the set C(r) defined in (43.1) when r_{1} = r_{2} = r. If t ∈ k \ {0}, then a ∈ C[r_{1}, r_{2}] if and only if t a ∈ C[r_{1} |t|, r_{2} |t|].

Suppose from now on in this section that N is a q-norm on A, and that A is complete with respect to the q-metric associated to N. As before, one can check that C[r_{1}, r_{2}] is a closed set in A with respect to the q-metric associated to N, using the remarks in Section 39. Similarly, if q = ∞, then C[r_{1}, r_{2}] is an open set in A with respect to the q-metric associated to N.

Let T be an indeterminate, and let a be an element of (49.1). Suppose for the moment that q < ∞, and let f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j} be an element of the space LS_{r_{1},r_{2}}^{q}(k) defined in (47.1). As before, we would like to define f(a) as an element of A by

f(a) = ∑_{j=−∞}^{∞} f_{j} a^{j}. (49.2)

Let w_{r_{1},r_{2}} be defined on Z as in (47.2) again, and observe that

N(a^{j}) ≤ w_{r_{1},r_{2}}(j) (49.3)

for every j ∈ Z. This implies that

∑_{j=−∞}^{∞} N(f_{j} a^{j})^{q} ≤ ∑_{j=−∞}^{∞} |f_{j}|^{q} w_{r_{1},r_{2}}(j)^{q} = ∥f(T)∥_{q,r_{1},r_{2}}^{q}, (49.4)

where ∥f(T)∥_{q,r_{1},r_{2}} is as in (47.3). It follows that (49.2) can be defined as an element of A, as a q-absolutely convergent doubly-infinite series, as in Section 17, or as a sum over Z, as in Section 18. As usual, we also get that

N(f(a)) ≤ ( ∑_{j=−∞}^{∞} N(f_{j} a^{j})^{q})^{1/q} = ∥f(T)∥_{q,r_{1},r_{2}}. (49.5)

If f(T) ∈ LS_{0,r_{1},r_{2}}(k), then

N(f_{j} a^{j}) = |f_{j}| N(a^{j}) → 0 as j → ±∞, (49.6)

because of (49.3). If q = ∞, then we can use this to define (49.2) as an element of A, as a convergent doubly-infinite series, as in Section 17, or as a sum over Z, as in Section 18. In this situation, we obtain that

(49.7) N(f(a)) ≤ max_{j∈Z} N(f_{j} a^{j}) ≤ max_{j∈Z}(|f_{j}| w_{r_{1},r_{2}}(j)) = ∥f(T)∥_{∞,r_{1},r_{2}}.

Suppose that q < ∞ and f(T) ∈ LS_{r_{1},r_{2}}^{q}(T), or that q = ∞ and f(T) is an element of LS_{0,r_{1},r_{2}}(k), as in the previous two paragraphs. In both cases,

∑_{j=−n}^{n} f_{j} a^{j} (49.8)

tends to f(a) as n → ∞ with respect to the q-metric on A associated to N. One can check that the convergence is uniform on C[r_{1}, r_{2}], using the same type of estimates as before. If j ∈ Z, then a ↦ a^{j} is uniformly continuous as a mapping from C[r_{1}, r_{2}] into A, with respect to the q-metric on A associated to N and its restriction to C[r_{1}, r_{2}], for the same reasons as in Section 44. It follows that f(a) defines a uniformly continuous mapping from C[r_{1}, r_{2}] into A under these conditions, because of uniform convergence.

Let t ∈ k \ {0} be given, so that t a ∈ C[r_{1} |t|, r_{2} |t|], as before. Suppose that either q < ∞ and f(T) ∈ LS_{r_{1} |t|,r_{2} |t|}^{q}(k), or that q = ∞ and f(T) is an element of LS_{0,r_{1} |t|,r_{2} |t|}(k). This implies that R_{t}(f(T)) ∈ LS_{r_{1},r_{2}}^{q}(k) when q < ∞, and that R_{t}(f(T)) ∈ LS_{0,r_{1},r_{2}}(k) when q = ∞. In both cases, f(t a) and (R_{t}(f))(a) can be defined as elements of A, as before. We also have that f(t a) = (R_{t}(f))(a), as in (45.4).

Suppose now that k is complete with respect to the q_{k}-metric associated to | · |. We may as well take q ≤ q_{k}, as in Section 7. Let f(T), g(T) be elements of LS_{r_{1},r_{2}}^{q}(k) when q < ∞, or elements of LS_{0,r_{1},r_{2}}(k) when q = ∞, so that f(T) g(T) = h(T) is defined as an element of the same space, as in the previous section. Thus f(a), g(a), and h(a) are defined as elements of A, as before, and in fact h(a) = f(a) g(a), as in (44.7). This implies that a ↦ f(a) is an algebra homomorphism from LS_{r_{1},r_{2}}^{q}(k) into A when q < ∞, and from LS_{0,r_{1},r_{2}}(k) into A when q = ∞.

50 Radii of convergence

Let k be a field with a q_{k}-absolute value function | · | for some q_{k} > 0, let T be an indeterminate, and let r_{0}, r_{1} be positive real numbers with r_{0} ≤ r_{1}. Suppose that f(T) is an element of the space LS_{r_{0},r_{1}}^{q}(k) defined in Section 47 for some q > 0. If r_{0} ≤ r ≤ r_{1}, then it is easy to see that f(T) is in the space LS_{r}^{q}(k) defined in Section 40, with

∥f(T)∥_{q,r} ≤ ∥f(T)∥_{q,r_{0},r_{1}}. (50.1)

Let t ∈ [0, 1] be given, and put r_{t} = r_{0}^{1−t} r_{1}^{t}, so that r_{0} ≤ r_{t} ≤ r_{1}. One can check that ∥f(T)∥_{q,r_{t}} ≤ ∥f(T)∥_{q,r_{0}}^{1−t} ∥f(T)∥_{q,r_{1}}^{t}, (50.2)

using (27.8) when q = ∞, and (27.9) when q < ∞.

Let r_{1}, r_{2} be positive real numbers with r_{1} < r_{2}, and suppose that f(T) is an element of the space LS_{r_{1},r_{2}}^{∞}(k). If r_{1} < r < r_{2}, then one can verify that f(T) ∈ LS_{r}^{q}(k) for every q > 0, using the same type of arguments as in Section 30. In particular, f(T) ∈ LS_{0,r}(k).

Suppose that 0 ≤ r_{1} < r_{2} ≤ ∞, and put

LS_{(r_{1},r_{2})}(k) = LS_{(r_{1},r_{2})}(k, T)
(50.3)  = {f(T) ∈ LS(k) : f(T) ∈ LS_{r}^{∞}(k) for every r_{1} < r < r_{2}}.

If f(T) is an element of (50.3) and r_{1} < r < r_{2}, then we have that f(T) ∈ LS_{r}^{q}(k) for every q > 0, and in particular that f(T) ∈ LS_{0,r}(k), by the remarks in the preceding paragraph. Note that (50.3) is a linear subspace of LS(k).

Let t ∈ k \ {0} be given, so that the rescaling operator R_{t} can be defined on LS(k) as in Section 45. If f(T) ∈ LS(k), then R_{t}(f(T)) ∈ LS_{(r_{1},r_{2})}(k) if and only if f(T) ∈ LS_{(r_{1} |t|,r_{2} |t|)}. This follows from the analogous statement for LS_{r}^{q}(k).

Let A be an algebra over k with a submultiplicative q-seminorm N with respect to | · | on k for some q > 0, and a multiplicative identity element e with N(e) = 1. Put

C(r_{1}, r_{2}) = C_{A}(r_{1}, r_{2}) = {a ∈ G(A) : N(a) < r_{2}, N(a^{−1}) < 1/r_{1}}, (50.4)

where G(A) is the group of invertible elements of A, as before. If t ∈ k \ {0}, then a ∈ C(r_{1}, r_{2}) if and only if t a ∈ C(r_{1} |t|, r_{2} |t|). Suppose from now on in this section that N is a q-norm on A, and that A is complete with respect to the q-metric associated to N. One can use the remarks in Section 39 to check that C(r_{1}, r_{2}) is an open set in A with respect to the q-metric associated to N, and a closed set when q = ∞.

If a ∈ C(r_{1}, r_{2}) and f(T) ∈ LS_{(r_{1},r_{2})}(k), then f(a) can be defined as an element of A, as in the previous section. The resulting mapping from C(r_{1}, r_{2}) into A is continuous with respect to the q-metric on A associated to N and its restriction to C(r_{1}, r_{2}), because of the analogous continuity property mentioned in the previous section.

Suppose that k is complete with respect to the q_{k}-metric associated to |·|, and let f(T), g(T) ∈ LS_{(r_{1},r_{2})}(k) be given. If q_{k} < ∞, then f(T), g(T) ∈ LS_{r}^{q_{k}}(k) for every r_{1} < r < r_{2}, so that f(T) g(T) = h(T) can be defined as an element of LS_{r}^{q_{k}}(k) as in Section 41. Similarly, if q_{k} = ∞, then f(T), g(T) ∈ LS_{0,r}(k) for every r_{1} < r < r_{2}, so that f(T) g(T) = h(T) can be defined as an element of LS_{0,r}(k), as before. As in Section 48, these definitions of h(T) do not depend on r. Thus h(T) is an element of LS_{(r_{1},r_{2})}(k) in both cases.

51 The complex case

Let us take k to be the field C of complex numbers with the standard absolute value function, and let 0 ≤ r_{1} < r_{2} ≤ ∞ be given again. Thus

C(r_{1}, r_{2}) = {z ∈ C : r_{1} < |z| < r_{2}} (51.1)

is an open set in C, which corresponds to (50.4), with A = C. Let T be an indeterminate, and let f(T) = ∑_{j=−∞}^{∞} f_{j} T^{j} be a formal Laurent series in T with complex coefficients. If f(T) is an element of the space LS_{(r_{1},r_{2})}(C) defined in the previous section and z ∈ C(r_{1}, r_{2}), then

f(z) = ∑_{j=−∞}^{∞} f_{j} z^{j} (51.2)

can be defined as a complex number, as an absolutely convergent doubly-infinite series. It is well known that this defines a holomorphic function on C(r_{1}, r_{2}), and that every holomorphic function on C(r_{1}, r_{2}) corresponds to a unique Laurent series in this way.

Let r_{1}, r_{2} be positive real numbers with r_{1} ≤ r_{2}. Note that

C[r_{1}, r_{2}] = {z ∈ C : r_{1} ≤ |z| ≤ r_{2}} (51.3)

is a closed set in C, which corresponds to (49.1), with A = C. Let f(T) be an element of the space LS_{r_{1},r_{2}}^{1}(C) defined in Section 47. If z ∈ C[r_{1}, r_{2}], then f(z) can be defined as a complex number as in (51.2), where the sum is an absolutely convergent doubly-infinite series. This defines a continuous complexvalued function on C[r_{1}, r_{2}], as in Section 49. If r_{1} < r_{2}, then the restriction of f(z) to C(r_{1}, r_{2}) is holomorphic, as in the preceding paragraph. Of course, C(r_{1}, r_{2}) is the interior of C[r_{1}, r_{2}] in this case.

Let f(z) be a holomorphic function on C(r_{1}, r_{2}) for some 0 ≤ r_{1} < r_{2} ≤ ∞. If r_{1} < r < r_{2}, then put

M_{∞}(f, r) = sup{|f(z)| : z ∈ C, |z| = r}. (51.4)

Let r_{1} < ρ_{0} ≤ ρ_{1} < r_{2} and t ∈ [0, 1] be given, and put

ρ_{t} = ρ_{0}^{1−t} ρ_{1}^{t}, (51.5)

so that ρ_{0} ≤ ρ_{t} ≤ ρ_{1}. The maximum principle implies that

M_{∞}(f, ρ_{t}) ≤ max(M_{∞}(f, ρ_{0}), M_{∞}(f, ρ_{1})). (51.6)

In fact, it is well known that

M_{∞}(f, ρ_{t}) ≤ M_{∞}(f, ρ_{0})^{1−t} M_{∞}(f, ρ_{1})^{t}, (51.7)

by Hadamard’s three-circles theorem.

Let q be a positive real number, and put

M_{q}(f, r) = (\frac{1}{2 π r} ∫_{|z|=r} |f(z)|^{q} |dz|)^{1/q} (51.8)

for r_{1} < r < r_{2}, where the integral is taken over the set of z ∈ C with |z| = r, with respect to arclength. If r_{1} < ρ_{0} ≤ ρ_{1} < r_{2}, t ∈ [0, 1], and ρ_{t} is as in (51.5), then it is well known that

M_{q}(f, ρ_{t}) ≤ M_{q}(f, ρ_{0})^{1−t} M_{q}(f, ρ_{1})^{t}. (51.9)

If f is given by the Laurent expansion (51.2), then

M_{2}(f, r) = ( ∑_{j=−∞}^{∞} |f_{j}|^{2} r^{2 j})^{1/2}, (51.10)

by a standard argument. This is the same as ∥f(T)∥_{2,r} in (40.5), where T is an indeterminate, and f(T) is the formal Laurent series in T corresponding to (51.2), as before. Thus (51.9) corresponds to (50.2) when q = 2.

Part V

Some additional topics

52 Cesaro means

Let V be a vector space over the real or complex numbers, and let N be a norm on V, with respect to the standard absolute value function on R or C, as appropriate. Also let {s_{n}}_{n=0}^{∞} be a sequence of elements of V, and put

σ_{n} = \frac{1}{n + 1} ∑_{l=0}^{n} s_{l} (52.1)

for each nonnegative integer n. If {σ_{n}}_{n=0}^{∞} converges to an element s of V with respect to the metric associated to N, then s is said to be the Cesaro limit of the sequence {s_{n}}_{n=0}^{∞}. It is well known that if {s_{n}}_{n=0}^{∞} converges to s ∈ V with respect to the metric associated to N, then {σ_{n}}_{n=0}^{∞} converges to s as well.

Suppose that {σ_{n}}_{n=0}^{∞} converges to s ∈ V with respect to the metric associated to N. If a ∈ R or C, as appropriate, then

a σ_{n} = \frac{1}{n + 1} ∑_{l=0}^{n} a s_{l} (52.2)

for each n ≥ 0. Hence the Cesaro limit of {a s_{n}}_{n=0}^{∞} is equal to a s. Let {t_{n}}_{n=0}^{∞} be another sequence of elements of V, and let

τ_{n} = \frac{1}{n + 1} ∑_{l=0}^{n} t_{l} (52.3)

be the corresponding sequence of Cesaro means. Thus

σ_{n} + τ_{n} = \frac{1}{n + 1} ∑_{l=0}^{n}(s_{l} + t_{l}) (52.4)

for each n ≥ 0. If {τ_{n}}_{n=0}^{∞} also converges to t ∈ V with respect to the metric associated to N, then {σ_{n} + τ_{n}}_{n=0}^{∞} converges to s + v. This implies that the Cesaro limit of {s_{n} + t_{n}}_{n=0}^{∞} is equal to the sum of the Cesaro limits of {s_{n}}_{n=0}^{∞} and {t_{n}}_{n=0}^{∞}, when they exist.

Let ∑_{j=0}^{∞} v_{j} be an infinite series with terms in V, and let

s_{n} = ∑_{j=0}^{n} v_{j} (52.5)

be the corresponding partial sums, for each nonnegative integer n. If the sequence of Cesaro means (52.1) converges to s ∈ V with respect to the metric associated to N, then ∑_{j=0}^{∞} v_{j} is said to be Cesaro summable, with Cesaro sum equal to s. If ∑_{j=0}^{∞} v_{j} converges in the usual sense with respect to N, then ∑_{j=0}^{∞} v_{j} is Cesaro summable with the same sum, by the analogous property for convergent sequences. In this situation,

σ_{n} = \frac{1}{n + 1} ∑_{l=0}^{n} ∑_{j=0}^{l} v_{j} = \frac{1}{n + 1} ∑_{j=0}^{n} ∑_{l=j}^{n} v_{j} = ∑_{j=0}^{n} (\frac{n − j + 1}{n + 1}) v_{j} (52.6)

for every n ≥ 0.

Now let

∑_{j=−∞}^{∞} v_{j} (52.7)

be a doubly-infinite series of elements of V. As in Section 17, we say that (52.7) converges in V with respect to N if

∑_{j=0}^{∞} v_{j} and ∑_{j=1}^{∞} v_{−j} (52.8)

converge as ordinary infinite series with terms in V with respect to N, in which case the value of the sum (52.7) is defined by

∑_{j=−∞}^{∞} v_{j} = ∑_{j=0}^{∞} v_{j} + ∑_{j=1}^{∞} v_{−j}. (52.9)

If

∑_{j=1}^{∞}(v_{j} + v_{−j}) (52.10)

converges as an infinite series with terms in V, then one can use

v_{0} + ∑_{j=1}^{∞}(v_{j} + v_{−j}) (52.11)

to define the sum (52.7). If the series in (52.8) converge, then ∑_{j=1}^{∞} v_{j} converges, and hence (52.10) converges. Of course, (52.9) and (52.11) are the same in this situation.

Put

s_{n} = ∑_{j=−n}^{n} v_{j} (52.12)

for each nonnegative integer n. Thus s_{0} = v_{0}, and

s_{n} = v_{0} + ∑_{j=1}^{n}(v_{j} + v_{−j}) (52.13)

for every positive integer n. It follows that {s_{n}}_{n=0}^{∞} converges as a sequence of elements in V with respect to the metric associated to N if and only if (52.10) converges, with

lim_{n→∞} s_{n} = v_{0} + ∑_{j=1}^{∞}(v_{j} + v_{−j}). (52.14)

If the corresponding sequence of Cesaro means (52.1) converges to s ∈ V with respect to the metric associated to N, then we may say that (52.7) is Cesaro summable, with Cesaro sum equal to s. Observe that the Cesaro means can be expressed as

(52.15)  σ_{n} = \frac{1}{n + 1} ∑_{l=0}^{n} ∑_{j=−l}^{l} v_{j} = \frac{1}{n + 1} ∑_{j=−n}^{n} ∑_{l=|j|}^{n} v_{j}

= ∑_{j=−n}^{n} (\frac{n − |j| + 1}{n + 1}) v_{j}

for every n ≥ 0.

53 Abel sums

Let V be a vector space over the real or complex numbers again, and let N be a norm on V with respect to the standard absolute value function on R or C, as appropriate. In this section, we suppose that V is complete with respect to the metric associated to N. Let ∑_{j=0}^{∞} v_{j} be an infinite series with terms in V. Suppose that for each nonnegative real number r with r < 1, we have that

∑_{j=0}^{∞} N(v_{j}) r^{j} (53.1)

converges as an infinite series of nonnegative real numbers. This means that

∑_{j=0}^{∞} v_{j} r^{j} (53.2)

converges absolutely with respect to N, and hence that this series converges in V, by completeness. If the one-sided limit

lim_{r→1−} ∑_{j=0}^{∞} v_{j} r^{j} (53.3)

exists in V, with respect to the metric associated to N, then ∑_{j=0}^{∞} v_{j} is said to be Abel summable in V with respect to N. In this case, the Abel sum of ∑_{j=0}^{∞} v_{j} is defined to be (53.3). If ∑_{j=0}^{∞} v_{j} converges in the ordinary sense, then it is well known that ∑_{j=0}^{∞} v_{j} is Abel summable, with Abel sum equal to the ordinary sum. Note that (53.1) converges when 0 ≤ r < 1 and N(v_{j}) is bounded.

Suppose that ∑_{j=0}^{∞} v_{j} is Abel summable in V, which includes the convergence of (53.1) when 0 ≤ r < 1. If a is a real or complex number, as appropriate, then

∑_{j=0}^{∞} a v_{j} r^{j} = a ∑_{j=0}^{∞} v_{j} r^{j} (53.4)

for every 0 ≤ r < 1. It follows that

lim_{r→1−} ∑_{j=0}^{∞} a v_{j} r^{j} = a ( lim_{r→1−} ∑_{j=0}^{∞} v_{j} r^{j}), (53.5)

so that ∑_{j=0}^{∞} a v_{j} is Abel summable too, with Abel sum equal to a times the Abel sum of ∑_{j=0}^{∞} v_{j}. Let ∑_{j=0}^{∞} w_{j} be another infinite series with terms in V that is Abel summable with respect to N. In particular, this means that ∑_{j=0}^{∞} N(w_{j}) r^{j} converges when 0 ≤ r < 1. Thus ∑_{j=0}^{∞} N(v_{j} + w_{j}) r^{j} converges when 0 ≤ r < 1, because of the triangle inequality for N and the comparison test. Of course,

∑_{j=0}^{∞}(v_{j} + w_{j}) r^{j} = ∑_{j=0}^{∞} v_{j} r^{j} + ∑_{j=0}^{∞} w_{j} r^{j} (53.6)

for every 0 ≤ r < 1, so that

(53.7)  lim_{r→1−} ∑_{j=0}^{∞}(v_{j} + w_{j}) r^{j} = lim_{r→1−} ∑_{j=0}^{∞} v_{j} r^{j} + lim_{r→1−} ∑_{j=0}^{∞} w_{j} r^{j}.

This implies that ∑_{j=0}^{∞}(v_{j} +w_{j}) is Abel summable in V with respect to N, with Abel sum equal to the sum of the Abel sums of ∑_{j=0}^{∞} v_{j} and ∑_{j=0}^{∞} w_{j}.

Let ∑_{j=−∞}^{∞} v_{j} be a doubly-infinite series with terms in V. Suppose that

∑_{j=−∞}^{∞} N(v_{j}) r^{|j|} (53.8)

converges as a doubly-infinite series of nonnegative real numbers for every nonnegative real number r < 1. Equivalently, this means that

∑_{j=0}^{∞} N(v_{j}) r^{j} and ∑_{j=1}^{∞} N(v_{−j}) r^{j} (53.9)

converge as infinite series of nonnegative real numbers when 0 ≤ r < 1, so that ∑_{j=0}^{∞} v_{j} r^{j} and ∑_{j=1}^{∞} v_{−j} r^{j} converge in V with respect to N. As in the previous section, we put

∑_{j=−∞}^{∞} v_{j} r^{|j|} = ∑_{j=0}^{∞} v_{j} r^{j} + ∑_{j=1}^{∞} v_{−j} r^{j} (53.10)

for every 0 ≤ r < 1. Let us say that ∑_{j=−∞}^{∞} v_{j} is Abel summable in V with respect to N if the one-sided limit

lim_{r→1−} ∑_{j=−∞}^{∞} v_{j} r^{|j|} (53.11)

exists in V, with respect to the metric associated to N. As before, the Abel sum of ∑_{j=−∞}^{∞} v_{j} is defined to be (53.11) in this case. Note that ∑_{j=0}^{∞} v_{j} is Abel summable when ∑_{j=0}^{∞} v_{j} and ∑_{j=1}^{∞} v_{−j} are Abel summable as ordinary infinite series.

Put w_{0} = v_{0} and w_{j} = v_{j} + v_{−j} (53.12)

for every positive integer j. If the series in (53.9) converge, then

∑_{j=0}^{∞} N(w_{j}) r^{j} (53.13)

converges as an infinite series of nonnegative real numbers when 0 ≤ r < 1, by the triangle inequality for N and the comparison test. Under these conditions, the Abel sums

∑_{j=0}^{∞} w_{j} r^{j} (53.14)

converge in V for each 0 ≤ r < 1, and are equal to (53.10). If ∑_{j=−∞}^{∞} v_{j} is Abel summable in V with respect to N, as in the preceding paragraph, then it follows that ∑_{j=0}^{∞} w_{j} is Abel summable in V as an ordinary infinite series, and that the corresponding Abel sums are the same.

54 Abel sums and Cauchy products

Let A be an algebra over the real or complex numbers, and let N be a submultiplicative norm on A, with respect to the standard absolute value function on R or C, as appropriate. Also let ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} be infinite series with terms in A, and put

c_{n} = ∑_{j=0}^{n} a_{j} b_{n−j} (54.1)

for each nonnegative integer n. Thus ∑_{n=0}^{∞} c_{n} is the Cauchy product of the series ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l}, as in Section 20. We would like to look at the Abel summability of ∑_{n=0}^{∞} c_{n} in terms of the Abel summability of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l}. Remember that

N(c_{n}) ≤ ∑_{j=0}^{n} N(a_{j}) N(b_{n−j}) (54.2)

for every nonnegative integer n, by the triangle inequality and submultiplicativity of N.

Let r be a nonnegative real number with r < 1, and suppose that

∑_{j=0}^{∞} N(a_{j}) r^{j} and ∑_{l=0}^{∞} N(b_{l}) r^{l} (54.3)

converge as infinite series of nonnegative real numbers. Observe that

N(c_{n}) r^{n} ≤ ∑_{j=0}^{n} N(a_{j}) N(b_{n−j}) r^{n} = ∑_{j=0}^{n}(N(a_{j}) r^{j}) (N(b_{n−j}) r^{n−j}) (54.4)

for every nonnegative integer n. The right side of (54.4) is the same as the nth term of the Cauchy product of the series in (54.3). It follows that the sum over n ≥ 0 of the right side of (54.4) converges, and is equal to the product of the sums in (54.3), as in Section 20. This implies that

∑_{n=0}^{∞} N(c_{n}) r^{n} (54.5)

converges, with sum less than or equal to the product of the sums in (54.3). Thus

∑_{j=0}^{∞} a_{j} r^{j}, ∑_{l=0}^{∞} b_{l} r^{l}, and ∑_{n=0}^{∞} c_{n} r^{n} (54.6)

converge absolutely with respect to N. Suppose that A is complete with respect to the metric associated to N, so that these series converge in A with respect to N. We also get that

∑_{n=0}^{∞} c_{n} r^{n} = ( ∑_{j=0}^{∞} a_{j} r^{j}) ( ∑_{l=0}^{∞} b_{l} r^{l}), (54.7)

as in Section 20.

Suppose that the series in (54.3) converge for every 0 ≤ r < 1. This implies that (54.5) converges for every 0 ≤ r < 1, as before. Hence the series in (54.6) converge in A for every 0 ≤ r < 1, and satisfy (54.7).

Suppose that ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} are Abel summable in A with respect to N, so that the one-sided limits

(54.8)  lim_{r→1−} ∑_{j=0}^{∞} a_{j} r^{j} and lim_{r→1−} ∑_{l=0}^{∞} b_{l} r^{l}

exist in A with respect to the metric associated to N. This implies that

lim_{r→1−} ∑_{n=0}^{∞} c_{n} r^{n} (54.9)

is equal to the product of the limits in (54.8), because of (54.7). This means that ∑_{n=0}^{∞} c_{n} is Abel summable as well, with Abel sum equal to the product of the Abel sums of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l}.

55 Summing Fourier series

Let r be a positive real number, and let C(r) be the circle in the complex plane centered at 0 with radius r with respect to the standard absolute value function on C, as in Section 46. If f ∈ L^{1}(C(r)), then the jth Fourier coefficient f̂(j) of f is defined for every j ∈ Z as in (46.2). Thus the Fourier series

∑_{j=−∞}^{∞} f̂(j) z^{j} (55.1)

is a doubly-infinite series of complex numbers for every z ∈ C(r). Put

s_{n}(f, z) = ∑_{j=−n}^{n} f̂(j) z^{j} (55.2)

for every nonnegative integer n and z ∈ C(r), and let

σ_{n}(f, z) = \frac{1}{n + 1} ∑_{l=0}^{n} s_{l}(f, z) = ∑_{j=−n}^{n} (\frac{n − |j| + 1}{n + 1}) f̂(j) z^{j} (55.3)

be the corresponding Cesaro means, as in Section 52. It is well known that (55.3) converges to f(z) as n → ∞ for almost every z ∈ C(r) with respect to arclength measure. If 1 ≤ q < ∞ and f ∈ L^{q}(C(r)), then (55.3) converges to f as n → ∞ with respect to the L^{q} norm. If f is a continuous complex-valued function on C(r), then (55.3) converges to f uniformly on C(r) as n → ∞.

Let f ∈ L^{1}(C(r)) be given again, and put

A_{r_{1}}(f, z) = ∑_{j=−∞}^{∞} f̂(j) r_{1}^{|j|} z^{j} (55.4)

for every nonnegative real number r_{1} with r_{1} < 1 and z ∈ C(r). This is the Abel sum associated to the Fourier series (55.1) and r_{1}. Remember that

|f̂(j) z^{j}| = |f̂(j)| r^{j} ≤ ∥f∥_{L^{1}(C(r))} (55.5)

for every j ∈ Z and z ∈ C(r), so that

∑_{j=−∞}^{∞} |f̂(j)| r_{1}^{|j|} r^{j} ≤ ∥f∥_{L^{1}(C(r))} ∑_{j=−∞}^{∞} r_{1}^{|j|} < ∞ (55.6)

when 0 ≤ r_{1} < 1. Thus the doubly-infinite series in (55.4) converges absolutely for every z ∈ C(r) and 0 ≤ r_{1} < 1. As before, it is well known that (55.4) converges to f(z) as r_{1} → 1− for almost every z ∈ C(r) with respect to arclength measure. If 1 ≤ q < ∞ and f ∈ L^{q}(C(r)), then (55.4) converges to f as r_{1} → 1− with respect to the L^{q} norm. If f is a continuous complex-valued function on C(r), then (55.4) converges to f uniformly on C(r) as r_{1} → 1−.

Suppose that f ∈ L^{1}(C(r)) again, and let 0 ≤ r_{1} < 1 be given. Consider the partial sum

∑_{j=−n}^{n} f̂(j) r_{1}^{|j|} z^{j} (55.7)

for each nonnegative integer n and z ∈ C(r). Note that (55.7) converges to (55.4) uniformly on C(r) as n → ∞, by a standard argument.

Suppose now that f is a continuous complex-valued function on C(r). It is well known that f(z) can be approximated uniformly on C(r) by finite linear combinations of the z^{j}’s. This can be obtained from the famous theorem of Lebesgue, Stone, and Weierstrass. This also follows from the uniform convergence of the Cesaro means (55.3) to f as n → ∞. Similarly, this can be obtained from the uniform convergence of the Abel sums (55.4) to f as r_{1} → 1−, using the uniform convergence of the partial sums (55.7) for each 0 ≤ r_{1} < 1, as in the preceding paragraph.

56 Continuous functions and unitary operators

Let (V, ⟨v, w⟩) be a complex Hilbert space, and let ∥v∥ be the norm associated to the inner product. Remember that the space BL(V) of bounded linear mappings from V into itself is an algebra over the complex numbers with respect to composition of mappings, and that the corresponding operator norm ∥ · ∥_{op} is a submultiplicative norm on BL(V). Of course, BL(V) is complete with respect to the metric associated to the operator norm, because V is complete with respect to the metric associated to ∥ · ∥.

Let T be an indeterminate, and let

f(T) = ∑_{j=−n}^{n} a_{j} T^{j} (56.1)

be a formal Laurent polynomial in T with complex coefficients. If z ∈ C and z ≠ 0, then

f(z) = ∑_{j=−n}^{n} a_{j} z^{j} (56.2)

is defined as a complex number. Similarly, if A is an invertible element of BL(V), then

f(A) = ∑_{j=−n}^{n} a_{j} A^{j} (56.3)

defines an element of BL(V). Let A be a unitary operator from V onto itself. It is well known that

∥f(A)∥_{op} ≤ sup_{z∈C(1)} |f(z)|, (56.4)

where C(1) is the unit circle in the complex plane, as before.

Remember that continuous functions on C(1) corresponding to Laurent polynomials as in (56.2) are dense in the space of all continuous complex-valued functions on C(1) with respect to the supremum metric, as in the previous section. It follows that there is a unique extension of

f ↦ f(A) (56.5)

to a bounded linear mapping from the space of continuous complex-valued functions on C(1) equipped with the supremum norm into BL(V), as in Section 10. More precisely, this extension satisfies (56.4) for all continuous complex-valued functions f on C(1). Of course, the space of continuous complex-valued functions on C(1) is a commutative algebra with respect to pointwise multiplication of functions. The extension of (56.5) to continuous complex-valued functions f on C(1) is an algebra homomorphism, because of the analogous property for Laurent polynomials.

If f is a continuous complex-valued function on C(1), then the complexconjugate \overline{f} is continuous on C(1) too. Similarly, if B is a bounded linear mapping from V into itself, then the Hilbert space adjoint B^{∗} of B is a bounded linear mapping as well, as in Section 23. The hypothesis that A be a unitary operator on V is the same as saying that A is invertible, with inverse equal to A^{∗}. In fact, we have that \overline{f}(A) = f(A)^{∗} (56.6)

for every continuous complex-valued function f on C(1). To see this, suppose first that f(z) = z^{j} on C(1) for some j ∈ Z. In this case,

\overline{f(z)} = \overline{z}^{j} = z^{−j} (56.7)

on C(1), so that \overline{f}(A) = A^{−j} = (A^{j})^{∗} = f(A)^{∗}. (56.8)

It follows that (56.6) holds when f is a linear combination of z^{j}’s on C(1). If f is any continuous complex-valued function on C(1), then one can get (56.6) by approximating f uniformly by linear combinations of z^{j}’s, as before.

57 Analytic type

Let r be a positive real number, and let C(r) be the circle in the complex plane centered at 0 with radius r with respect to the standard absolute value function on C, as before. Also let f ∈ L^{1}(C(r)) be given, so that the jth Fourier coefficient f̂(j) can be defined for every j ∈ Z as in (46.2). If

f̂(j) = 0 (57.1)

for every j < 0, then f is said to be of analytic type. This means that the corresponding Fourier series reduces to

∑_{j=0}^{∞} f̂(j) z^{j}. (57.2)

If T is an indeterminate, then

∑_{j=0}^{∞} f̂(j) T^{j} (57.3)

may be considered as a formal power series in T with complex coefficients.

Remember that |f̂(j)| r^{j} ≤ ∥f∥_{L^{1}(C(r))} (57.4)

for each j. If z ∈ C satisfies |z| < r, then (57.2) converges absolutely. This defines a holomorphic function on the open disk U_{r} in C centered at 0 with radius r, as in Section 34.

Let z ∈ C(r) and 0 ≤ r_{1} < 1 be given, so that the corresponding Abel sum A_{r_{1}}(f, z) of the Fourier series of f can be defined as in (55.4). If f is of analytic type, then this reduces to

A_{r_{1}}(f, z) = ∑_{j=0}^{∞} f̂(j) r_{1}^{j} z^{j}. (57.5)

Suppose now that f is a continuous complex-valued function on C(r) of analytic type. If z ∈ U_{r}, then let f(z) be the value of the sum in (57.2). This defines f as a complex-valued function on the closed disk \overline{U_{r}} in C centered at 0 with radius r. One can check that f is continuous on \overline{U_{r}} under these conditions. This uses the fact that (57.5) converges to f uniformly on C(r) as r_{1} → 1−, as in Section 55.

Conversely, if f is a continuous complex-valued function on \overline{U_{r}} that is holomorphic on U_{r}, then the restriction of f to C(r) is of analytic type. Indeed, if 0 < ρ < r, then

∫_{C(ρ)} f(z) z^{j} dz = 0 (57.6)

for every nonnegative integer j, by Cauchy’s theorem. Equivalently, this means that ∫_{C(ρ)} f(z) z^{j+1} |dz| = 0 (57.7)

for every nonnegative integer j. If 0 < r_{1} < 1, then we can apply this to ρ = r_{1} r, to get that

∫_{C(r)} f(r_{1} z) z^{j+1} |dz| = 0 (57.8)

for every nonnegative integer j. Note that f is uniformly continuous on \overline{U_{r}}, because f is continuous on \overline{U_{r}}, and \overline{U_{r}} is compact. This implies that f(r_{1} z) tends to f(z) uniformly on \overline{U_{r}} as r_{1} → 1−. It follows that

∫_{C(r)} f(z) z^{j+1} |dz| = 0 (57.9)

for every nonnegative integer j, by taking the limit as r_{1} → 1− in (57.8).

58 Hilbert space contractions

Let (V, ⟨v, w⟩) be a complex Hilbert space with norm ∥v∥ associated to the inner product, and let ∥·∥_{op} be the corresponding operator norm on the algebra BL(V) of bounded linear mappings from V into itself. Also let T be an indeterminate, and let

f(T) = ∑_{j=0}^{n} f_{j} T^{j} (58.1)

be a formal polynomial in T with complex coefficients. If z ∈ C, then

f(z) = ∑_{j=0}^{n} f_{j} z^{j} (58.2)

is defined as a complex number, as usual. Similarly, if A is a bounded linear mapping from V into itself, then

f(A) = ∑_{j=0}^{n} f_{j} A^{j} (58.3)

is defined as an element of BL(V).

Suppose that A is a contraction on V, in the sense that

∥A∥_{op} ≤ 1. (58.4)

Let \overline{U_{1}} be the closed unit disk in C, as before. A famous theorem of von Neumann and Heinz states that

∥f(A)∥_{op} ≤ sup_{z∈\overline{U_{1}}} |f(z)|. (58.5)

Let A(U_{1}) be the space of continuous complex-valued functions on \overline{U_{1}} that are holomorphic on the open unit disk U_{1}, as in Section 34. Remember that elements of A(U_{1}) can be approximated uniformly on \overline{U_{1}} by polynomials in z. Hence there is a unique extension of

f ↦ f(A) (58.6)

to a bounded linear mapping from A(U_{1}) into BL(V), as in Section 10. Of course, this uses the supremum norm on A(U_{1}), and this extension satisfies (58.5) for every f ∈ A(U_{1}).

If f, g ∈ A(U_{1}), then their product f g is an element of A(U_{1}) too. One can check that (f g)(A) = f(A) g(A), (58.7)

so that (58.6) is an algebra homomorphism. More precisely, this can be verified directly when f and g are polynomials in z, and otherwise one can reduce to that case by approximation.

References

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Index

A(U_{r}), 60 Abel summability, 88, 90 absolute convergence, 19, 30 absolute value functions, 13

archimedean, 24 discrete, 25 non-archimedean, 24 ultrametric, 13

algebras, 4 analytic type, 94

Banach algebras, 21 Banach spaces, 18 BL(V), 21 BL(V, W), 20 bounded functions, 23, 47 bounded linear mappings, 20

C, 13 C(r), 77 C(r), 73 C(r_{1}, r_{2}), 84 C(r_{1}, r_{2}), 84 C(X, V), 23 c(X, V), 4 C[r_{1}, r_{2}], 85 C[r_{1}, r_{2}], 81 C^{∗} algebras, 46 c^{+}(Z, k), 8 c_{0}(X, V), 27 c_{0,w}(X, V), 48 c_{00}(X, V), 4 c_{00}^{+}(Z, k), 8 Cauchy products, 34, 37 Cauchy–Schwarz inequality, 42, 49 Cesaro limits, 86 Cesaro summability, 87, 88 closed balls, 12 commutative algebras, 4 complex-linear mappings, 41 conjugate-linear mappings, 41 convergent series, 18

discrete metric, 12 equivalent absolute value functions, 15

formal Laurent polynomials, 7 formal Laurent series, 8 formal polynomials, 5 formal power series, 5 Fourier coefficients, 77 Fourier series, 77

G(A), 66

H(U), 59 H^{∞}(U), 60 H^{q}(U_{r}), 62 Hardy spaces, 62 Hilbert space adjoints, 43 Hilbert spaces, 42 Hölder’s inequality, 49 homomorphisms, 5

inner products, 41 invertibility, 5 involutions, 45 isometric linear mappings, 44

k((T)), 9 k[[T]], 5 k[T], 5 k_{r}^{∞}[[T]], 52 k_{r}^{q}[[T]], 51 k_{0,r}[[T]], 52 k_{r}[[T]], 55

L(V), 5 L(V, W), 5 ℓ^{∞}(X, V), 23 ℓ_{w}^{∞}(X, V), 47 ℓ^{r}(X, V), 26 ℓ_{w}^{r}(X, V), 47 LP (k), 68 LS(k), 68 LS_{r}^{∞}(k), 69 LS_{r}^{q}(k), 68 LS_{r_{1},r_{2}}^{q}(k), 78 LS_{(r_{1},r_{2})}(k), 83 LS_{0,r}(k), 69 LS_{0,r_{1},r_{2}}(k), 79

measure spaces, 49 metrics, 11 Minkowski’s inequality, 11 multiplicative q-seminorms, 21

norms, 15

open balls, 12 open sets, 12 Ostrowski’s theorems, 24, 25

P (k), 68 p-adic absolute value, 14 p-adic metric, 14 p-adic numbers, 14 P S(k), 68 P S_{r}^{∞}(k), 69 P S_{r}^{q}(k), 69 P S_{0,r}(k), 70

Q, 14 Q_{p}, 14 q-absolute convergence, 19, 30 q-absolute value functions, 13 q-Banach algebras, 21 q-Banach spaces, 18 q-metrics, 11 q-norms, 15 q-semimetrics, 11 q-seminorms, 15

R, 13 R_{+}, 25 radius of convergence, 54 real-linear mappings, 41 r-summable functions, 26, 47

semi-ultrametrics, 11 semi-ultranorms, 15 semimetrics, 11 seminorms, 15 submultiplicative q-seminorms, 21 summable functions, 26 supports of functions, 4 supremum q-seminorms, 24

trivial absolute value function, 13 trivial ultranorm, 16

ultrametrics, 11 ultranorms, 15 unitary mappings, 45

vanishing at infinity, 27, 48

Z, 8 Z_{+}, 5