Some topics in analysis related to Banach algebras, 3

Stephen Semmes Rice University

Abstract Some spaces of linear mappings and matrices are discussed, as well as

some properties of functions defined by power series.

Contents

I Basic notions 3

1 Metrics and ultrametrics 3 2 Absolute value functions 5 3 Uniform continuity and completeness 7 4 The archimedean property and discreteness 8 5 Norms and ultranorms 9 6 Comparing topologies 11 7 Finite support 12 8 Bounded functions 13 9 Summable functions 14 10 Infinite series 16 11 Bounded linear mappings 18 12 Banach spaces 19 13 Sums of vectors 21 14 Banach algebras 22 15 Cauchy products 23

II Linear mappings, matrices, and involutions 25

16 Some linear mappings 26 17 Bounded linear functionals 27 18 Inner product spaces 30 19 Hilbert spaces 31 20 Isometric linear mappings 33 21 n × n Matrices 34 22 Involutions 36 23 Infinite matrices 38 24 Double sums 40 25 A summability condition 42 26 Another summability condition 45 27 A c_{0} condition 48 28 Another c_{0} condition 50

III Lipschitz mappings and power series 52

29 Lipschitz mappings 52 30 Lipschitz seminorms 53 31 Bounded continuous functions 55 32 Bounded Lipschitz functions 57 33 Polynomials 58 34 Power series 59 35 Lipschitz conditions 61 36 Some isometric mappings 62 37 Some contractions 64

IV The exponential function 66

38 The real and complex cases 67 39 Trivial absolute values on Q 68 40 Some preliminary facts 70 41 The p-adic case 72 References 74 Index 75

Part I

Basic notions

1 Metrics and ultrametrics

Let X be a set, and let d(x, y) be a nonnegative real-valued function d(x, y) defined for x, y ∈ X. As usual, d(·, ·) is said to be a metric on X if it satisfies the following three conditions: first, (1.1) d(x, y) = 0 if and only if x = y; second, d(x, y) = d(y, x) (1.2) for every x, y ∈ X; and third, d(x, z) ≤ d(x, y) + d(y, z) (1.3) for every x, y, z ∈ X. If d(·, ·) satisfies (1.1), (1.2), and d(x, z) ≤ max(d(x, y), d(y, z)) (1.4) for every x, y, z ∈ X, then d(·, ·) is said to be an ultrametric on X. Note that (1.4) implies (1.3), so that ultrametrics are metrics.

The discrete metric can be defined on any set X by putting d(x, y) equal to 1 when x ≠ y, and to 0 when x = y. It is easy to see that this defines an ultrametric on X.

If a is a positive real number with a ≤ 1, then it is well known that

(r + t)^{a} ≤ r^{a} + t^{a} (1.5)

for all nonnegative real numbers r, t. To see this, observe first that

max(r, t)^{a} = max(r^{a}, t^{a}) ≤ r^{a} + t^{a} (1.6)

for every a > 0, so that

max(r, t) ≤ (r^{a} + t^{a})^{1/a}. (1.7)

If a ≤ 1, then it follows that

(1.8)  r + t ≤ max(r, t)^{1−a} (r^{a} + t^{a})
≤ (r^{a} + t^{a})^{(1−a)/a} (r^{a} + t^{a})^{1/a} = (r^{a} + t^{a})^{1/a},

which implies (1.5).

Let d(x, y) be a metric on a set X, and let a be a positive real number. If a ≤ 1, then it is easy to see that

d(x, y)^{a} (1.9)

also defines a metric on X, using (1.5). If d(x, y) is an ultrametric on X, then (1.9) is an ultrametric on X for every a > 0.

Let d(x, y) be a metric on a set X again. As usual, the open ball in X centered at a point x ∈ X with radius r > 0 with respect to d(·, ·) is defined by

B(x, r) = B_{d}(x, r) = {y ∈ X : d(x, y) < r}. (1.10)

Similarly, \overline{B}(x, r) = \overline{B}_{d}(x, r) = \{y ∈ X : d(x, y) ≤ r\} (1.11)

is the closed ball in X centered at x ∈ X with radius r ≥ 0 with respect to d(·, ·). Remember that one can define a topology on X using d(·, ·) in a standard way. It is well known that open balls in X with respect to d(·, ·) are open sets with respect to this topology. More precisely, the collection of open balls in X with respect to d(·, ·) forms a base for this topology on X. One can also verify that closed balls in X with respect to d(·, ·) are closed sets with respect to this topology.

If (1.9) is a metric on X as well for some a > 0, then the corresponding open and closed balls in X are given by

B_{d^{a}}(x, r^{a}) = B_{d}(x, r) (1.12)

for every x ∈ X and r > 0, and

\overline{B}_{d^{a}}(x, r^{a}) = \overline{B}_{d}(x, r) (1.13) for every x ∈ X and r ≥ 0, respectively. It follows from (1.12) that the topologies determined on X by d(·, ·) and (1.9) are the same.

Suppose for the moment that d(x, y) is an ultrametric on X. If x, y ∈ X satisfy d(x, y) < r for some r > 0, then one can check that

B(x, r) ⊆ B(y, r), (1.14)

using the ultrametric version of the triangle inequality. This is symmetric in x and y, so that B(x, r) = B(y, r) (1.15)

when d(x, y) < r. Similarly, if d(x, y) ≤ r for some r ≥ 0, then

\overline{B}(x, r) ⊆ \overline{B}(y, r), (1.16)

and hence \overline{B}(x, r) = \overline{B}(y, r). (1.17)

In particular, this implies that closed balls in X with positive radius are open sets with respect to the topology determined by d(·, ·). One can verify that open balls in X are closed sets too. Of course, the topology determined on X by the discrete metric is the discrete topology.

2 Absolute value functions

Let k be a field. A nonnegative real-valued function |x| defined on k is said to be an absolute value function on k if it satisfies the following three conditions: first, for each x ∈ k,

(2.1) |x| = 0 if and only if x = 0;

second, |x y| = |x| |y| (2.2)

for every x, y ∈ k; and third,

|x + y| ≤ |x| + |y| (2.3)

for every x, y ∈ k. Of course, the standard absolute value functions on the fields R of real numbers and C of complex numbers are absolute value functions in this sense.

Let k be a field, and let | · | be a nonnegative real-valued function on k that satisfies (2.1) and (2.2). 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 facts that |1| > 0 by (2.1), and |1| = |1^{2}| = |1|^{2}, by (2.2). Similarly, if x ∈ k satisfies x^{n} = 1 for some positive integer n, then |x| = 1. It follows that | − 1| = 1, where −1 is the additive inverse of 1 in k, because (−1)^{2} = 1. If | · | is an absolute value function on k, then

d(x, y) = |x − y| (2.4) defines a metric on k. More precisely, (2.4) is symmetric in x and y, because | − 1| = 1.

A nonnegative real-valued function |·| on a field k is said to be an ultrametric absolute value function on k if it satisfies (2.1), (2.2), and

|x + y| ≤ max(|x|, |y|) (2.5)

for every x, y ∈ k. Clearly (2.5) implies (2.3), so that an ultrametric absolute value function is an ordinary absolute value function. If | · | is an ultrametric absolute value function on k, then (2.4) is an ultrametric on k.

If k is any field, then the trivial absolute value function is defined on k by putting |x| = 1 for every x ∈ k with x ≠ 0, and |0| = 0. It is easy to see that this defines an ultrametric absolute value function on k. The ultrametric on k associated to the trivial absolute value function as in (2.4) is the discrete metric on k.

The p-adic absolute value function |x|_{p} is defined on the field Q of rational numbers for each prime number p as follows. Let x ∈ Q be given, and suppose that x ≠ 0, since we should put |0|_{p} = 0. We can express x as p^{j} (a/b), where a, b, and j are integers, a, b ≠ 0, and neither a nor b is a multiple of p. Note that j is uniquely determined by x, and put

|x|_{p} = p^{−j}. (2.6)

It is not difficult to verify that this defines an ultrametric absolute value function on Q.

Let k be any field again, and let | · | be an absolute value function on k. If a is a positive real number less than or equal to 1, then

|x|^{a} (2.7)

also defines an absolute value function on k. This uses (1.5) to get that (2.7) satisfies the triangle inequality on k. If | · | is an ultrametric absolute value function on k, then (2.7) is an ultrametric absolute value function on k for every a > 0. If | · | is the standard Euclidean absolute value function on Q, then (2.7) does not satisfy the triangle inequality on Q when a > 1.

Let | · |_{1}, | · |_{2} be absolute value functions on a field k. If there is a positive real number a such that |x|_{2} = |x|_{1}^{a} (2.8)

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 metrics associated to | · |_{1} and | · |_{2} are the same. Conversely, if the topologies determined on k by the metrics associated to | · |_{1} and | · |_{2} are the same, then it is well known that | · |_{1} and | · |_{2} are equivalent on k in the sense of (2.8). A famous theorem of Ostrowski says that any absolute value function on Q is either trivial, or equivalent to the standard Euclidean absolute value function, or equivalent to the p-adic absolute value function for some prime number p.

3 Uniform continuity and completeness

Let (X, d(x, y)) be a metric space, let a be a positive real number, and suppose that d(x, y)^{a} also defines a metric on X. As in Section 1, the topologies determined on X by d(x, y) and d(x, y)^{a} are the same. More precisely, the identity mapping on X is uniformly continuous as a mapping from X equipped with d(x, y) into X equipped with d(x, y)^{a}, and from X equipped with d(x, y)^{a} into X equipped with d(x, y). Note that a sequence of elements of X is a Cauchy sequence with respect to d(x, y) if and only if it is a Cauchy sequence with respect to d(x, y)^{a}. It follows that X is complete as a metric space with respect to d(x, y) if and only if X is complete with respect to d(x, y)^{a}.

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces, and suppose that d_{X}^{a}, d_{Y}^{b} are also metrics on X, Y, respectively, for some positive real numbers a, b. If a mapping f from X into Y is uniformly continuous with respect to d_{X}, d_{Y}, respectively, then f is uniformly continuous with respect to d_{X}^{a}, d_{Y}^{b}, respectively. This can be verified directly, or using the remark about uniform continuity in the previous paragraph, and the fact that compositions of uniformly continuous mappings are uniformly continuous. Remember that uniformly continuous mappings send Cauchy sequences in the domain to Cauchy sequences in the range, by a simple argument.

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces again, let E be a dense subset of X, and suppose that f is a uniformly continuous mapping from E into Y. If Y is complete, then it is well known that f has a unique extension of a uniformly continuous mapping from X into Y. Of course, uniqueness of the extension only requires ordinary continuity.

If a metric space (X, d) is not already complete, then it is well known that X has a completion, which can be given as an isometric embedding of X onto a dense subset of a complete metric space. The completion is unique up to isometric equivalence, because of the extension theorem mentioned in the previous paragraph. Note that a closed subset of a complete metric space is also complete with respect to the restriction of the metric to the subset, by a standard argument.

Let (X, d) be a metric space, and let E be a subset of X. Of course, the restriction of d(x, y) to x, y ∈ E defines a metric on E. Similarly, if d(x, y) is an ultrametric on X, then its restriction to E is an ultrametric on E. If d(x, y) is any metric on X, E ⊆ X is dense in X with respect to d(x, y), and the restriction of d(x, y) to x, y ∈ E is an ultrametric on E, then one can check that d(x, y) is an ultrametric on X. In particular, if d(x, y) is an ultrametric on X, then the corresponding metric on a completion of X is an ultrametric as well.

Let k be a field with an absolute value function |·|, and let k_{0} be a subfield of k. As before, the restriction of |x| to x ∈ k_{0} defines an absolute value function on k_{0}. If |x| is an ultrametric absolute value function on k, then its restriction to x ∈ k_{0} is an ultrametric absolute value function on k_{0}. Let |·| be any absolute value function on k again, let k_{0} be a subfield of k, and suppose that k_{0} is dense in k with respect to the metric associated to | · |. If the restriction of |x| to x ∈ k_{0} is an ultrametric absolute value function on k_{0}, then one can verify that | · | is an ultrametric absolute value function on k.

Let k be a field with an absolute value function | · | again, which leads to a metric on k as in (2.4). If k is not already complete with respect to this metric, then one can pass to a completion of k as a metric space, as before. One can show that the field operations on k can be extended continuously to the completion, so that the completion is also a field. Similarly, | · | can be extended continuously to the completion, which corresponds to the distance to 0 in the completion. This extension of | · | defines an absolute value on the completion, and the metric on the completion corresponds to this extension of | · | in the usual way. This completion of k with respect to | · | is unique up to isomorphic isometric equivalence, as before. If | · | is an ultrametric absolute value function on k, then the corresponding absolute value function on the completion is an ultrametric absolute value function as well.

In particular, one can apply this to the p-adic absolute value function |·|_{p} on Q, for any prime number p. This leads to the field Q_{p} of p-adic numbers. The extension of |x|_{p} to x ∈ Q_{p} is denoted the same way, and defines an ultrametric absolute value function on Q_{p}.

4 The archimedean property and discreteness

Let k be a field, and let | · | be an absolute value function on k. Also let Z_{+} be the set of positive integers, as usual. If x ∈ k and n ∈ Z_{+}, then we let n · x be the sum of n x’s in k. Suppose that there is an n ∈ Z_{+} such that |n · 1| > 1, where the first 1 is the multiplicative identity element in k, and the second 1 is the multiplicative identity element in R. If j ∈ Z_{+}, then one can check that n^{j} · 1 = (n · 1)^{j}, so that

|n^{j} · 1| = |(n · 1)^{j}| = |n · 1|^{j} → ∞ as j → ∞. (4.1)

In this case, | · | is said to be archimedean on k. Otherwise, if |n · 1| ≤ 1 for every n ∈ Z_{+}, then | · | is said to be non-archimedean on k. Equivalently, if there is a finite upper bound for |n · 1|, n ∈ Z_{+}, then | · | is non-archimedian on k, by the previous remark. If | · | is an ultrametric absolute value function on k, then it is easy to see that | · | is non-archimedean on k. Conversely, if | · | is a non-archimedean absolute value function on k, then it is well known that | · | is an ultrametric absolute value function on k.

Let | · | be an absolute value function on a field k again, and observe that

{|x| : x ∈ k, x ≠ 0} (4.2)

is a subgroup of the multiplicative group R_{+} of positive real numbers. If the real number 1 is a limit point of (4.2) with respect to the standard topology on R, then one can check that (4.2) is dense in R_{+} with respect to the topology induced by the standard topology on R. Otherwise, | · | is said to be discrete on k if 1 is not a limit point of (4.2) with respect to the standard topology on R.

Let ρ_{1} be the nonnegative real number defined by

ρ_{1} = sup{|x| : x ∈ k, |x| < 1}, (4.3)

and note that ρ_{1} ≤ 1. One can check that ρ_{1} = 0 if and only if | · | is the trivial absolute value function on k, and that ρ_{1} < 1 if and only if | · | is discrete on k. If | · | is nontrivial and discrete on k, so that 0 < ρ_{1} < 1, then one can verify that the supremum in (4.3) is attained. More precisely, (4.2) consists of integer powers of ρ_{1} in this case.

If k has positive characteristic, then there are only finitely many elements of k of the form n · 1, with n ∈ Z_{+}. This implies that any absolute value function on k is non-archimedean. Suppose that k has characteristic 0, so that there is a natural embedding of Q into k. Let | · | be an absolute value function on k, which leads to an absolute value function on Q, using the natural embedding of Q into k. If | · | is a archimedean on k, then it is easy to see that the corresponding absolute value function on Q is archimedean. This implies that the corresponding absolute value function on Q is equivalent to the standard Euclidean absolute value function on Q, by Ostrowski’s theorem, as in Section 2. In particular, this means that the corresponding absolute value function on Q is not discrete on Q. It follows that | · | is not discrete on k when | · | is archimedean on k. If | · | is a discrete absolute value function on a field k, then we get that | · | is non-archimedean on k.

Suppose that | · | is an archimedean absolute value function on a field k, and that k is complete with respect to the metric associated to | · |. Under these conditions, another famous theorem of Ostrowski states that k is isomorphic to R of C, in such a way that | · | corresponds to an absolute value function on R or C that is equivalent to the standard one.

5 Norms and ultranorms

Let k be a field, and let | · | be an absolute value function on k. Also let V be a vector space over k. A nonnegative real-valued function N on V is said to be a seminorm on V with respect to | · | on k if it satisfies the following two conditions: first, N(t v) = |t| N(v) (5.1)

for every t ∈ k and v ∈ V; and second,

N(v + w) ≤ N(v) + N(w) (5.2)

for every v, w ∈ V. Note that (5.1) implies that N(0) = 0, by taking t = 0. If N also satisfies N(v) > 0 for every v ∈ V with v ≠ 0, then N is said to be a norm on V with respect to | · | on k.

Similarly, a nonnegative real-valued function N on V is said to be a semiultranorm with respect to | · | if it satisfies (5.1) and

N(v + w) ≤ max(N(v), N(w)) (5.3) for every v, w ∈ V. Clearly (5.3) implies (5.2), so that semi-ultranorms are seminorms. A semi-ultranorm that is also a norm is called an ultranorm. If N is a semi-ultranorm on V with respect to | · | on k, and if N(v) > 0 for some v ∈ V, then it is easy to see that |·| is an ultrametric absolute value function on k. More precisely, one can get the ultrametric version of the triangle inequality (2.5) for | · | on k using (5.1) and (5.3) in this case.

If N is a norm on V with respect to | · | on k, then

d(v, w) = N(v − w) (5.4)

defines a metric on k. This uses the fact that | − 1| = 1, as in Section 2, to get that (5.4) is symmetric in v and w. If N is an ultranorm on V, then (5.4) is an ultrametric on V.

Of course, k may be considered as a one-dimensional vector space over itself. Similarly, | · | may be considered as a norm on k, with respect to itself. If | · | is an ultrametric absolute value function on k, then | · | may be considered as an ultranorm on k.

Suppose for the moment that | · | is the trivial absolute value function on a field k. If V is any vector space over k, then 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. The ultrametric on V corresponding to N as in (5.4) is the discrete metric.

Let | · | be any absolute value function on a field k again, and let n be a positive integer. Consider the space k^{n} of n-tuples of elements of k, which is the Cartesian product of n copies of k. This is a vector space over k, with respect to coordinatewise addition and scalar multiplication, as usual. If v = (v_{1}, …, v_{n}) ∈ k^{n}, then put

∥v∥_{1} = ∑_{j=1}^{n} |v_{j}| (5.5)

and

∥v∥_{∞} = max_{1≤j≤n} |v_{j}|. (5.6)

One can check that (5.5) and (5.6) define norms on k^{n}, with respect to | · | on k. If | · | is an ultrametric absolute value function on k, then (5.6) is an ultranorm on k^{n}. Note that ∥v∥_{∞} ≤ ∥v∥_{1} ≤ n ∥v∥_{∞} (5.7)

for every v ∈ k^{n}.

Let v ∈ k^{n} and a positive real number be given, and consider the open ball in k^{n} centered at v with radius r with respect to the metric associated to (5.6). This is the same as the Cartesian product of the balls in k centered at v_{j} with radius r with respect to the metric associated to | · | for j = 1, …, n. Using this, it is easy to see that the topology determined on k^{n} by the metric associated to (5.6) is the same as the product topology corresponding to the topology determined on k by the metric associated to | · | on each factor. The topology determined on k^{n} by the metric associated to (5.5) is the same as the topology determined by the metric associated to (5.6), because of (5.7).

6 Comparing topologies

Let k be a field with an absolute value function | · |, and let V be a vector space over k. If N is a norm on V, then put

B_{N}(v, r) = {w ∈ V : N(v − w) < r} (6.1)

for each v ∈ V and positive real number r. This is the same as the open ball in V centered at v with radius r with respect to the metric (5.4) associated to N, as in (1.10).

Let N_{1}, N_{2} be norms on V with respect to |·| on k. Suppose for the moment that there is a positive real number C such that

N_{1}(v) ≤ C N_{2}(v) (6.2)

for every v ∈ V. This implies that

B_{N_{2}}(v, r) ⊆ B_{N_{1}}(v, C r) (6.3)

for every v ∈ V and r > 0, using the notation (6.1) for open balls in V with respect to N_{1} and N_{2}. If U ⊆ V is an open set with respect to the topology determined by the metric associated to N_{2}, then it follows from (6.3) that U is also an open set with respect to the topology determined by the metric associated to N_{1}.

Conversely, suppose that the topology determined on V by the metric associated to N_{2} is at least as strong as the topology determined on V by the metric associated to N_{1}. Let r_{1} > 0 be given, and remember that B_{N_{1}}(0, r_{1}) is an open set in V with respect to the topology determined by the metric associated to N_{1}. Thus B_{N_{1}}(0, r_{1}) is also an open set in V with respect to the topology determined by the metric associated to N_{2}, by hypothesis. This implies that there is an r_{2} > 0 such that

B_{N_{2}}(0, r_{2}) ⊆ B_{N_{1}}(0, r_{1}), (6.4)

because 0 is an element of B_{N_{1}}(0, r_{1}). Of course, we could simply take r_{1} = 1 here.

Let v ∈ V be given, and suppose that t ∈ k satisfies

N_{2}(v) < |t| r_{2}. (6.5)

Equivalently, this means that t ≠ 0, and that N_{2}(t^{−1} v) = |t|^{−1} N_{2}(v) < r_{2}. Thus (6.4) implies that

|t|^{−1} N_{1}(v) = N_{1}(t^{−1} v) < r_{1}, (6.6)

so that N_{1}(v) < |t| r_{1}.

Suppose that | · | is not trivial on k, which implies that there is a t_{0} ∈ k such that |t_{0}| > 1. If v ∈ V and v ≠ 0, then there is an integer j such that

|t_{0}|^{j−1} r_{2} ≤ N_{2}(v) < |t_{0}|^{j} r_{2}. (6.7)

Thus we can apply the remarks in the previous paragraph to t = t_{0}^{j}, to get that

N_{1}(v) < |t_{0}|^{j} r_{1} ≤ |t_{0}| (r_{1}/r_{2}) N_{2}(v), (6.8)

using the first inequality in (6.7) in the second step. This shows that (6.2) holds with C = |t_{0}| (r_{1}/r_{2}) when v ≠ 0, and of course (6.2) is trivial when v = 0.

7 Finite support

Let k be a field, and let V be a vector space over k. Also let X be a nonempty set, and let c(X, V) be the space of V -valued functions on X. This is a vector space over k with respect to pointwise addition and scalar multiplication. If f ∈ c(X, V), then the support of f is defined to be the set of x ∈ X such that f(x) ≠ 0. Let c_{00}(X, V) be the subset of c(X, V) consisting of V -valued functions f on X whose supports have only finitely many elements. It is easy to see that c_{00}(X, V) is a linear subspace of c(X, V). Of course, if X has only finitely many elements, then c_{00}(X, V) is equal to c(X, V).

Let | · | be an absolute value function on k, and let N be a norm on V with respect to | · | on k. If f ∈ c_{00}(X, V), then put

∥f∥_{1} = ∑_{x∈X} N(f(x)), (7.1)

where the sum on the right reduces to a finite sum of nonnegative real numbers. Similarly,

∥f∥_{∞} = max_{x∈X} N(f(x)) (7.2)

reduces to the maximum of finitely many nonnegative real numbers. One can check that (7.1) and (7.2) define norms on c_{00}(X, V) with respect to | · | on k. If N is an ultranorm on V, then (7.2) is an ultranorm on c_{00}(X, V).

Clearly ∥f∥_{∞} ≤ ∥f∥_{1} (7.3)

for every f ∈ c_{00}(X, V). Thus the topology determined on c_{00}(X, V) by the metric associated to ∥f∥_{1} is at least as strong as the topology determined on c_{00}(X, V) by the metric associated to ∥f∥_{∞}, as in the previous section. If X has only finitely many elements, then

∥f∥_{1} ≤ (#X) ∥f∥_{∞} (7.4)

for every f ∈ c_{00}(X, V), where #X denotes the number of elements in X. In this case, the topologies determined on c_{00}(X, V) by the metrics associated to ∥f∥_{1} and ∥f∥_{∞} are the same.

Suppose for the moment that | · | is the trivial absolute value function on k, and that N is the trivial ultranorm on V. Observe that ∥f∥_{∞} is the same as the trivial ultranorm on c_{00}(X, V) in this situation. In particular, the ultrametric on c_{00}(X, V) corresponding to ∥f∥_{∞} is the discrete metric, which determines the discrete topology on c_{00}(X, V). It follows that the topology determined on c_{00}(X, V) by the metric associated to ∥f∥_{1} is at least as strong as the discrete topology, as in the previous paragraph. Of course, this means that the topology determined on c_{00}(X, V) by the metric associated to ∥f∥_{1} is the discrete topology too.

Let | · | be any absolute value function on k again, and let N be any norm with respect to | · | on k. If f ∈ c_{00}(X, V), then

∑_{x∈X} f(x) (7.5)

can be defined as an element of V, by reducing to a finite sum. It is easy to see that this defines a linear mapping from c_{00}(X, V) into V. Observe that

N( ∑_{x∈X} f(x)) ≤ ∥f∥_{1} (7.6)

for every f ∈ c_{00}(X, V), because of the triangle inequality. Similarly, if N is an ultranorm on V, then

N( ∑_{x∈X} f(x)) ≤ ∥f∥_{∞} (7.7)

for every f ∈ c_{00}(X, V).

8 Bounded functions

Let k be a field with an absolute value function | · |, and let V be a vector space over k with a norm N with respect to | · | on k. Also let X be a nonempty set, and let f be a V -valued function on X. As usual, f is said to be bounded on X with respect to N on V if there is a finite upper bound for N(f(x)), x ∈ X. Let ℓ^{∞}(X, V) = ℓ_{N}^{∞}(X, V) be the space of bounded V -valued functions f on X, and put

∥f∥_{∞} = ∥f∥_{ℓ^{∞}(X,V)} = sup_{x∈X} N(f(x)) (8.1)

for each such function f. 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 (8.1) defines a norm on ℓ^{∞}(X, V) with respect to | · | on k. If N is an ultranorm on V, then (8.1) defines an ultranorm on ℓ^{∞}(X, V). If | · | is the trivial absolute value function on k, and N is the trivial ultranorm on V, then (8.1) is the trivial ultranorm on ℓ^{∞}(X, V).

A V -valued function f on X is said to vanish at infinity with respect to N if for each ϵ > 0, N(f(x)) < ϵ (8.2)

for all but finitely many x ∈ X. Let c_{0}(X, V) = c_{0,N}(X, V) be the space of these functions on X. It is easy to see that

c_{0}(X, V) ⊆ ℓ^{∞}(X, V), (8.3) and that c_{0}(X, V) is a linear subspace of ℓ^{∞}(X, V). One can also check that c_{0}(X, V) is a closed set in ℓ^{∞}(X, V), with respect to the metric associated to the supremum norm. Of course, if X has only finitely many elements, then every V -valued function on X vanishes at infinity.

Clearly c_{00}(X, V) ⊆ c_{0}(X, V). (8.4)

If f ∈ c_{0}(X, V), then f can be approximated by V -valued functions on X with finite support uniformly on X. 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 metric. If | · | is the trivial absolute value function on k, N is the trivial ultranorm on V, and f ∈ c_{0}(X, V), then f has finite support in X.

If N is any norm on V, and f ∈ c_{0}(X, V), then the support of f has at most finitely or countably many elements. More precisely, for each n ∈ Z_{+}, there are at most finitely many x ∈ X such that N(f(x)) ≥ 1/n, and the support of f is the union of this sequence of finite sets. Note that (8.1) is the same as (7.2) when f ∈ c_{00}(X, V). If f ∈ c_{0}(X, V), then the supremum on the right side of (8.1) is attained. This is trivial when f(x) = 0 for every x ∈ X, and otherwise the supremum can be reduced to the maximum over a finite subset of X.

If V is complete with respect to the metric associated to N, then ℓ^{∞}(X, V) is complete with respect to the metric associated to (8.1), by standard arguments. Indeed, if {f_{j}}_{j=1}^{∞} is a Cauchy sequence in ℓ^{∞}(X, V) with respect to the supremum metric, then {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in V for each x ∈ X, with respect to the metric associated to N. If V is complete, then it follows that {f_{j}(x)}_{j=1}^{∞} converges to an element f(x) of V for each x ∈ X. The Cauchy condition for {f_{j}}_{j=1}^{∞} implies in particular that {f_{j}}_{j=1}^{∞} is bounded in ℓ^{∞}(X, V), which can be used to show that f is bounded on X. One can use the Cauchy condition for {f_{j}}_{j=1}^{∞} in ℓ^{∞}(X, V) again to get that {f_{j}}_{j=1}^{∞} converges to f with respect to the supremum metric, as desired.

9 Summable functions

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

∑_{x∈X} f(x) (9.1)

is defined as a nonnegative extended real number as the supremum of the sums

∑_{x∈A} f(x) (9.2)

over all nonempty finite subsets A of X. If (7.5) is finite, then f is said to be summable on X. Of course, (9.1) reduces to a finite sum when X has only finitely many elements, or when f has finite support in X. If f is summable on X, then it is easy to see f vanishes at infinity on X, with respect to the standard absolute value function on R.

If f is summable on X, and t is a nonnegative real number, then one can check that t f(x) is summable on X as well, with

∑_{x∈X} t f(x) = t ∑_{x∈X} f(x). (9.3)

Similarly, if g is another nonnegative real-valued summable function on X, then f + g is summable on X, with

∑_{x∈X}(f(x) + g(x)) = ∑_{x∈X} f(x) + ∑_{x∈X} g(x). (9.4)

Both statements can be obtained directly from the definitions, using the analogous properties of finite sums. There are also versions of (9.3) and (9.4) for arbitrary nonnegative real-valued functions on X, with suitable interpretations for extended real numbers. In particular, if f is not summable on X, then one may apply (9.3) to positive real numbers t, or interpret the right side of (9.3) as being equal to 0 when t = 0.

If {f_{j}}_{j=1}^{∞} is a sequence of nonnegative real-valued functions on X that converges pointwise to a nonnegative real-valued function f on X, then

∑_{x∈X} f(x) ≤ sup_{j≥1} ( ∑_{x∈X} f_{j}(x)). (9.5)

This is a simplified version of Fatou’s lemma for sums. More precisely, the supremum on the right side of (9.5) is defined as a nonnegative extended real number, and of course (9.5) is trivial when the supremum is +∞. To get (9.5), let A be a nonempty finite subset of X, and observe that

(9.6) ∑_{x∈A} f(x) = lim_{j→∞} ( ∑_{x∈A} f_{j}(x)) ≤ sup_{j≥1} ( ∑_{x∈A} f_{j}(x)) ≤ sup_{j≥1} ( ∑_{x∈X} f_{j}(x)).

This implies (9.5), by the definition of the sum (9.1).

Let k be a field with an absolute value function | · |, let V be a vector space over k, and let N be a norm on V with respect to | · | on k. A V -valued function f on X is said to be summable on X with respect to N if N(f(x)) is summable as a nonnegative real-valued function on X. Let ℓ^{1}(X, V) = ℓ_{N}^{1}(X, V) be the space of V -valued functions f on X that are summable with respect to N, and put

∥f∥_{1} = ∥f∥_{ℓ^{1}(X,V)} = ∑_{x∈X} N(f(x)) (9.7)

for all such functions f. One can verify that ℓ^{1}(X, V) is a linear subspace of the space c(X, V) of all V -valued functions on X, and that (9.7) defines a norm on ℓ^{1}(X, V) with respect to | · | on k. Clearly

c_{00}(X, V) ⊆ ℓ^{1}(X, V), (9.8) and (9.7) is the same as (7.1) when f ∈ c_{00}(X, V). One can check that c_{00}(X, V) is dense in ℓ^{1}(X, V) with respect to the metric associated to the ℓ^{1} norm. This uses nonempty finite subsets A of X such that

∑_{x∈A} N(f(x)) (9.9)

approximates ∥f∥_{1}. If f ∈ ℓ^{1}(X, V), then N(f(x)) is summable as a nonnegative real-valued function on X, and hence N(f(x)) vanishes at infinity on X, as before. This is the same as saying that f vanishes at infinity as a V -valued function on X, so that ℓ^{1}(X, V) ⊆ c_{0}(X, V). (9.10)

Note that ∥f∥_{∞} ≤ ∥f∥_{1} (9.11)

for every f ∈ ℓ^{1}(X, V).

If V is complete with respect to the metric associated to N, then ℓ^{1}(X, V) is complete with respect to the metric associated to (9.7). To see this, let {f_{j}}_{j=1}^{∞} be a Cauchy sequence with respect to the ℓ^{1} metric. This implies that {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in V with respect to the metric associated to N for each x ∈ X, so that {f_{j}(x)}_{j=1}^{∞} converges to an element f(x) of V for each x ∈ X, because V is complete. It follows that {N(f_{j}(x))}_{j=1}^{∞} converges to N(f(x)) in R for every x ∈ X, by standard arguments. Thus

∑_{x∈X} N(f(x)) ≤ sup_{j≥1} ( ∑_{x∈X} N(f_{j}(x))), (9.12)

as in (9.5). The Cauchy condition for {f_{j}}_{j=1}^{∞} in ℓ^{1}(X, V) implies that the right side of (9.12) is finite, so that f ∈ ℓ^{1}(X, V). Similarly, if l ∈ Z_{+}, then

∥f − f_{l}∥_{1} ≤ sup_{j≥l} ∥f_{j} − f_{l}∥_{1}, (9.13)

because {f_{j} − f_{l}}_{j=l}^{∞} converges to f − f_{l} pointwise on X. This implies that {f_{l}}_{l=1}^{∞} converges to f with respect to the ℓ^{1} metric, using the Cauchy condition for {f_{j}}_{j=1}^{∞} in ℓ^{1}(X, V) again, as desired.

10 Infinite series

Let k be a field with an absolute value function | · |, and let V be a vector space over k with a norm N with respect to | · | on k. As usual, an infinite series

∑_{j=1}^{∞} v_{j} (10.1)

with terms in V is said to converge in V with respect to N if the corresponding sequence of partial sums ∑_{j=1}^{n} v_{j} converges in V with respect to the metric associated to N. Of course, the value of the sum (10.1) is defined to be the limit of the sequence of partial sums in this case. If (10.1) converges in V, then the corresponding sequence of partial sums is bounded, and

N( ∑_{j=1}^{∞} v_{j}) ≤ sup_{n≥1} N( ∑_{j=1}^{n} v_{j}). (10.2)

This follows from the analogous statements for convergent sequences.

If ∑_{j=1}^{∞} a_{j} is an infinite series of nonnegative real numbers, then the corresponding sequence of partial sums increases monotonically. It follows that ∑_{j=1}^{∞} a_{j} converges in R with respect to the standard absolute value function if and only if the sequence of partial sums is bounded, in which case

∑_{j=1}^{∞} a_{j} = sup_{n≥1} ( ∑_{j=1}^{n} a_{j}), (10.3)

by the analogous statements for monotonically increasing sequences in R. Otherwise, the right side of (10.3) may be interpreted as being +∞, and one may take the left side of (10.3) to be +∞ as well. One can check that this is equivalent to the sum ∑_{j∈Z_{+}} a_{j}, as defined in the previous section.

Let k, V, and N be as before, and let (10.1) be an infinite series with terms in V again. Note that the corresponding sequence of partial sums is a Cauchy sequence in V with respect to the 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 (10.4)

for every n ≥ l ≥ 1. This implies that

lim_{j→∞} N(v_{j}) = 0, (10.5)

by taking l = n, as usual.

If ∑_{j=1}^{∞} N(v_{j}) converges as an infinite series of nonnegative real numbers, then (10.1) is said to converge absolutely with respect to N. Observe that

N( ∑_{j=l}^{n} v_{j}) ≤ ∑_{j=l}^{n} N(v_{j}) (10.6)

for every n ≥ l ≥ 1, by the triangle inequality. If (10.1) converges absolutely with respect to N, then it follows that the corresponding sequence of partial sums is a Cauchy sequence in V with respect to the metric associated to N, because of (10.4). If V is complete with respect to the metric associated to N, then we get that (10.1) converges in V. We also have that

N( ∑_{j=1}^{∞} v_{j}) ≤ ∑_{j=1}^{∞} N(v_{j}) (10.7)

under these conditions, by (10.2).

Suppose now that N is an ultranorm on V, so that

N( ∑_{j=l}^{n} v_{j}) ≤ max_{l≤j≤n} N(v_{j}) (10.8)

for every n ≥ l ≥ 1. If (10.5) holds, then the sequence of partial sums corresponding to (10.1) is a Cauchy sequence in V with respect to N, by (10.4). This implies that (10.1) converges in V with respect to N when V is complete with respect to the ultrametric associated to N. In this situation, we also have that

N( ∑_{j=1}^{∞} v_{j}) ≤ max_{j≥1} N(v_{j}), (10.9)

by (10.2). The maximum on the right side of (10.9) is attained when (10.5) holds, as before.

11 Bounded linear mappings

Let k be a field with an absolute value function | · |, and let V, W be vector spaces over k. Also let N_{V}, N_{W} be norms on V, W, respectively, with respect to | · | on k. A linear mapping T from V into W is said to be bounded with respect to N_{V}, N_{W} if there is a nonnegative real number C such that

N_{W} (T (v)) ≤ C N_{V} (v) (11.1)

for every v ∈ V. In this case, we have that

N_{W} (T (u) − T (v)) = N_{W} (T (u − v)) ≤ C N_{V} (u − v) (11.2)

for every u, v ∈ V. This implies that T is uniformly continuous with respect to the metrics on V, W associated to N_{V}, N_{W}, respectively.

Let T be a linear mapping from V into W again, and suppose that there are positive real numbers r_{V}, r_{W} such that

N_{W} (T (v)) < r_{W} (11.3)

for every v ∈ V with N_{V} (v) < r_{V}. Of course, if T is continuous at 0 with respect to the metrics on V, W associated to N_{V}, N_{W}, respectively, then for each r_{W} > 0 there is an r_{V} > 0 with this property. Suppose that | · | is not trivial on k, and let t_{0} be an element of k with |t_{0}| > 1. If v ∈ V and v ≠ 0, then there is an integer j such that

|t_{0}|^{j−1} r_{V} ≤ N_{V} (v) < |t_{0}|^{j} r_{V} . (11.4)

Thus N_{V} (t_{0}^{−j} v) = |t_{0}|^{−j} N_{V} (v) < r_{V}, so that

|t_{0}|^{−j} N_{W} (T (v)) = N_{W} (T (t_{0}^{−j} v)) < r_{W} , (11.5) by (11.3). This implies that

N_{W} (T (v)) < |t_{0}|^{j} r_{W} ≤ |t_{0}| (r_{W} /r_{V}) N_{V} (v), (11.6)

using the first inequality in (11.4) in the second step. Hence (11.1) holds with C = |t_{0}| (r_{W} /r_{V}) when v ≠ 0, and (11.1) is trivial when v = 0.

If T is a bounded linear mapping from V into W with respect to N_{V}, N_{W}, respectively, then we put

(11.7) ∥T ∥_{op} = ∥T ∥_{op,V W} = inf{C ≥ 0 : (11.1) holds}.

More precisely, the infimum is taken over all nonnegative real numbers C for which (11.1) holds. One can check that the infimum is attained, so that (11.1) holds with C = ∥T ∥_{op}. Let BL(V, W) be the space of bounded linear mappings from V into W with respect to N_{V}, N_{W}, respectively. It is easy to see that BL(V, W) is a vector space over k with respect to pointwise addition and scalar multiplication. Similarly, (11.7) defines a norm on BL(V, W), which is the operator norm associated to N_{V}, N_{W}. If N_{W} is an ultranorm on W, then (11.7) is an ultranorm on BL(V, W).

Let Z be another vector space over k, and let N_{Z} be a norm on Z with respect to | · | on k. Suppose that T_{1} is a bounded linear mapping from V into W with respect to N_{V}, N_{W}, and that T_{2} is a bounded linear mapping from W into Z with respect to N_{W}, N_{Z}. Observe that the composition T_{2} ∘T_{1} of T_{1} and T_{2} is a bounded linear mapping from V into Z with respect to N_{V}, N_{Z}, with

∥T_{2} ∘ T_{1}∥_{op,V Z} ≤ ∥T_{1}∥_{op,V W} ∥T_{2}∥_{op,W Z}. (11.8)

Let V_{0} be a linear subspace of V, and suppose that V_{0} is dense in V with respect to the metric associated to N_{V}. Also let T_{0} be a bounded linear mapping from V_{0} into W, using the restriction of N_{V} to V_{0}. Thus T_{0} is uniformly continuous with respect to the metrics associated to N_{V} and N_{W}, as before. If W is complete with respect to the metric associated to N_{W}, then there is a unique extension of T_{0} to a uniformly continuous mapping from V into W, as in Section 3. One can check that this extension is a bounded linear mapping from V into W under these conditions, with the same operator norm as on V_{0}.

12 Banach spaces

Let k be a field with an absolute value function | · |, let V be a vector space over k, and let N be a norm on V with respect to | · | on k. If V is complete with respect to the metric associated to N, then V is said to be a Banach space with respect to N. Otherwise, one can pass to a completion of V, as usual. The vector space operations on V can be extended continuously to the completion, so that the completion is a vector space over k. Similarly, N can be extended continuously to the completion of V, which corresponds to the distance to 0 in the completion. This extension of N defines a norm on the completion, which corresponds to the metric on the completion in the usual way. The completion of V with respect to N is unique up to isometric linear equivalence, because of the existence of extensions of bounded linear mappings from dense linear subspaces into Banach spaces mentioned in the previous section.

If V_{0} is a linear subspace of V, then the restriction of N to V_{0} defines a norm on V_{0} with respect to | · | on k. If N is an ultranorm on V, then the restriction of N to V_{0} is an ultranorm on V_{0}. If N is any norm on V with respect to | · | on k, V_{0} is a dense linear subspace of V, and the restriction of N to V_{0} is an ultranorm on V_{0}, then one can check that N is an ultranorm on V. In particular, if N is an ultranorm on V, then the extension of N to the completion of V mentioned in the preceding paragraph is an ultranorm as well. Note that closed linear subspaces of Banach spaces are Banach spaces too, with respect to the restriction of the norm, because closed subsets of complete metric spaces are complete with respect to the restriction of the metric to the subset.

Remember that k may be considered as a one-dimensional vector space over itself, and that | · | may be considered as a norm on k with respect to itself, as in Section 5. If v ∈ V, then t ↦ t v (12.1)

defines a bounded linear mapping from k into V, with operator norm equal to N(v). If V is complete with respect to the metric associated to N, but k is not complete with respect to the metric associated to | · |, then (12.1) extends to a bounded linear mapping from the completion of k into V, as before. One can verify that V is a vector space over the completion of k with respect to this extension of scalar multiplication on V to the completion of k, and that N is a norm on V as a vector space over the completion of k. One could also include completeness of k in the definition of a Banach space.

Suppose for the moment that k is complete with respect to the metric associated to | · |, so that k may be considered as a one-dimensional Banach space over itself. If n is a positive integer, then it is easy to see that k^{n} is a Banach space with respect to the norms ∥v∥_{1} and ∥v∥_{∞} defined in (5.5) and (5.6), respectively. In both cases, a sequence of elements of k^{n} is a Cauchy sequence with respect to the metric associated to the norm if and only if the corresponding n sequences of coordinates in k are Cauchy sequences with respect to the metric associated to | · |. If X is a nonempty set, then the spaces ℓ^{∞}(X, k) and ℓ^{1}(X, k) discussed in Sections 8 and 9, respectively, are Banach spaces with respect to the corresponding norms defined earlier. It follows that c_{0}(X, k) is a Banach space with respect to the supremum norm as well, because it is a closed subspace of ℓ^{∞}(X, k).

Let V, W be vector spaces over k, with norms N_{V}, N_{W} with respect to | · | on k, respectively. If W is complete with respect to the metric associated to N_{W}, then BL(V, W) is complete with respect to the metric associated to the corresponding operator norm. More precisely, let {T_{j}}_{j=1}^{∞} be a Cauchy sequence in BL(V, W) with respect to the operator norm. If v ∈ V, then it is easy to see that {T_{j}(v)}_{j=1}^{∞} is a Cauchy sequence in W with respect to the metric associated to N_{W}. It follows that {T_{j}(v)}_{j=1}^{∞} converges to an element T (v) of W for every v ∈ V, because W is complete. One can check that T is a linear mapping from V into W, because the T_{j}’s are linear. The Cauchy condition for {T_{j}}_{j=1}^{∞} with respect to the operator norm implies that the operator norms of the T_{j}’s are bounded, which can be used to show that T is bounded as a linear mapping from V into W. One can use the Cauchy condition for {T_{j}}_{j=1}^{∞} with respect to the operator norm again to get that {T_{j}}_{j=1}^{∞} converges to T with respect to the operator norm, as desired.

13 Sums of vectors

Let k be a field, let V be a vector space over k, and let X be a nonempty set. As in Section 7,

f ↦ ∑_{x∈X} f(x) (13.1)

defines a linear mapping from c_{00}(X, V) into V. Let | · | be an absolute value function on k, and let N be a norm on V. Using (7.6), we get that (13.1) is a bounded linear mapping from c_{00}(X, V) into V, with respect to the ℓ^{1} norm on c_{00}(X, V) as in (9.7), and N on V. Similarly, if N is an ultranorm on V, then (7.7) implies that (13.1) is a bounded linear mapping from c_{00}(X, V) into V, with respect to the supremum ultranorm on c_{00}(X, V) as in (8.1), and N on V. More precisely, (7.6) and (7.7) say that the corresponding operator norms of (13.1) are less than or equal to 1. In both cases, it is easy to see that the operator norm is equal to 1, unless V = {0}.

Suppose now that V is complete with respect to the metric associated to N, so that V is a Banach space. As in the Section 11, there is a unique extension of (13.1) to a bounded linear mapping from ℓ^{1}(X, V) into V. This uses the fact that c_{00}(X, V) is dense in ℓ^{1}(X, V), as in Section 9. If N is an ultranorm on V, then there is a unique extension of (13.1) to a bounded linear mapping from c_{0}(X, V) into V, using the supremum ultranorm (8.1) on c_{0}(X, V). This uses the fact that c_{00}(X, V) is dense in c_{0}(X, V) with respect to the supremum metric, as in Section 8. These extensions can be used to define (13.1) in these situations. In both cases, the operator norm of the extension of (13.1) is equal to 1 when V ≠ {0}.

Of course, if X has only finitely many elements, then c_{00}(X, V) is the same as ℓ^{1}(X, V) and c_{0}(X, V). Suppose that X has infinitely many elements, and let {x_{j}}_{j=1}^{∞} be a sequence of distinct elements in X. If f is a V -valued function on X whose support is contained in the set of x_{j}’s, then ∑_{x∈X} f(x) corresponds formally to the infinite series

∑_{j=1}^{∞} f(x_{j}). (13.2)

If f ∈ ℓ^{1}(X, V), then one can check that (13.2) converges absolutely with respect to N. This implies that (13.2) converges in V when V is complete with respect to the metric associated to N, as in Section 10. One can verify that the value of (13.2) is the same as what one gets by extending (13.1) to a bounded linear mapping from ℓ^{1}(X, V) into V. If f is any element of ℓ^{1}(X, V), then f vanishes at infinity on X with respect to N on V, and hence the support of f has only finitely or countably many elements.

Let f be a V -valued function on X whose support is contained in the set of x_{j}’s again. If f ∈ c_{0}(X, V), then it is easy to see that N(f(x_{j})) → 0 as j → ∞. This implies that (13.2) converges in V with respect to N when N is an ultranorm on V and V is complete with respect to the corresponding ultrametric, as in Section 10. In this case, one can check that the value of (13.2) is the same as what one gets by extending (13.1) to a bounded linear mapping from c_{0}(X, V) into V, as before. If f is any element of c_{0}(X, V), then the support of f has only finitely or countable many elements, and hence can be enumerated by a finite or infinite sequence.

If f is a summable real-valued function on X, then f can be expressed as a difference of nonnegative real-valued summable functions on X. This permits one to define ∑_{x∈X} f(x) using the definition of the sum for nonnegative realvalued functions in Section 9. Similarly, if f is a summable complex-valued function on X, then one can apply the previous remark to the real and imaginary parts of f. This approach to the sum is also compatible with the ones just mentioned in this situation.

14 Banach algebras

Let k be a field, and let A be an (associative) algebra over k. This means that A is a vector space over k equipped with a binary operation of multiplication. Multiplication on A should be bilinear as a mapping from A × A into A, and satisfy the associative law. Let |·| be an absolute value function on k, and let N be a seminorm on A with respect to |·| on k. We say that N is submultiplicative on A if N(x y) ≤ N(x) N(y) (14.1)

for every x, y ∈ A, and that N is multiplicative on A if

N(x y) = N(x) N(y) (14.2)

for every x, y ∈ A.

Let ∥ · ∥ be a submultiplicative norm on A with respect to | · | on k. One can check that multiplication on A is continuous as a mapping from A × A into A, using the topology determined on A by the metric associated to ∥ · ∥, and that corresponding product topology on A × A. If A is complete with respect to the metric associated to ∥ · ∥, then A is said to be a Banach algebra with respect to ∥ · ∥ over k. Otherwise, one can pass to a completion of A, as in Section 12. One can verify that multiplication extends continuously to the completion of A, so that the completion of A is an algebra over k, and so that the extension of the norm to the completion is submultiplicative on the completion.

Similarly, if A is complete with respect to the metric associated to the norm ∥ · ∥, but k is not complete with respect to the metric associated to | · |, then scalar multiplication on A can be extended to the completion of k, as in Section 12. It is easy to see that multiplication on A is also bilinear with respect to this extension of scalar multiplication on A to the completion of k. One can include completeness of k in the definition of a Banach algebra, as in the case of Banach spaces. One may also ask that A have a multiplicative identity element e with ∥e∥ = 1, and we shall normally do that here.

Let X be a nonempty set, and observe that the space c(X, k) is a commutative algebra over k with respect to pointwise multiplication of functions. Also let 1_{X} be the function on X whose value at every point is the multiplicative identity element 1 in k, which is the multiplicative identity element in c(X, k). Remember that ℓ^{∞}(X, k) denotes the space of bounded k-valued functions on X, as in Section 8, and let ∥f∥_{∞} be the supremum norm on ℓ^{∞}(X, k) corresponding to | · | on k. If f, g ∈ ℓ^{∞}(X, k), then their product f g is bounded on X too, with ∥f g∥_{∞} ≤ ∥f∥_{∞} ∥g∥_{∞}. (14.3)

Thus ℓ^{∞}(X, k) is a subalgebra of c(X, k), and ∥ · ∥_{∞} is submultiplicative on ℓ^{∞}(X, k). Of course, constant functions on X are bounded, and

∥1_{X}∥_{∞} = |1| = 1. (14.4)

If k is complete with respect to the metric associated to | · |, then ℓ^{∞}(X, k) is complete with respect to the corresponding supremum metric, as in Section 8.

Let V be a vector space over k, and let N_{V} be a norm on V with respect to | · | on V. Also let BL(V) = BL(V, V) be the space of bounded linear mappings from V into itself, with respect to N_{V} on the domain and range. This is an algebra over k, with composition of linear mappings as multiplication. The corresponding operator norm ∥T ∥_{op} = ∥T ∥_{op,V V} is submultiplicative on BL(V), as in (11.8). The identity mapping I = I_{V} on V is the multiplicative identity element in BL(V). If V ≠ {0}, then it is easy to see that ∥I∥_{op} = 1. If V is complete with respect to the metric associated to N_{V}, then BL(V) is complete with respect to the metric associated to the operator norm, as in Section 12.

15 Cauchy products

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. Thus

c_{n} = ∑_{j=0}^{n} a_{j} b_{n−j} (15.1)

is defined as an element of A for every nonnegative integer n, and the corresponding 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}) (15.2)

formally. More precisely, both sides of (15.2) correspond formally to the sum of a_{j} b_{l} over all nonnegative integers j, l.

Let us look at the partial sums corresponding to these infinite series. If N_{1}, N_{2} are nonnegative integers, then

( ∑_{j=0}^{N_{1}} a_{j}) ( ∑_{l=0}^{N_{2}} b_{l}) = ∑{a_{j} b_{l} : j = 0, …, N_{1}, l = 0, …, N_{2}}. (15.3)

Similarly, if N is a nonnegative integer, then

∑_{n=0}^{N} c_{n} = ∑_{n=0}^{N} ( ∑_{j=0}^{n} a_{j} b_{n−j}) = ∑{a_{j} b_{l} : j, l ∈ Z, j, l ≥ 0, j + l ≤ N}. (15.4)

If N_{1} + N_{2} ≤ N, then each of the terms in the right side of (15.3) occur in the right side of (15.4). If N ≤ N_{1}, N_{2}, then each of the terms on right side of (15.4) also occurs in the right side of (15.3).

Suppose for the moment that there are nonnegative integers J, L such that a_{j} = 0 when j ≥ J and b_{l} = 0 when l ≥ L. In this case, one can check that c_{n} = 0 when n ≥ J + L − 1. Thus the infinite series ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l}, and ∑_{n=0}^{∞} c_{n} reduce to finite sums in A, and one can check that (15.2) holds in this situation. More precisely, (15.3) is equal to (15.4) when J −1 ≤ N_{1}, L−1 ≤ N_{2}, and J + L − 2 ≤ N. This is because the terms a_{j} b_{l} that occur in one of (15.3) or (15.4) and not the other are equal to 0.

Suppose now that a_{j}, b_{l} are nonnegative real numbers for every j, l ≥ 0, so that c_{n} is a nonnegative real number for every n ≥ 0. Using the earlier remarks about (15.3) and (15.4), we get that

( ∑_{j=0}^{N_{1}} a_{j}) ( ∑_{l=0}^{N_{2}} b_{l}) ≤ ∑_{n=0}^{N} c_{n} (15.5)

when N_{1} + N_{2} ≤ N. Similarly,

∑_{n=0}^{N} c_{n} ≤ ( ∑_{j=0}^{N_{1}} a_{j}) ( ∑_{l=0}^{N_{2}} b_{l}) (15.6)

when N ≤ N_{1}, N_{2}. In this case, ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l}, and ∑_{n=0}^{∞} c_{n} can be defined as nonnegative extended real numbers. The product of the first two series can also be defined as a nonnegative extended real number when both sums are finite or both sums are positive. One can use (15.5) and (15.6) to get that (15.2) holds as an equality between nonnegative extended real numbers when the right side of (15.2) is defined. Otherwise, if a_{j} = 0 for every j ≥ 0, or if b_{l} = 0 for every l ≥ 0, then c_{n} = 0 for every n ≥ 0.

Let k be a field with an absolute value function | · |, and let A be an algebra over k with a submultiplicative norm ∥ · ∥ with respect to | · | on k. Thus

∥a_{j} b_{l}∥ ≤ ∥a_{j}∥ ∥b_{l}∥ (15.7) for every j, l ≥ 0, so that

∥c_{n}∥ ≤ ∑_{j=0}^{n} ∥a_{j} b_{n−j}∥ ≤ ∑_{j=0}^{n} ∥a_{j}∥ ∥b_{n−j}∥ (15.8)

for every n ≥ 0. Suppose that ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge absolutely with respect to ∥·∥ on A, so that ∑_{j=0}^{∞} ∥a_{j}∥ and ∑_{l=0}^{∞} ∥b_{l}∥ converge as infinite series of nonnegative real numbers. Observe that

∑_{n=0}^{∞} ∥c_{n}∥ ≤ ∑_{n=0}^{∞} ( ∑_{j=0}^{n} ∥a_{j}∥ ∥b_{n−j}∥) = ( ∑_{j=0}^{∞} ∥a_{j}∥) ( ∑_{l=0}^{∞} ∥b_{l}∥), (15.9)

using (15.8) in the first step. The second step in (15.9) uses the fact that the right side of (15.8) is the same as the nth term of the Cauchy product of ∑_{j=0}^{∞} ∥a_{j}∥ and ∑_{l=0}^{∞} ∥b_{l}∥, as infinite series of nonnegative real numbers. Hence the sum over n is equal to the product of the two sums, as in the preceding paragraph. In particular, ∑_{n=0}^{∞} c_{n} converges absolutely with respect to ∥ · ∥ under these conditions.

If A is complete with respect to the metric associated to ∥ · ∥, then absolute convergence of these series implies convergence in A, as in Section 10. One can check that (15.2) also holds in this situation. Basically, one can compare (15.3) and (15.4) as before, and verify that the norms of the errors are small when N, N_{1}, N_{2} are sufficiently large.

Suppose now that ∥ · ∥ is an ultranorm on A, so that

∥c_{n}∥ ≤ max_{0≤j≤n} ∥a_{j} b_{n−j}∥ ≤ max_{0≤j≤n}(∥a_{j}∥ ∥b_{n−j}∥) (15.10)

for every n ≥ 0. If

lim_{j→∞} ∥a_{j}∥ = lim_{l→∞} ∥b_{l}∥ = 0, (15.11)

then one can check that

lim_{n→∞} ∥c_{n}∥ = 0, (15.12)

using (15.10). If A is also complete with respect to the ultrametric associated to ∥ · ∥, then it follows that ∑_{j=0}^{∞} a_{j}, ∑_{l=0}^{∞} b_{l}, and ∑_{n=0}^{∞} c_{n} converge in A with respect to ∥ · ∥, as in Section 10. Under these conditions, one can verify that (15.2) holds again. This is analogous to the previous case, but using the ultranorm version of the triangle inequality to estimate the errors.

Part II

Linear mappings, matrices, and involutions

16 Some linear mappings

Let k be a field, and let X be a nonempty set. If x, y ∈ X, then put

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

This defines δ_{x}(y) as a k-valued function of y on X for each x ∈ X. Of course, δ_{x} is in the space c_{00}(X, k) of k-valued functions on X with finite support for every x ∈ X. It is easy to see that the collection of δ_{x}, x ∈ X, is a basis for c_{00}(X, k) as a vector space over k.

Let V be a vector space over k, and let a be a V -valued function on X, so that a ∈ c(X, V). If f ∈ c_{00}(X, k), then

T_{a}(f) = ∑_{x∈X} a(x) f(x) (16.2)

is defined as an element of V, because a(x) f(x) is a V -valued function on X with finite support. Clearly T_{a} defines a linear mapping from c_{00}(X, k) into V. Note that T_{a}(δ_{x}) = a(x) (16.3)

for every x ∈ X. One can check that every linear mapping from c_{00}(X, k) into V is of this form, because the δ_{x}’s form a basis for c_{00}(X, k).

Let | · | be an absolute value function on k, and remember that k may be considered as a one-dimensional vector space over itself, with | · | as a norm on k. Thus ℓ^{∞}(X, k) and c_{0}(X, k) may be defined as in Section 8, and ℓ^{1}(X, k) may be defined as in Section 9. Of course,

∥δ_{x}∥_{ℓ^{1}(X,k)} = ∥δ_{x}∥_{ℓ^{∞}(X,k)} = 1 (16.4)

for each x ∈ X.

Let N be a norm on V with respect to | · | on k, and suppose that a is bounded on X with respect to N. If f ∈ ℓ^{1}(X, k), then it is easy to see that a f ∈ ℓ^{1}(X, V), with

∥a f∥_{ℓ^{1}(X,V)} ≤ ∥a∥_{ℓ^{∞}(X,V)} ∥f∥_{ℓ^{1}(X,k)}. (16.5)

Similarly, if f ∈ ℓ^{∞}(X, k), then a f ∈ ℓ^{∞}(X, V), with

∥a f∥_{ℓ^{∞}(X,V)} ≤ ∥a∥_{ℓ^{∞}(X,V)} ∥f∥_{ℓ^{∞}(X,k)}. (16.6)

If f ∈ c_{0}(X, k), then a f ∈ c_{0}(X, V).

If f ∈ c_{00}(X, k), then a f ∈ c_{00}(X, V), and T_{a}(f) ∈ V is defined as in (16.2). We also have that

N(T_{a}(f)) ≤ ∥a f∥_{ℓ^{1}(X,V)} ≤ ∥a∥_{ℓ^{∞}(X,V)} ∥f∥_{ℓ1(X,k)}, (16.7)

using (7.6) in the first step, and (16.5) in the second step. If N is an ultranorm on V, then

N(T_{a}(f)) ≤ ∥a f∥_{ℓ^{∞}(X,V)} ≤ ∥a∥_{ℓ^{∞}(X,V)} ∥f∥_{ℓ^{∞}(X,k)}, (16.8)

using (7.7) in the first step, and (16.6) in the second step.

Of course, (16.7) says that T_{a} is bounded as a linear mapping from c_{00}(X, k) into V, with respect to the ℓ^{1} norm on c_{00}(X, k). If N is an ultranorm on V, then (16.8) says that T_{a} is bounded as a linear mapping from c_{00}(X, k) into V, with respect to the supremum norm on c_{00}(X, k). These inequalities also imply that the corresponding operator norms of T_{a} are less than or equal to ∥a∥_{ℓ^{∞}(X,V)}. The operator norm of T_{a} is equal to ∥a∥_{ℓ^{∞}(X,V)} in both cases, because of (16.3) and (16.4).

Suppose now that V is complete with respect to the metric associated to N. If f ∈ ℓ^{1}(X, k), then a f ∈ ℓ^{1}(X, V), and the sum on the right side of (16.2) can be defined as an element of V as in Section 13. This defines T_{a} as a bounded linear mapping from ℓ^{1}(X, k) into V, with operator norm equal to ∥a∥_{ℓ^{∞}(X,V)}, as before. If N is an ultranorm on V, and f ∈ c_{0}(X, k), then the sum on the right side of (16.2) can be defined as an element of V again, as in Section 13. This defines T_{a} as a bounded linear mapping from c_{0}(X, k) into V in this situation, with operator norm equal to ∥a∥_{ℓ^{∞}(X,V)}.

Every linear mapping from c_{00}(X, k) into V is of the form T_{a} for some a ∈ c(X, V), as mentioned earlier. If T_{a} is bounded with respect to the ℓ^{1} or supremum norm on c_{00}(X, V), then it is easy to see that a ∈ ℓ^{∞}(X, V), using (16.3) and (16.4). If T is a bounded linear mapping from ℓ^{1}(X, k), then T is determined by its restriction to c_{00}(X, k), because c_{00}(X, k) is dense in ℓ^{1}(X, k), as in Section 9. It follows that every bounded linear mapping from ℓ^{1}(X, k) into V is of the form T_{a} for some a ∈ ℓ^{∞}(X, V) when V is complete. Similarly, if N is an ultranorm on V, and T is a bounded linear mapping from c_{0}(X, k) into V with respect to the supremum norm on c_{0}(X, k), then T is of the form T_{a} for some a ∈ ℓ^{∞}(X, V).

17 Bounded linear functionals

Let k be a field, and let V be a vector space over k. As usual, a linear functional on V is a linear mapping from V into k, where k is considered as a one-dimensional vector space over itself. Let | · | be an absolute value function on k, and let N be a norm on V with respect to | · | on k. A bounded linear functional is a linear functional on V that is bounded as a linear mapping from V into k, using | · | as a norm on k. The space of bounded linear functionals on V is denoted V′, which is the same as BL(V, k) in the notation of Section 11. This is the dual space associated to V, and it is a vector space over k with respect to pointwise addition and scalar multiplication. The dual norm N′ on V′ associated to N on V is the same as the operator norm in this situation. Thus if λ ∈ V′, then |λ(v)| ≤ N′(λ) N(v) (17.1)

for every v ∈ V, and N′ is the smallest nonnegative real number with this property. If | · | is an ultrametric absolute value function on k, then N′ is an ultranorm on V′, as in Section 11. If k is complete with respect to the metric associated to | · |, then V′ is complete with respect to the metric associated to the dual norm, as in Section 12.

Let us continue to ask that k be complete with respect to the metric associated to | · |. Let X be a nonempty set, and let a be a bounded k-valued function on X. If f ∈ ℓ^{1}(X, k), then put

λ_{a}(f) = ∑_{x∈X} a(x) f(x), (17.2)

where the sum on the right side of (17.2) can be defined as in Section 13. This is the same as in the previous section, with V = k. This defines a bounded linear functional on ℓ^{1}(X, k), with dual norm equal to ∥a∥_{ℓ^{∞}(X,k)}. Every bounded linear functional on ℓ^{1}(X, k) is of this form, so that the dual of ℓ^{1}(X, k) can be identified with ℓ^{∞}(X, k). The cases where k = R or C with the standard absolute value functions are of particular concern here.

Suppose that | · | is an ultrametric absolute value function on a field k, and that k is complete with respect to the corresponding ultrametric, as before. If a ∈ ℓ^{∞}(X, k), then (17.2) defines a bounded linear functional on c_{0}(X, k), with respect to the supremum norm on c_{0}(X, k). This follows from the remarks in the previous section again, with V = k. As before, the dual norm of (17.2) on c_{0}(X, k) is equal to ∥a∥_{ℓ^{∞}(X,k)}, and every bounded linear functional on c_{0}(X, k) is of this form. Thus the dual of c_{0}(X, k) can be identified with ℓ^{∞}(X, k) in this situation.

Let us now take k = R or C, with the standard absolute value function. If a ∈ ℓ^{1}(X, k) and f ∈ ℓ^{∞}(X, k), then a f ∈ ℓ^{1}(X, k), with

∥a f∥_{ℓ^{1}(X,k)} ≤ ∥a∥_{ℓ^{1}(X,k)} ∥f∥_{ℓ^{∞}(X,k)}, (17.3)

as in (16.5). This implies that λ_{a}(f) can be defined as an element of k as in (17.2), as discussed in Section 13, with

|λ_{a}(f)| ≤ ∥a f∥_{ℓ^{1}(X,k)} ≤ ∥a∥_{ℓ^{1}(X,k)} ∥f∥_{ℓ^{∞}(X,k)}. (17.4)

Thus λ_{a} defines a bounded linear functional on ℓ^{∞}(X, k), with dual norm less than or equal to ∥a∥_{ℓ^{1}(X,k)}. The dual norm of λ_{a} on ℓ^{∞}(X, k) is equal to ∥a∥_{ℓ^{1}(X,k)} in this case, because there is an f ∈ ℓ^{∞}(X, k) such that ∥f∥_{ℓ^{∞}(X,k)} = 1, a(x) f(x) = |a(x)| (17.5) for every x ∈ X, and hence

λ_{a}(f) = ∑_{x∈X} |a(x)| = ∥a∥_{ℓ^{1}(X,k)}. (17.6)

The restriction of λ_{a} to c_{0}(X, k) defines a bounded linear functional on c_{0}(X, k), with respect to the restriction of the supremum norm to c_{0}(X, k). The dual norm of the restriction of λ_{a} to c_{0}(X, k) is clearly less than or equal to ∥a∥_{ℓ^{1}(X,k)}, and we would like to verify that they are equal. To do this, let E be a nonempty finite subset of X, and let f be a k-valued function on X with support contained in E such that ∥f∥_{ℓ^{∞}(X,k)} = 1 and (17.5) holds for every x ∈ E. Observe that

λ_{a}(f) = ∑_{x∈E} |a(x)|, (17.7)

and that this is less than or equal to the dual norm of the restriction of λ_{a} to c_{0}(X, k). This implies that the dual norm of the restriction of λ_{a} to c_{0}(X, k) is equal to ∥a∥_{ℓ^{1}(X,k)}, by taking the supremum over all nonempty finite subsets E of X.

Let λ be any bounded linear functional on c_{0}(X, k), with respect to the restriction of the supremum norm to c_{0}(X, k). Put

a(x) = λ(δ_{x}) (17.8)

for each x ∈ X, where δ_{x} is as in (16.1). This defines a k-valued function on X, which determines a linear functional λ_{a} on c_{00}(X, k) as in (17.2). If f ∈ c_{00}(X, k), then it is easy to see that

λ_{a}(f) = λ(f), (17.9)

by expressing f as a linear combination of the δ_{x}’s. Using this, one can check that a ∈ ℓ^{1}(X, k), with ∥a∥_{ℓ^{1}(X,k)} less than or equal to the dual norm of λ on c_{0}(X, k), by the same type of argument as in the preceding paragraph. In particular, λ_{a} also defines a bounded linear functional on c_{0}(X, k), as before. It follows that (17.9) holds for every f ∈ c_{0}(X, k), because c_{00}(X, k) is dense in c_{0}(X, k) with respect to the supremum metric. This shows that the dual of c_{0}(X, k) can be identified with ℓ^{1}(X, k) when k = R or C, with the standard absolute value function.

Let k be a field with an ultrametric absolute value function | · | again, and suppose that k is complete with respect to the ultrametric associated to | · |. If a ∈ c_{0}(X, k) and f ∈ ℓ^{∞}(X, k), then a f ∈ c_{0}(X, k), as in the previous section. This implies that the sum on the right side of (17.2) can be defined as an element of k, as in Section 13. Thus λ_{a} defines a linear functional on ℓ^{∞}(X, k), with

|λ_{a}(f)| ≤ ∥a f∥_{ℓ^{∞}(X,k)} ≤ ∥a∥_{ℓ^{∞}(X,k)} ∥f∥_{ℓ^{∞}(X,k)} (17.10)

for every f ∈ ℓ^{∞}(X, k). This means that λ_{a} is a bounded linear functional on ℓ^{∞}(X, k), with dual ultranorm less than or equal to ∥a∥_{ℓ^{∞}(X,k)}. The dual ultranorm of λ_{a} on ℓ^{∞}(X, k) is equal to ∥a∥_{ℓ^{∞}(X,k)}, because

λ_{a}(δ_{x}) = a(x) (17.11) for every x ∈ X. More precisely, the dual ultranorm of the restriction of λ_{a} to c_{0}(X, k) is equal to ∥a∥_{ℓ^{∞}(X,k)} too, using the restriction of the supremum norm to c_{0}(X, k).

18 Inner product spaces

Suppose for the moment that V and W are vector spaces over the complex numbers, and remember that V, W can also be considered as vector spaces over the real numbers. A mapping T from V into W is said to be real or complex linear if T is linear when V, W are considered to be vector spaces over the real or complex numbers, respectively. Note that a real-linear mapping T from V into W is complex linear exactly when

T (i v) = i T (v) (18.1)

for every v ∈ V. A real-linear mapping T from V into V is said to be conjugate linear if T (i v) = −i T (v) (18.2)

for every v ∈ V. This implies that

T (a v) = \overline{a} T (v) (18.3)

for every a ∈ C and v ∈ V, where \overline{a} is the complex-conjugate of a.

Now let V be a vector space over the real or complex numbers, and let ⟨v, w⟩ be a real or complex-valued function, as appropriate, defined for v, w ∈ V. As usual, ⟨v, w⟩ is said to be an inner product on V if it satisfies the following three conditions. The first condition asks that

λ_{w}(v) = ⟨v, w⟩ (18.4)

be a linear functional on V, as a function of v, for every w ∈ V. The second condition asks that ⟨v, w⟩ = ⟨w, v⟩ (18.5)

for every v, w ∈ V in the real case, and that

⟨v, w⟩ = \overline{⟨v, w⟩} (18.6)

for every v, w ∈ V in the complex case. It follows that ⟨v, w⟩ is linear as a function of w for every v ∈ V in the real case, and conjugate-linear in the complex case. Note that ⟨v, v⟩ ∈ R (18.7)

for every v ∈ V in the complex case, by (18.6). The third condition asks that

⟨v, v⟩ > 0 (18.8)

for every v ∈ V with v ≠ 0.

Suppose that ⟨v, w⟩ is an inner product on V, and put

∥v∥ = ⟨v, v⟩^{1/2} (18.9)

for every v ∈ V. The Cauchy–Schwarz inequality states that

|⟨v, w⟩| ≤ ∥v∥ ∥w∥ (18.10)

for every v, w ∈ V. One can use this to show that (18.9) defines a norm on V with respect to the standard Euclidean absolute value function on R or C, as appropriate. The standard inner product on R^{n} is defined for each positive integer n by

⟨v, w⟩ = ∑_{j=1}^{n} v_{j} w_{j} (18.11)

for every v, w ∈ R^{n}. Similarly, the standard inner product is defined on C^{n} by

⟨v, w⟩ = ∑_{j=1}^{n} v_{j} \overline{w_{j}} (18.12)

for every v, w ∈ C^{n}.

Let ⟨v, w⟩ be an inner product on a vector space V over the real or complex numbers again. Also let w ∈ V be given, so that (18.4) defines a linear functional λ_{w} on V. The Cauchy–Schwarz inequality implies that λ_{w} is a bounded linear functional on V with respect to the norm (18.9), with dual norm less than or equal to ∥w∥. It is easy to see that the dual norm of λ_{w} is equal to ∥w∥, because

λ_{w}(w) = ⟨w, w⟩ = ∥w∥^{2}. (18.13)

Similarly, one can check that

∥v∥ = sup{|⟨v, w⟩| : w ∈ V, ∥w∥ ≤ 1} (18.14)

for every v ∈ V.

19 Hilbert spaces

Let (V, ⟨·, ·⟩_{V}) be a real or complex inner product space, and let ∥ · ∥_{V} be the corresponding norm, as in (18.9). If V is complete with respect to the metric associated to this norm, then V is said to be a Hilbert space. Otherwise, one can pass to a completion, as in Section 12. More precisely, ⟨·, ·⟩_{V} can be extended continuously to the completion of V, and this extension defines an inner product on the completion of V. The norm associated to the extension of ⟨·, ·⟩_{V} to the completion of V is the same as the continuous extension of ∥ · ∥_{V} to the completion of V, so that the completion of V is a Hilbert space too.

Remember that (18.4) defines a bounded linear functional on V for every w ∈ V. If V is a Hilbert space, then it is well known that every bounded linear functional on V is of this form for some w ∈ V. It is easy to see that this representation is unique.

Suppose that (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) are Hilbert spaces, both real or both complex, and with the corresponding norms ∥ · ∥_{V} and ∥ · ∥_{W}, respectively. If T is a bounded linear mapping from V into W, and if w ∈ W, then v ↦ ⟨T (v), w⟩_{W} (19.1) defines a bounded linear functional on V. Indeed, |⟨T (v), w⟩_{W} | ≤ ∥T (v)∥_{V} ∥w∥_{W} ≤ ∥T ∥_{op,V W} ∥v∥_{V} ∥w∥_{W} (19.2) for every v ∈ V, where ∥T ∥_{op,V W} is the operator norm of T with respect to ∥ · ∥_{V} and ∥ · ∥_{W}. This implies that (19.1) is a bounded linear functional on V with respect to ∥ · ∥_{V}, with dual norm less than or equal to ∥T ∥_{op,V W} ∥w∥_{W}. It follows that there is a unique element T^{∗}(w) of V such that ⟨T (v), w⟩_{W} = ⟨v, T^{∗}(w)⟩_{V} (19.3) for every v ∈ V, because V is a Hilbert space. We also have that ∥T^{∗}(w)∥_{V} ≤ ∥T ∥_{op,V W} ∥w∥_{W} , (19.4) because the dual norm of (19.1) is equal to ∥T^{∗}(w)∥_{V}, and less than or equal to the right side of (19.4). One can check that T^{∗} defines a linear mapping from W into V, which is known as the adjoint of T. More precisely, T^{∗} is a bounded linear mapping from W into V, with ∥T^{∗}∥_{op,W V} ≤ ∥T ∥_{op,V W} , (19.5) by (19.4). Similarly, |⟨T (v), w⟩_{W} | = |⟨v, T^{∗}(w)⟩_{V} | ≤ ∥v∥ ∥T^{∗}(w)∥_{V} ≤ ∥v∥ ∥T^{∗}∥_{op,W V} ∥w∥ (19.6) for every v ∈ V and w ∈ W, which implies that ∥T (v)∥_{W} ≤ ∥v∥ ∥T^{∗}∥_{op,W V} (19.7) for every v ∈ V. This shows that ∥T ∥_{op,V W} ≤ ∥T^{∗}∥_{op,W V}, so that ∥T^{∗}∥_{op,W V} = ∥T ∥_{op,V W} . (19.8)

The adjoint (T^{∗})^{∗} of T^{∗} is defined as a bounded linear mapping from V into W in the same way, and one can verify that (T^{∗})^{∗} = T. (19.9) The mapping T ↦ T^{∗} is linear as a mapping from BL(V, W) into BL(W, V) in the real case, and conjugate-linear in the complex case. The identity mapping I_{V} on V defines a bounded linear mapping from V into itself, and it is easy to see that I_{V}^{∗} = I_{V} (19.10)

as bounded linear mappings from V into itself.

Let (Z, ⟨·, ·⟩_{Z}) be another Hilbert space, which is real or complex depending on whether V and W are real or complex. If T_{1} is a bounded linear mapping from V into W, and T_{2} is a bounded linear mapping from W into Z, then T_{2} ∘T_{1} is a bounded linear mapping from V into Z, as in Section 11. One can check that (T_{2} ∘ T_{1})^{∗} = T_{1}^{∗} ∘ T_{2}^{∗} (19.11)

as bounded linear mappings from Z into V.

20 Isometric linear mappings

Let k be a field with an absolute value function | · |, and let V, W be vector spaces over k with norms N_{V}, N_{W}, respectively. A linear mapping T from V into W is said to be an isometry with respect to N_{V}, N_{W} if

N_{W} (T (v)) = N_{V} (v) (20.1)

for every v ∈ V. In particular, this implies that the kernel of T is trivial, so that T is injective. Of course, (20.1) implies that T is a bounded linear mapping from V into W, with operator norm equal to 1 when V ≠ {0}.

Suppose that T is a one-to-one linear mapping from V onto W, so that the inverse mapping T^{−1} is defined as a linear mapping from W onto V. In this case, T is an isometry from V onto W if and only if T^{−1} is an isometry from W onto V. If T is a bounded linear mapping from V onto W, T^{−1} is a bounded linear mapping from W onto V, and

∥T ∥_{op,V W} , ∥T^{−1}∥_{op,W V} ≤ 1, (20.2)

then one can check that T is an isometry.

Let us now take k = R or C with the standard absolute value function for the rest of the section. Let (V, ⟨·, ·⟩_{V}) and (W, ⟨·, ·⟩_{W}) be Hilbert spaces again, both real or both complex, and with corresponding norms ∥ · ∥_{V} and ∥ · ∥_{W}, respectively. Suppose that a linear mapping T from V into W satisfies

⟨T (u), T (v)⟩_{W} = ⟨u, v⟩_{V} (20.3)

for every u, v ∈ V. This implies that T is an isometry with respect to ∥ · ∥_{V} and ∥ · ∥_{W}, by taking u = v. It is well known that the converse holds, because of polarization identities. If T also maps V onto W, then T is said to be unitary. In the real case, T may be called an orthogonal linear transformation.

Now let T be any bounded linear mapping from V into itself, so that the adjoint T^{∗} of T is defined as a bounded linear mapping from W into V, as in the previous section. If u, v ∈ V, then

⟨T^{∗}(T (u)), v⟩_{V} = ⟨T (u), T (v)⟩_{W} . (20.4)

Using this, one can check that (20.3) holds if and only if

T^{∗} ∘ T = I_{V} , (20.5)

where I_{V} denotes the identity mapping on V, as before. If T maps V onto W, then (20.5) implies that T is invertible as a linear mapping from V onto W, with T^{−1} = T^{∗}.

If T is any bounded linear mapping from V into W and v ∈ V, then

⟨T^{∗}(T (v)), v⟩_{V} = ∥T (v)∥_{W}^{2} , (20.6)

by taking u = v in (20.4). This implies that

∥T (v)∥_{W}^{2} ≤ ∥T^{∗}(T (v))∥_{V} ∥v∥_{V} ≤ ∥T^{∗} ∘ T ∥_{op,V V} ∥v∥_{V}^{2} (20.7)

for every v ∈ V, using the Cauchy–Schwarz inequality in the first step, and the definition of the operator norm in the second step. It follows that

∥T ∥_{op,V W}^{2} ≤ ∥T^{∗} ∘ T ∥_{op,V V} . (20.8)

We also have that

∥T^{∗} ∘ T ∥_{op,V V} ≤ ∥T ∥_{op,V W} ∥T^{∗}∥_{op,W V} = ∥T ∥_{op,V W}^{2} , (20.9)

using (11.8) in the first step, and (19.8) in the second step. Thus

∥T^{∗} ∘ T ∥_{op,V V} = ∥T ∥_{op,V W}^{2} , (20.10)

by combining (20.8) and (20.9).

21 n × n Matrices

Let R be a ring, and let n be a positive integer. The space of n×n matrices with entries in R is denoted M_{n}(R). Of course, M_{n}(R) is a commutative group with respect to addition of matrices, which is defined using addition on R in each entry. If A = [a_{j,l}] and B = [b_{j,l}] are elements of M_{n}(R), then their product is defined to be the element C = [c_{j,l}] of M_{n}(R) given by

c_{j,l} = ∑_{r=1}^{n} a_{j,r} b_{r,l} (21.1)

for every j, l = 1, …, n, as usual. It is well known and easy to see that M_{n}(R) is a ring with respect to matrix multiplication. If R has a multiplicative identity element e, then the identity matrix I in M_{n}(R) is defined to be the matrix with diagonal entries equal to e, and off-diagonal entries equal to 0. This is the multiplicative identity element in M_{n}(R) in this case.

Let k be a field, and let A be an algebra over k. Note that M_{n}(A) is a vector space over k, where addition and scalar multiplication of matrices is defined entry-wise. More precisely, M_{n}(A) is an algebra over k with respect to matrix multiplication. Let | · | be an absolute value function on k, and let N be a seminorm on A with respect to | · | on k. If A = [a_{j,l}] ∈ M_{n}(A), then put

N_{∞}(A) = max_{1≤j,l≤n} N(a_{j,l}). (21.2)

It is easy to see that this defines a seminorm on M_{n}(A) with respect to | · | on k, which is a norm when N is a norm on A. If N is a semi-ultranorm on A, then N_{∞} is a semi-ultranorm on M_{n}(A) with respect to | · | on k. If N is a submultiplicative semi-ultranorm on A, then one can check that N_{∞} is submultiplicative on M_{n}(A).

Similarly, if A = [a_{j,l}] ∈ M_{n}(A), then put

N_{1,∞}(A) = max_{1≤l≤n} ( ∑_{j=1}^{n} N(a_{j,l})). (21.3)

One can verify that this also defines a seminorm on M_{n}(A) with respect to | · | on k, which is a norm when N is a norm on A. Observe that

N_{∞}(A) ≤ N_{1,∞}(A) ≤ n N_{∞}(A) (21.4)

for every A ∈ M_{n}(A). Suppose that N is submultiplicative on A, and let us check that N_{1,∞} is submultiplicative on M_{n}(A). Let A = [a_{j,l}] and B = [b_{j,l}] be elements of M_{n}(A), and let C = [c_{j,l}] be as in (21.1). Thus

N(c_{j,l}) ≤ ∑_{r=1}^{n} N(a_{j,r} b_{r,l}) ≤ ∑_{r=1}^{n} N(a_{j,r}) N(b_{r,l}) (21.5)

for each j, l = 1, …, n. Summing over j, we get that

(21.6) ∑_{j=1}^{n} N(c_{j,l}) ≤ ∑_{j=1}^{n} ( ∑_{r=1}^{n} N(a_{j,r}) N(b_{r,l}))

= ∑_{r=1}^{n} ( ∑_{j=1}^{n} N(a_{j,r}) N(b_{r,l}))

≤ N_{1,∞}(A) ( ∑_{r=1}^{n} N(b_{r,l})) ≤ N_{1,∞}(A) N_{1,∞}(B)

for each l = 1, …, n. This implies that

N_{1,∞}(C) ≤ N_{1,∞}(A) N_{1,∞}(B), (21.7)

as desired, by taking the maximum over l = 1, …, n.

If A = [a_{j,l}] ∈ M_{n}(A) again, then put

N_{∞,1}(A) = max_{1≤j≤n} ( ∑_{l=1}^{n} N(a_{j,l})). (21.8)

As before, this is a seminorm on M_{n}(A) with respect to | · | on k, and a norm when N is a norm on A. We also have that

N_{∞}(A) ≤ N_{∞,1}(A) ≤ n N_{∞}(A) (21.9)

for every A ∈ M_{n}(A), as in (21.4). Let us verify that N_{∞,1} is submultiplicative on M_{n}(A) when N is submultiplicative on A. Let A = [a_{j,l}], B = [b_{j,l}] ∈ M_{n}(A) be given, let C = [c_{j,l}] be as in (21.1), and remember that (21.5) holds for every j, l = 1, …, n under these conditions. In this case, we can sum over l to get that

(21.10) ∑_{l=1}^{n} N(c_{j,l}) ≤ ∑_{l=1}^{n} ( ∑_{r=1}^{n} N(a_{j,r}) N(b_{r,l}))

= ∑_{r=1}^{n} ( ∑_{l=1}^{n} N(a_{j,r}) N(b_{r,l}))

≤ ( ∑_{r=1}^{n} N(a_{j,r})) N_{∞,1}(B) ≤ N_{∞,1}(A) N_{∞,1}(B)

for every j = 1, …, n. Hence

N_{∞,1}(C) ≤ N_{∞,1}(A) N_{∞,1}(B), (21.11)

as desired, by taking the maximum over j = 1, …, n.

22 Involutions

Let R be a ring, so that R is a commutative group with respect to addition in particular. Also let x ↦ x^{∗} (22.1)

be a mapping from R into itself that is a group homomorphism with respect to addition. If (x y)^{∗} = y^{∗} x^{∗} (22.2)

for every x, y ∈ R, and (x^{∗})^{∗} = x (22.3)

for every x ∈ R, then (22.1) is said to be an (ring) involution on R. Note that the identity mapping on R is a ring involution when R is commutative.

Let n be a positive integer, and let M_{n}(R) be the ring of n × n matrices with entries in R, as in the previous section. If A = [a_{j,l}] ∈ M_{n}(R), then the transpose of A is the matrix A^{t} = [a_{j,l}^{t}] ∈ M_{n}(R) defined by

a_{j,l}^{t} = a_{l,j} (22.4)

for j, l = 1, …, n, as usual. Observe that

A ↦ A^{t} (22.5) is a group isomorphism from M_{n}(R) into itself with respect to addition, and that (A^{t})^{t} = A (22.6)

for every A ∈ M_{n}(R).

Let (22.1) be an involution on R. If A = [a_{j,l}] ∈ M_{n}(R), then let A^{t∗} = [a_{j,l}^{t∗}] ∈ M_{n}(R) be defined by

a_{j,l}^{t∗} = (a_{j,l}^{t})^{∗} = (a_{l,j})^{∗} (22.7)

for j, l = 1, …, n. It is well known and easy to see that

A ↦ A^{t∗} (22.8)

defines an involution on M_{n}(R). If R is commutative, then we can take (22.1) to be the identity mapping on R, so that (22.5) is an involution on M_{n}(R).

Let k be a field, and let A be an algebra over k. A linear mapping from A into itself is said to be an (algebra) involution on A if it is an involution on A as a ring. If k = C, then a conjugate-linear mapping on A is said to be a conjugatelinear involution on A if it is an involution on A as a ring. In particular, complex-conjugation may be considered as a conjugate-linear involution on C.

Let | · | be an absolute value function on k, and let N be a seminorm on A with respect to | · | on k. Also let (22.1) be an algebra involution on A, which may be conjugate-linear when k = C. A basic compatibility condition between (22.1) and N is that (22.1) be bounded with respect to N, so that

N(x^{∗}) ≤ C N(x) (22.9)

for some C ≥ 0 and every x ∈ A. This implies that

N(x) = N((x^{∗})^{∗}) ≤ C N(x^{∗}) (22.10)

for every x ∈ A, because of (22.3). In particular, if (22.9) holds with C = 1, then N(x^{∗}) = N(x) (22.11)

for every x ∈ A.

Let (V, ⟨v, w⟩_{V}) be a real or complex Hilbert space, and consider the algebra BL(V) of bounded linear mappings from V into itself. The mapping from T ∈ BL(V) to its adjoint T^{∗} defines an involution on BL(V), which is conjugate-linear in the complex case, as in Section 19. We have also seen that this involution preserves the operator norm, as in (19.8).

Let A be an algebra over a field k again, let n be a positive integer, and let M_{n}(A) be the algebra of n × n matrices with entries in A, as in the previous section. Also let |·| be an absolute value function on k, and let N be a seminorm on A with respect to | · | on k. Note that (22.5) defines a linear mapping from M_{n}(A) into itself. If A ∈ M_{n}(A), then

N_{∞}(A^{t}) = N_{∞}(A), (22.12) where N_{∞} is as defined in (21.2). Similarly,

N_{1,∞}(A^{t}) = N_{∞,1}(A), (22.13)

where N_{1,∞}, N_{∞,1} are as defined in (21.3), (21.8), respectively.

Let (22.1) be an algebra involution on A, which may be conjugate-linear when k = C, and suppose that (22.9) holds for some C ≥ 0. If A ∈ M_{n}(A), and A^{t∗} is as in (22.7), then

(22.14)  N_{∞}(A^{t∗}) ≤ C N_{∞}(A),
(22.15)  N_{1,∞}(A^{t∗}) ≤ C N_{∞,1}(A),

and N_{∞,1}(A^{t∗}) ≤ C N_{1,∞}(A). (22.16)

If C = 1, so that (22.11) holds, then equality holds in (22.14), (22.15), and (22.16) as well.

23 Infinite matrices

Let R be a ring, let X be a nonempty set, and let c(X, R) be the space of Rvalued functions on X. This is a commutative group with respect to pointwise addition of functions, and a ring with respect to pointwise multiplication. As before, the support of f ∈ c(X, R) is the set of x ∈ X such that f(x) ≠ 0, and we let c_{00}(X, R) be the set of f ∈ c(X, R) whose support has only finitely many elements. In particular, c_{00}(X, R) is a subgroup of c(X, R) with respect to addition.

An element a of c(X × X, R) may be considered as a matrix with entries in R, using the elements of X as indices. Let a, b ∈ c_{00}(X × X, R) be given, and put

c(x, y) = ∑_{w∈X} a(x, w) b(w, y) (23.1)

for every x, y ∈ X. More precisely, for each x, y ∈ X, all but finitely many terms in the sum on the right side of (23.1) are equal to 0, so that the sum defines an element of R. Thus (23.1) defines an R-valued function on X × X, and one can check that this function has finite support too. The product of a and b in c_{00}(X × X, R) is defined to be c in this discussion, in analogy with matrix multiplication. In particular, if n is a positive integer and X = {1, …, n}, then c_{00}(X × X, R) = c(X × X, R) corresponds to the space M_{n}(R) of n × n matrices with entries in R, and (23.1) corresponds to (21.1). One can check that c_{00}(X × X, R) is a ring with respect to this definition of multiplication, and pointwise addition of functions.

Put

c_{00,1}(X × X, R) = {a ∈ c(X × X, R) : for each y ∈ X, a(x, y) = 0

(23.2) for all but finitely many x ∈ X}.

Equivalently, c_{00,1}(X × X, R) consists of a ∈ c(X × X, R) such that for each y ∈ X, a(x, y) has finite support as an R-valued function of x ∈ X. Note that c_{00,1}(X × X, R) is a subgroup of c(X × X, R) with respect to pointwise addition of functions, and that

c_{00}(X × X, R) ⊆ c_{00,1}(X × X, R). (23.3)

Let a, b ∈ c_{00,1}(X × X, R) be given, and let us check that (23.1) defines an element of c_{00,1}(X × X, R) as well. To do this, let y ∈ X be given, and put

B_{y} = {w ∈ X : b(w, y) ≠ 0}, (23.4)

so that B_{y} has only finitely many elements, because b ∈ c_{00,1}(X × X, R). Thus (23.1) reduces to

c(x, y) = ∑_{w∈B_{y}} a(x, w) b(w, y), (23.5)

which defines an element of R for every x ∈ X, and is interpreted as being equal to 0 when B_{y} = ∅. We also have that (23.5) is equal to 0 for all but finitely many x ∈ X, because a ∈ c_{00,1}(X × X, R). This shows that (23.1) defines an element of c_{00,1}(X ×X, R), and we take c to be the product of a and b, as before. One can verify that c_{00,1}(X × X, R) is a ring with respect to pointwise addition of functions and this definition of multiplication.

Similarly, put

c_{00,2}(X × X, R) = {a ∈ c(X × X, R) : for each x ∈ X, a(x, y) = 0

(23.6) for all but finitely many y ∈ X}.

This is a subgroup of c(X×X, R) with respect to pointwise addition of functions, and c_{00}(X × X, R) ⊆ c_{00,2}(X × X, R), (23.7)

as before. If a, b ∈ c_{00,2}(X × X, R), then one can check that (23.1) defines an element of c_{00,2}(X ×X, R), using the same type of argument as in the preceding paragraph. This makes c_{00,2}(X × X, R) into a ring, with respect to pointwise addition of functions, and this definition of multiplication.

If a ∈ c(X × X, R), then let a^{t} ∈ c(X × X, R) be defined by

a^{t}(x, y) = a(y, x) (23.8)

for every x, y ∈ X. Clearly a ↦ a^{t} (23.9)

is a group isomorphism from c(X × X, R) onto itself with respect to addition, with (a^{t})^{t} = a (23.10)

for every a ∈ c(X × X, R). Of course, (23.9) corresponds to the transpose of a matrix in this situation. Note that (23.9) maps c_{00}(X × X, R) onto itself, and maps c_{00,1}(X × X, R) onto c_{00,2}(X × X, R).

Put c_{00,12}(X × X, R) = c_{00,1}(X × X, R) ∩ c_{00,2}(X × X, R), (23.11)

which is a subring of c_{00,1}(X × X, R) and c_{00,2}(X × X, R). Using (23.3) and (23.7), we get that

c_{00}(X × X, R) ⊆ c_{00,12}(X × X, R). (23.12)

We also have that (23.9) maps c_{00,12}(X × X, R) onto itself, because it maps c_{00,1}(X × X, R) onto c_{00,2}(X × X, R). If R has a multiplicative identity element e, then let δ ∈ c(X × X, R) be defined by

(23.13)  δ(x, y) = e when x = y
= 0 when x ≠ y.

This is an element of c_{00,12}(X × X, R), and the multiplicative identity element in c_{00,1}(X × X, R) and c_{00,2}(X × X, R).

Let r ↦ r^{∗} be an involution on R. If a ∈ c(X × X, R), then let a^{t∗} in c(X × X, R) be defined by

a^{t∗}(x, y) = (a^{t}(x, y))^{∗} = a(y, x)^{∗} (23.14)

for every x, y ∈ X. Observe that

a ↦ a^{t∗} (23.15)

is a group isomorphism from c(X × X, R) onto itself with respect to addition, with (a^{t∗})^{t∗} = a (23.16)

for every a ∈ c(X × X, R). As before, (23.15) maps c_{00}(X × X, R) onto itself, c_{00,1}(X × X, R) onto c_{00,2}(X × X, R), and c_{00,12}(X × X, R) onto itself. If a, b are elements of c_{00,1}(X × X, R), then it is easy to see that

(a b)^{t∗} = b^{t∗} a^{t∗}, (23.17)

where multiplication is defined as in (23.1). Of course, this also works when a, b ∈ c_{00,2}(X×X, R). In particular, (23.15) is an involution on c_{00,12}(X×X, R), and hence on c_{00}(X × X, R).

24 Double sums

Let X be a nonempty set, and let A be a commutative group, with the group operations expressed additively. The space c(X, A) of A-valued functions on X is also a commutative group with respect to pointwise addition of functions. As usual, the support of f ∈ c(X, A) is defined to be the set of x ∈ X such that f(x) ≠ 0. Let c_{00}(X, A) be the set of f ∈ c(X, A) with finite support in X, which is a subgroup of c(X, A). If f ∈ c_{00}(X, A), then ∑_{x∈X} f(x) can be defined as an element of A, and this defines a group homomorphism from c_{00}(X, A) into A.

Let X, Y be nonempty sets, so that their Cartesian product X × Y is a nonempty set as well. If f ∈ c_{00}(X × Y, A), then

f_{X}(x) = ∑_{y∈Y} f(x, y) (24.1)

is defined as an element of A for each x ∈ X, and

f_{Y} (y) = ∑_{x∈X} f(x, y) (24.2)

is defined as an element of A for each y ∈ Y. It is easy to see that f_{X} and f_{Y} have finite support as A-valued functions on X and Y, respectively, so that

∑_{x∈X} f_{X}(x) (24.3)

and

∑_{y∈Y} f_{Y} (y) (24.4)

are defined as elements of A too. The double sum

∑_{(x,y)∈X×Y} f(x, y) (24.5)

can also be defined as an element of A. Of course, the iterated sums (24.3) and (24.4) are equal to (24.5).

Now let f be a nonnegative real-valued function on X ×Y. The sums (24.1), (24.2), and (24.5) can be defined as nonnegative extended real numbers, as in Section 9. The iterated sums (24.3) and (24.4) can also be defined as nonnegative extended real numbers, with the convention that the sum is automatically +∞ when any of the terms is +∞. One can check that (24.3), (24.4), and (24.5) are the same in this situation too, by approximating all of these sums by finite subsums.

Let k be a field with an absolute value function | · |, let V be a vector space over k, and let N be a norm on V with respect to | · | on k. Suppose that V is complete with respect to the metric associated to N. Let f be a V -valued function on X × Y that is summable with respect to N. Thus the double sum (24.5) can be defined as an element of V, as in Section 13. Observe that f(x, y) is summable as a V -valued function on x ∈ X for each y ∈ Y, and similarly that f(x, y) is summable as a V -valued function of y ∈ Y for each x ∈ X. This permits us to define f_{X}(x) and f_{Y} (y) as V -valued functions on X and Y, respectively, as in (24.1) and (24.2), and using the remarks in Section 13 again. We also have that

N(f_{X}(x)) ≤ ∑_{y∈Y} N(f(x, y)) (24.6)

for every x ∈ X, and

N(f_{Y} (y)) ≤ ∑_{x∈X} N(f(x, y)) (24.7)

for every y ∈ Y. It follows that

∑_{x∈X} N(f_{X}(x)) ≤ ∑_{x∈X} ( ∑_{y∈Y} N(f(x, y))) = ∑_{(x,y)∈X×Y} N(f(x, y)) (24.8)

and

∑_{y∈Y} N(f_{Y} (y)) ≤ ∑_{y∈Y} ( ∑_{x∈X} N(f(x, y))) = ∑_{(x,y)∈X×Y} N(f(x, y)), (24.9)

using the previous remarks about double sums of nonnegative real numbers. In particular, f_{X} is summable as a V -valued function on X, and f_{Y} is summable as a V -valued function on Y. This means that the sums (24.3) and (24.4) can also be defined as elements of V, as in Section 13. Under these conditions, one can check that the iterated sums (24.3) and (24.4) are equal to the double sum (24.5). More precisely, this was mentioned earlier when f has finite support in X × Y, and otherwise one can approximate f by V -valued functions with finite support in X × Y with respect to the ℓ^{1} norm.

Suppose now that N is an ultranorm on V, and that V is still complete with respect to the ultrametric associated to N. Let f be a V -valued function on X × Y that vanishes at infinity with respect to N. In this case, the double sum (24.5) can be defined as an element of V, as in Section 13 again. It is easy to see that f(x, y) vanishes at infinity as a V -valued function of x ∈ X for every y ∈ Y, and as a function of y ∈ Y for every x ∈ X. Hence f_{X}(x) and f_{Y} (y) may be defined as V -valued functions on X and Y, respectively, as in (24.1) and (24.2), using the remarks in Section 13. In this situation,

N(f_{X}(x)) ≤ max_{y∈Y} N(f(x, y)) (24.10)

for every x ∈ X, and

N(f_{Y} (y)) ≤ max_{x∈X} N(f(x, y)) (24.11)

for every y ∈ Y. One can verify that the right side of (24.10) vanishes at infinity as a nonnegative real-valued function of x ∈ X, because f(x, y) vanishes at infinity on X × Y, by hypothesis. Similarly, the right side of (24.11) vanishes at infinity as a nonnegative real-valued function of y ∈ Y. This implies that the iterated sums (24.3) and (24.4) can be defined as elements of V, as in Section 13. One can check that the iterated sums (24.3) and (24.4) are equal to the double sum (24.5), by approximating f by V -valued functions with finite support in X × Y with respect to the supremum norm.

25 A summability condition

Let k be a field with an absolute value function | · |, let A be an algebra over k with a submultiplicative norm N with respect to | · | on k, and suppose that A is complete with respect to the metric associated to N. Also let X be a nonempty set, and remember that the space c(X × X, A) of A-valued functions on X × X is a vector space over k with respect topointwise addition and scalar multiplication. If a is an element of c(X × X, A), then

N_{∞}(a) = sup_{x,y∈X} N(a(x, y)) (25.1)

and

N_{1,∞}(a) = sup_{y∈X} ( ∑_{x∈X} N(a(x, y))) (25.2)

are defined as nonnegative extended real numbers. Note that (25.1) and (25.2) correspond to (21.2) and (21.3) when X = {1, …, n} for some positive integer n. Of course, (25.1) is finite exactly when a is bounded on X × X, in which case (25.1) is the same as the supremum norm of a. Put

M_{X}^{1,∞}(A) = {a ∈ c(X × X, A) : N_{1,∞}(a) < ∞}. (25.3)

One can check that (25.3) is a linear subspace of c(X × X, A), and that (25.2) defines a norm on (25.3) with respect to | · | on k. Clearly

N_{∞}(a) ≤ N_{1,∞}(a) (25.4)

for every a ∈ c(X × X, A), so that (25.3) is contained in ℓ^{∞}(X × X, A). One can also verify that (25.3) is complete with respect to the metric associated to (25.2), because A is complete by hypothesis.

Let a, b ∈ M_{X}^{1,∞}(A) be given. We would like to put

c(x, y) = ∑_{w∈X} a(x, w) b(w, y) (25.5)

for every x, y ∈ X, as in Section 23. Of course,

∑_{w∈X} N(a(x, w) b(w, y)) ≤ ∑_{w∈X} N(a(x, w)) N(b(w, y)) (25.6)

for every x, y ∈ X, because N is submultiplicative on A. The right side of (25.6) is finite for every x, y ∈ X, because b ∈ M_{X}^{1,∞}(A), and N(a(x, w)) is bounded. Thus the right side of (25.5) may be defined as an element of A for every x, y ∈ X, as in Section 13. Note that the norm of (25.5) with respect to N is less than or equal to the left side of (25.6) for every x, y ∈ X. This implies that

∑_{x∈X} N(c(x, y)) ≤ ∑_{x∈X} ( ∑_{w∈X} N(a(x, w)) N(b(w, y))) (25.7)

for every y ∈ X. It follows that

∑_{x∈X} N(c(x, y)) ≤ ∑_{w∈X} ( ∑_{x∈X} N(a(x, w)) N(b(w, y))) (25.8)

for every y ∈ X, by interchanging the order of summation on the right side of (25.7), as in the previous section. Hence

∑_{x∈X} N(c(x, y)) ≤ N_{1,∞}(a) ∑_{w∈X} N(b(w, y)) ≤ N_{1,∞}(a) N_{1,∞}(b) (25.9)

for every y ∈ X, by the definition (25.2) of N_{1,∞}. This shows that c ∈ M_{X}^{1,∞}(A), with N_{1,∞}(c) ≤ N_{1,∞}(a) N_{1,∞}(b). (25.10)

As before, we take c to be the product of a and b, which may be expressed as

c = a b. (25.11)

It is easy to see that this operation of multiplication is bilinear on M_{X}^{1,∞}(A).

Now let a, b, c be any elements of M_{X}^{1,∞}(A). In order to show that multiplication is associative on M_{X}^{1,∞}(A), one would like to verify that

∑_{z∈X} ( ∑_{w∈X} a(x, w) b(w, z)) c(z, y) = ∑_{w∈X} a(x, w) ( ∑_{z∈X} b(w, z) c(z, y)) (25.12)

for every x, y ∈ X. More precisely, the left side of (25.12) is (a b) c evaluated at x, y, and the right side is a (b c) evaluated at x, y. Equivalently, this means that

∑_{z∈X} ( ∑_{w∈X} a(x, w) b(w, z) c(z, y)) = ∑_{w∈X} ( ∑_{z∈X} a(x, w) b(w, z) c(z, y)) (25.13)

for every x, y ∈ X, so that one would like to interchange the order of summation. If x, y ∈ X, then

∑_{z∈X} ( ∑_{w∈X} N(a(x, w) b(w, z) c(z, y))) (25.14)

≤ ∑_{z∈X} ( ∑_{w∈X} N(a(x, w)) N(b(w, z)) N(c(z, y)))
≤ N_{∞}(a) ∑_{z∈X} ( ∑_{w∈X} N(b(w, z)) N(c(z, y)))
≤ N_{∞}(a) N_{1,∞}(b) ∑_{z∈X} N(c(z, y)) ≤ N_{∞}(a) N_{1,∞}(b) N_{1,∞}(c).

This implies that for each x, y ∈ X,

a(x, w) b(w, z) c(z, y) (25.15)

is summable with respect to N as an A-valued function of w and z on X × X. This permits one to interchange the order of summation in (25.13), by the remarks in the previous section.

Suppose that A has a multiplicative identity element e, and let δ be the A-valued function on X × X defined by

(25.16)  δ(x, y) = e when x = y
= 0 when x ≠ y.

Observe that N_{1,∞}(δ) = N(e). (25.17)

In particular, δ ∈ M_{X}^{1,∞}(A), and it is easy to see that δ is the multiplicative identity element in M_{X}^{1,∞}(A), as in Section 23.

26 Another summability condition

Let us continue with the same notation and hypotheses as in the previous section. If a ∈ c(X × X, A), then

N_{∞,1}(a) = sup_{x∈X} ( ∑_{y∈X} N(a(x, y))) (26.1)

is defined as a nonnegative extended real number, which corresponds to (21.8) when X = {1, …, n} for some positive integer n. As before, one can verify that

M_{X}^{∞,1}(A) = {a ∈ c(X × X, A) : N_{∞,1}(a) < ∞} (26.2)

is a linear subspace of c(X × X, A), and that (26.1) defines a norm on (26.2) with respect to | · | on k. Note that

N_{∞}(a) ≤ N_{∞,1}(a) (26.3)

for every a ∈ c(X×X, A), where N_{∞}(a) is as in (25.1), so that (26.2) is contained in ℓ^{∞}(X × X, A). One can check that (26.2) is complete with respect to the metric associated to (26.1), because A is supposed to be complete, as in the preceding section.

If a, b ∈ M_{X}^{∞,1}(A), then we would like to define c(x, y) as an element of A for every x, y ∈ X as in (25.5). In this situation, the right side of (25.6) is finite for every x, y ∈ X, because a ∈ M_{X}^{∞,1}(A) and N(b(w, y)) is bounded. This permits us to define c(x, y) as an element of A as in (25.5) for every x, y ∈ X, using the remarks in Section 13 again. As before, the norm of c(x, y) with respect to N is less than or equal to the left side of (25.6) for every x, y ∈ X. Thus

∑_{y∈X} N(c(x, y)) ≤ ∑_{y∈X} ( ∑_{w∈X} N(a(x, w)) N(b(w, y))) (26.4)

for every x ∈ X. We can interchange the order of summation on the right side, as in Section 24, to get that

∑_{y∈Y} N(c(x, y)) ≤ ∑_{w∈Y} ( ∑_{y∈X} N(a(x, w)) N(b(w, y))) (26.5)

for every x ∈ X. It follows that

∑_{y∈Y} N(c(x, y)) ≤ N_{∞,1}(b) ∑_{w∈X} N(a(x, w)) ≤ N_{∞,1}(a) N_{∞,1}(b) (26.6)

for every x ∈ X, by the definition (26.1) of N_{∞,1}. Hence c ∈ M_{X}^{∞,1}(A), with

N_{∞,1}(c) ≤ N_{∞,1}(a) N_{∞,1}(b). (26.7)

We take c to be the product of a and b again, which may be expressed as a b. This operation of multiplication is clearly bilinear on M_{X}^{∞,1}(A).

In order to show that multiplication is associative on M_{X}^{∞,1}(A), one should check that (25.12) holds for every a, b, c ∈ M_{X}^{∞,1}(A) and x, y ∈ X. This is the same as (25.13), as before. In this situation, we observe that

∑_{w∈X} ( ∑_{z∈X} N(a(x, w) b(w, z) c(z, y))) (26.8)

≤ ∑_{w∈X} ( ∑_{z∈X} N(a(x, w)) N(b(w, z)) N(c(z, y)))
≤ N_{∞}(c) ∑_{w∈X} ( ∑_{z∈X} N(a(x, w)) N(b(w, z)))
≤ N_{∞,1}(b) N_{∞}(c) ∑_{w∈X} N(a(x, w)) ≤ N_{∞,1}(a) N_{∞,1}(b) N_{∞}(c)

for every x, y ∈ X. It follows that (25.15) is summable with respect to N as an A-valued function of w, z ∈ X for every x, y ∈ X. Thus one can interchange the order of summation in (25.13), as in Section 24.

If A has a multiplicative identity element e, then we define δ ∈ c(X × X, A) as in (25.16). Clearly N_{∞,1}(δ) = N(e), (26.9)

so that δ ∈ M_{X}^{∞,1}(A). It is easy to see that δ is the multiplicative identity element in M_{X}^{∞,1}(A), as before.

If a ∈ c(X × X, A), then a^{t} ∈ c(X × X, A) is defined by

a^{t}(x, y) = a(y, x) (26.10)

for every x, y ∈ X, as in Section 23. Of course,

a ↦ a^{t} (26.11)

is a linear mapping from c(X × X, A) onto itself, and

(a^{t})^{t} = a (26.12)

for every a ∈ c(X × X, A), as before. Observe that (26.11) maps ℓ^{∞}(X × X, A) onto itself, with N_{∞}(a^{t}) = N_{∞}(a) (26.13) for every a ∈ c(X × X, A). Similarly, (26.11) maps M_{X}^{1,∞}(A) onto M_{X}^{∞,1}(A), with N_{∞,1}(a^{t}) = N_{1,∞}(a) (26.14)

for every a ∈ c(X × X, A).

Let x ↦ x^{∗} be an involution on A, which may be conjugate-linear when k = C. If a ∈ c(X × X, A), then a^{t∗} ∈ c(X × X, A) is defined by

a^{t∗}(x, y) = (a^{t}(x, y))^{∗} = a(y, x)^{∗} (26.15)

for every x, y ∈ X, as in Section 23 again. Clearly

a ↦ a^{t∗} (26.16)

is a linear mapping from c(X × X, A) onto itself when x ↦ x^{∗} is linear, and otherwise (26.16) is conjugate-linear when k = C and x ↦ x^{∗} is conjugatelinear. We also have that (a^{t∗})^{t∗} = a (26.17)

for every a ∈ c(X × X, A), as before.

Suppose that there is a positive real number C such that

N(x^{∗}) ≤ C N(x) (26.18)

for every x ∈ A. In this case, we have that

(26.19)  N_{∞}(a^{t∗}) ≤ C N_{∞}(a),
(26.20)  N_{∞,1}(a^{t∗}) ≤ C N_{1,∞}(a),

and N_{1,∞}(a^{t∗}) ≤ C N_{∞,1}(a) (26.21)

for every a ∈ c(X ×X, A). In particular, (26.16) maps ℓ^{∞}(X ×X, A) onto itself, and M_{X}^{1,∞}(A) onto M_{X}^{∞,1}(A). If C = 1, then equality holds in (26.18), as in Section 22. This implies that equality holds in (26.19), (26.20), and (26.21) as well.

If a, b ∈ M_{X}^{1,∞}(A), then

(a b)^{t∗} = b^{t∗} a^{t∗}, (26.22)

where mutliplication is defined as in (25.5), as before. This also works when a, b ∈ M_{X}^{∞,1}(A). Note that

M_{X}^{1,∞}(A) ∩ M_{X}^{∞,1}(A) (26.23)

is a subalgebra of M_{X}^{1,∞}(A) and M_{X}^{∞,1}(A), and that (26.16) maps (26.23) onto itself, by the remarks in the preceding paragraph. It follows that (26.16) defines an algebra involution on (26.23), which is conjugate-linear when k = C and x ↦ x^{∗} is conjugate-linear on A.

27 A c_{0} condition

Let k be a field with an absolute value function | · | again, let A be an algebra over k with a submultiplicative norm N with respect to | · | on k, and let X be a nonempty set. Put

c_{0,1}(X × X, A) = {a ∈ ℓ^{∞}(X × X, A) : for each y ∈ X, a(x, y) vanishes

(27.1) at infinity as a function of x ∈ X}.

More precisely, this means that a(x, y) vanishes at infinity with respect to N as an A-valued function of x ∈ X for each y ∈ X, so that a(·, y) ∈ c_{0}(X, A) for every y ∈ X. It is easy to see that (27.1) is a closed linear subspace of ℓ^{∞}(X × X, A) with respect to the supremum metric, because c_{0}(X, A) is a closed linear subspace of ℓ^{∞}(X, A). If a is an element of the space M_{X}^{1,∞}(A) defined in (25.3), then a ∈ ℓ^{∞}(X×X, A), and a(·, y) ∈ ℓ^{1}(X, A) for every y ∈ X. This implies that a(·, y) ∈ c_{0}(X, A) for every y ∈ X, so that a ∈ c_{0,1}(X × X, A). Of course, if X has only finitely many elements, then (27.1) is the same as the space c(X × X, A) of all A-valued functions on X × X.

Let us suppose from now on in this section that N is an ultranorm on A, and that A is complete with respect to the corresponding ultrametric. If a, b ∈ c_{0,1}(X × X, A), then we would like to put

c(x, y) = ∑_{w∈X} a(x, w) b(w, y) (27.2)

for every x, y ∈ X, as before. Observe that a(x, w) b(w, y) vanishes at infinity as a function of w ∈ X for every x, y ∈ X, because a(x, w) is bounded, and b(w, y) vanishes at infinity as a function of w ∈ X, by hypothesis. This implies that the sum on the right side of (27.2) may be defined as an element of A for every x, y ∈ X, as in Section 13. We also have that

N(c(x, y)) ≤ max_{w∈X} N(a(x, w) b(w, y)) ≤ max_{w∈X}(N(a(x, w)) N(b(w, y))) (27.3)

for every x, y ∈ X in this situation. In particular, it follows that c(x, y) is bounded on X × X, with

N_{∞}(c) ≤ N_{∞}(a) N_{∞}(b), (27.4)

where N_{∞} is the supremum ultranorm, as in (25.1). We would like to check that c ∈ c_{0,1}(X × X, A). To do this, let y ∈ X be given, and let us verify that c(x, y) vanishes at infinity as a function of x ∈ X.

If E is a nonempty finite subset of X, then

c_{E}(x, y) = ∑_{w∈E} a(x, w) b(w, y) (27.5)

vanishes at infinity as a function of x ∈ X, because a(x, w) vanishes at infinity as a function of x ∈ X for every w ∈ X, by hypothesis. Of course,

c(x, y) − c_{E}(x, y) = ∑_{w∈X\E} a(x, w) b(w, y), (27.6)

where the sum on the right is defined as an element of A for the same reasons as before. It follows that

(27.7)  N(c(x, y) − c_{E}(x, y)) ≤ max_{w∈X\E} N(a(x, w) b(w, y))
≤ max_{w∈X\E}(N(a(x, w)) N(b(w, y)))
≤ N_{∞}(a) ( max_{w∈X\E} N(b(w, y)))

for every x ∈ X. We can choose E so that the right side of (27.7) is as small as we want, because b(w, y) vanishes at infinity as a function of w ∈ X. This implies that c(x, y) vanishes at infinity as a function of x ∈ X, because it can be approximated uniformly by functions that vanish at infinity on X.

As in the earlier situations, we take (27.2) to be the product of a and b in c_{0,1}(X ×X, A), which may be expressed as a b. This defines a bilinear operation of multiplication on c_{0,1}(X × X, A), which we would like to show is associative. Let a, b, c ∈ c_{0,1}(X × X, A) be given, and let us verify that

∑_{z∈X} ( ∑_{w∈X} a(x, w) b(w, z)) c(z, y) = ∑_{w∈X} ( ∑_{z∈X} b(w, z) c(z, y)) c(z, y) (27.8)

for every x, y ∈ X. As before, this is the same as saying that

∑_{z∈X} ( ∑_{w∈X} a(x, w) b(w, z) c(z, y)) = ∑_{w∈X} ( ∑_{z∈X} a(x, w) b(w, z) c(z, y)) (27.9)

for every x, y ∈ X, so that we want to interchange the order of summation. Let x, y ∈ X be given, and let us check that

a(x, w) b(w, z) c(z, y) (27.10)

vanishes at infinity with respect to N as an A-valued function of w and z on X × X. Observe that

(27.11) N(a(w, x) b(w, z) c(z, y)) ≤ N(a(x, w)) N(b(w, z)) N(c(z, y))
≤ N_{∞}(a) N(b(w, z)) N(c(z, y))

for every w, z ∈ X. Because c(z, y) vanishes at infinity as a function of z ∈ X, there are finite subsets E of X such that N(c(z, y)) is as small as we want when z ∈ X \ E. This implies that the right side of (27.11) is as small as we want for every w ∈ X and z ∈ X \ E, because b(w, z) is bounded on X × X. If z ∈ E, then the right side of (27.11) is as small as we want for all but finitely many w ∈ X, because b(w, z) vanishes at infinity as a function of w ∈ X. It follows that (27.10) vanishes at infinity as a function of w and z on X × X, as desired. This permits us to interchange the order of summation in (27.9), as in Section 24.

If A has a multiplicative identity element e, then we let δ(x, y) be the Avalued function on X × X equal to e when x = y, and to 0 when x ≠ y, as before. Note that δ ∈ c_{0,1}(X × X, A), with

N_{∞}(δ) = N(e). (27.12)

In fact, δ is the multiplicative identity element in c_{0,1}(X × X, A).

28 Another c_{0} condition

Let k be a field with an absolute value function | · |, let A be an algebra over k with a submultiplicative norm N with respect to | · | on k, and let X be a nonempty set, as in the preceding section. Put

c_{0,2}(X × X, A) = {a ∈ ℓ^{∞}(X × X, A) : for each x ∈ X, a(x, y) vanishes

(28.1) at infinity as a function of y ∈ X}.

As before, this means that a(x, y) vanishes at infinity with respect to N as an A-valued function of y for each x ∈ X, so that a(x, ·) ∈ c_{0}(X, A) for every x ∈ X. This also defines a closed linear subapce of ℓ^{∞}(X ×X, A) with respect to the supremum metric, because c_{0}(X, A) is a closed linear subspace of ℓ^{∞}(X, A). The space M_{X}^{∞,1}(A) defined in (26.2) is contained in (28.1), because ℓ^{1}(X, A) is contained in c_{0}(X, A).

As in the previous section, we suppose from now on in this section that N is an ultranorm on A, and that A is complete with respect to the corresponding ultrametric. Let a, b ∈ c_{0,2}(X × X, A) be given, and let us define c(x, y) as an element of A for every x, y ∈ X as in (27.2) again. More precisely, for each x, y ∈ X, a(x, w) b(w, y) vanishes at infinity as a function of w ∈ X, because a(x, w) vanishes at infinity as a function of w ∈ X, and b(w, y) is bounded, by hypothesis. Hence the sum on the right side of (27.2) may be defined as an element of A, as in Section 13. The simple estimates (27.3) and (27.4) also hold in this situation, for the same reasons as before.

In order to check that c ∈ c_{0,2}(X × X, A), let x ∈ X be given, and let us verify that c(x, y) vanishes at infinity as a function of y ∈ X. If E is a nonempty finite subset of X and c_{E}(x, y) is defined as in (27.5), then c_{E}(x, y) vanishes at infinity as a function of y ∈ X, because b(w, y) vanishes at infinity as a function of y ∈ X for every w ∈ X, by hypothesis. Observe that

N(c(x, y) − c_{E}(x, y)) ≤ ( max_{w∈X\E} N(a(x, w))) N_{∞}(b) (28.2)

for every y ∈ X, for essentially the same reasons as for (27.7). As before, we can choose E so that the right side of (28.2) is as small as we want, because a(x, w) vanishes at infinity as a function of w ∈ X, by hypothesis. It follows that c(x, y) vanishes at infinity as a function of y ∈ X, because it can be approximated uniformly by functions that vanish at infinity on X.

As usual, we take (27.2) to be the product of a and b in c_{0,2}(X ×X, A), which may be expressed as a b. To show that this bilinear operation of multiplication is associative, we should check that (27.8) holds for every a, b, c ∈ c_{0,2}(X ×X, A) and x, y ∈ X. It suffices to verify that

a(x, w) b(w, z) c(z, y) (28.3)

vanishes at infinity with respect to N as an A-valued function of w, z ∈ X, as before. In this case, we use the fact that

(28.4) N(a(x, w) b(w, z) c(z, y)) ≤ N(a(x, w)) N(b(w, z)) N(c(z, y))
≤ N(a(x, w)) N(b(w, z)) N_{∞}(c)

for every w, z ∈ X. Because a(x, w) vanishes at infinity as a function of w ∈ X, there are finite sets E ⊆ X such that N(a(x, w)) is as small as we want when w ∈ X \ E, which implies that the right side of (28.4) is as small as we want when w ∈ X \E and z ∈ X, because b(w, z) is bounded. If w ∈ E, then the right side of (28.4) is as small as we want for all but finitely many z ∈ X, because b(w, z) vanishes at infinity as a function of z ∈ X. Hence (28.3) vanishes at infinity as a function of w, z ∈ X, which permits us to interchange the order of summation in (27.9), as in Section 24.

If A has a multiplicative identity element e, then we define δ ∈ c(X × X, A) as in the previous section. It is easy to see that δ ∈ c_{0,2}(X × X, A), and that δ is the multiplicative identity element in c_{0,2}(X × X, A).

As before, if a ∈ c(X × X, A), then a^{t} ∈ c(X × X, A) is defined by

a^{t}(x, y) = a(y, x) (28.5)

for every x, y ∈ X. Remember that

a ↦ a^{t} (28.6)

is a linear mapping from c(X × X, A) onto itself, and that

(a^{t})^{t} = a (28.7)

for every a ∈ c(X × X, A). We have also seen that (28.6) maps ℓ^{∞}(X × X, A) isometrically onto itself. Observe that (28.6) maps c_{0,1}(X × X, A) as defined in (27.1) onto c_{0,2}(X × X, A).

Let x ↦ x^{∗} be an involution on A. If a ∈ c(X×X, A), then a^{t∗} ∈ c(X×X, A) is defined by a^{t∗}(x, y) = (a^{t}(x, y))^{∗} = a(y, x)^{∗} (28.8)

for every x, y ∈ X, as before. Remember that

a ↦ a^{t∗} (28.9)

is a linear mapping from c(X × X, A) onto itself, and that

(a^{t∗})^{t∗} = a (28.10) for every a ∈ c(X × X, A).

Suppose that there is a positive real number C such that

N(x^{∗}) ≤ C N(x) (28.11)

for every x ∈ A, so that N_{∞}(a^{t∗}) ≤ C N_{∞}(a) (28.12)

for every a ∈ c(X × X, A), as in (26.19). Thus (28.9) maps ℓ^{∞}(X × X, A) onto itself, and it is easy to see that (28.9) maps c_{0,1}(X × X, A) onto c_{0,2}(X × X, A). If a, b ∈ c_{0,1}(X × X, A), then

(a b)^{t∗} = b^{t∗} a^{t∗}, (28.13)

where multiplication is defined as in (27.2), and this also works when a and b are elements of c_{0,2}(X × X, A). Put

c_{0,12}(X × X, A) = c_{0,1}(X × X, A) ∩ c_{0,2}(X × X, A), (28.14)

which is a subalgebra of c_{0,1}(X × X, A) and c_{0,2}(X × X, A). Under these conditions, (28.10) maps (28.14) onto itself, and defines an algebra involution on (28.14).

Part III

Lipschitz mappings and power series

29 Lipschitz mappings

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces. A mapping f from X into Y is said to be Lipschitz if there is a nonnegative real number C such that

d_{Y} (f(x), f(x′)) ≤ C d_{X}(x, x′) (29.1)

for every x, x′ ∈ X. In this case, we may also say that f is Lipschitz with constant C. Note that Lipschitz mappings are uniformly continuous. A mapping f from X to Y is Lipschitz with constant C = 0 if and only if f is constant as a mapping on X.

Let x_{0} ∈ X be given, and put

f_{0}(x) = d_{X}(x, x_{0}) (29.2)

for every x ∈ X. Observe that

f_{0}(x) ≤ f_{0}(x′) + d_{X}(x, x′) (29.3) for every x, x′ ∈ X, by the triangle inequality. Equivalently,

f_{0}(x) − f_{0}(x′) ≤ d_{X}(x, x′) (29.4)

for every x, x′ ∈ X, and the same inequality holds with the roles of x and x′ interchanged. This implies that

|f_{0}(x) − f_{0}(x′)| ≤ d_{X}(x, x′) (29.5)

for every x, x′ ∈ X, using the standard absolute value function on R on the left side of the inequality. Thus f_{0} is Lipschitz with constant C = 1 as a mapping from X into R, using the standard Euclidean metric on R.

Let (Z, d_{Z}) be another metric space, and suppose that f_{1} is a Lipschitz mapping from X into Y with constant C_{1} ≥ 0, and that f_{2} is a Lipschitz mapping from Y into Z with constant C_{2} ≥ 0. If x, x′ ∈ X, then

d_{Z}(f_{2}(f_{1}(x)), f_{2}(f_{1}(x′))) ≤ C_{2} d_{Y} (f_{1}(x), f_{1}(x′)) ≤ C_{1} C_{2} d_{X}(x, x′). (29.6)

Thus the composition f_{2} ∘ f_{1} of f_{1} and f_{2} is Lipschitz as a mapping from X into Z, with constant C_{1} C_{2}.

Let f be a Lipschitz mapping from X into Y, and consider

(29.7) inf{C ≥ 0 : (29.1) holds},

where more precisely the infimum is taken over all nonnegative real numbers C for which (29.1) holds. This is the infimum of a nonempty set of nonnegative real numbers, by hypothesis, so that the infimum exists and is nonnegative too. One can check that f is Lipschitz with constant equal to (29.7), so that the infimum is automatically attained. Equivalently, if X has at least two elements, then (29.7) is equal to

sup \{\frac{d_{Y} (f(x), f(x′))}{d_{X}(x, x′)} : x, x′ ∈ X, x ≠ x′\}. (29.8)

If X has only one element, then (29.7) is always equal to 0.

Let k be a field with an absolute value function |·|. Note that |x| is Lipschitz with constant C = 1 as a real-valued function on k, with respect to the metric on k associated to | · |, and the standard Euclidean metric on R. This follows from the earlier remarks about (29.2), with X = k and x_{0} = 0. Similarly, if V is a vector space over k with norm N with respect to | · | on k, then N is Lipschitz with constant C = 1 as a real-valued function on V with respect to the metric associated to N.

30 Lipschitz seminorms

Let (X, d_{X}) be a nonempty metric space, let k be a field with an absolute value function | · |, and let W be a vector space over k with a seminorm N_{W} with respect to |·| on k. As before, a mapping f from X into W is said to be Lipschitz if there is a nonnegative real number C such that

N_{W} (f(x) − f(x′)) ≤ C d_{X}(x, x′) (30.1)

for every x, x′ ∈ X. If N_{W} is a norm on W, then this corresponds exactly to (29.1), with Y = W, and d_{Y} taken to be the metric associated to N_{W}. We may also say that f is Lipschitz with constant C when (30.1) holds. In particular, if f is a constant mapping from X into W, then f is Lipschitz with constant C = 0.

Let Lip(X, W) be the space of Lipschitz mappings from X into W. If f is an element of Lip(X, W), then put

(30.2) ∥f∥_{Lip} = ∥f∥_{Lip(X,W)} = inf{C ≥ 0 : (30.1) holds},

where more precisely the infimum is taken over all nonnegative real numbers C for which (30.1) holds. One can check that f is Lipschitz with constant equal to (30.2), as in the previous section. Equivalently, if X has at least two elements, then

∥f∥_{Lip} = sup \{\frac{N_{W} (f(x) − f(y))}{d_{X}(x, x′)} : x, x′ ∈ X, x ≠ x′\}, (30.3)

as before. If X has only one element, then (30.2) is automatically equal to 0.

One can verify that Lip(X, W) is a vector space over k, with respect to pointwise addition and scalar multiplication. Moreover, (30.2) defines a seminorm on Lip(X, W) with respect to | · | on k. If N_{W} is a semi-ultranorm on W, then (30.2) is a semi-ultranorm on Lip(X, W). If N_{W} is a norm on W, then ∥f∥_{Lip} = 0 if and only if f is a constant mapping from X into W.

Let V be another vector space over k, and let N_{V} be a norm on V with respect to | · | on k. If N_{W} is a norm on W and T is a bounded linear mapping from V into W, then T is Lipschitz with respect to the metrics on V and W associated to N_{V} and N_{W}, respectively. More precisely, if (11.1) holds for some C ≥ 0, then (11.2) says exactly that T is Lipschitz with constant C. In this situation, the operator norm ∥T ∥_{op,V W} of T corresponds exactly to the Lipschitz seminorm of T, as in (30.2).

If X and Y are topological spaces, then the space of continuous mappings from X into Y may be denoted C(X, Y), as usual. Suppose that N_{W} is a norm on W, so that W may be considered as a topological space with respect to the topology determined by the metric associated to N_{W}. If X is a nonempty topological space, then the space C(X, W) of continuous mappings from X into W is a vector space over k with respect to pointwise addition and scalar multiplication. Similarly, if (X, d_{X}) and (Y, d_{Y}) are metric spaces, then the space of uniformly continuous mappings from X into Y may be denoted UC(X, Y), and the space of Lipschitz mappings from X into Y may be denoted Lip(X, Y). Thus Lip(X, Y) ⊆ UC(X, Y) ⊆ C(X, Y). (30.4) If X is a nonempty metric space, then it is easy to see that UC(X, W) is a linear subspace of C(X, W). In this case, Lip(X, W) may be considered as a linear subspace of UC(X, W).

31 Bounded continuous functions

Let (Y, d_{Y}) be a metric space, and let E be a subset of Y. As usual, E is said to be bounded in Y if there is a finite upper bound for d(y, z) with y, z ∈ E. If y_{0} is any element of Y, then E is bounded in Y if and only if E is contained in a ball in Y centered at y_{0} with finite radius. If E is compact with respect to the topology determined on Y by d_{Y}, then E is bounded in Y.

A mapping f from a set X into Y is said to be bounded if f(X) is a bounded set in Y. Let B(X, Y) be the set of bounded mappings from X into Y. If f, g ∈ B(X, Y), then d_{Y} (f(x), g(x)) is bounded as a nonnegative real-valued function on X. If X ≠ ∅, then it follows that

θ(f, g) = sup_{x∈X} d_{Y} (f(x), g(x)) (31.1)

is defined as a nonnegative real number. One can check that (31.1) defines a metric on B(X, Y), which is the supremum metric.

If {f_{j}}_{j=1}^{∞} is a sequence of elements of B(X, Y), and f ∈ B(X, Y), then {f_{j}}_{j=1}^{∞} converges to f with respect to the supremum metric if and only if {f_{j}}_{j=1}^{∞} converges to f uniformly on X with respect to d_{Y} on Y. Let {f_{j}}_{j=1}^{∞} be any sequence of mappings from X into Y that converges to a mapping f from X into Y uniformly on X. If f_{j} is bounded for each j, then it is easy to see that f is bounded as well.

If Y is complete with respect to d_{Y}, then it is well known that B(X, Y) is complete with respect to the supremum metric. More precisely, if {f_{j}}_{j=1}^{∞} is a Cauchy sequence in B(X, Y) with respect to the supremum metric, then {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in Y for each x ∈ X. This implies that {f_{j}(x)}_{j=1}^{∞} converges to an element f(x) in Y for each x ∈ X, because Y is complete. One can verify that {f_{j}}_{j=1}^{∞} converges to f uniformly on X, using the fact that {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to the supremum metric. It follows that f is bounded too, as in the preceding paragraph.

Let X be a nonempty topological space, and let

C_{b}(X, Y) = C(X, Y) ∩ B(X, Y) (31.2)

be the space of bounded continuous mappings from X into Y. It is well known that C_{b}(X, Y) is a closed set in B(X, Y) with respect to the supremum metric. If X is compact, then C_{b}(X, Y) is the same as C(X, Y), because continuous mappings send compact sets to compact sets.

Similarly, if (X, d_{X}) is a nonempty metric space, then let

UC_{b}(X, Y) = UC(X, Y) ∩ B(X, Y) (31.3)

be the space of bounded uniformly continuous mappings from X into Y. It is also well known that UC_{b}(X, Y) is a closed set in B(X, Y) with respect to the supremum metric. If X is compact, then another well-known theorem states that continuous mappings from X into Y are uniformly continuous.

Let k be a field with an absolute value function | · |, and let W be a vector space over k with a norm N_{W} with respect to | · | on k. If X is a nonempty set, then B(X, W) is the same as the space ℓ^{∞}(X, W) discussed in Section 8. This is a vector space over k with respect to pointwise addition and scalar multiplication, and the supremum metric on this space is the same as the metric associated to the supremum norm discussed earlier. If X is a nonempty topological space, then C_{b}(X, W) is a linear subspace of ℓ^{∞}(X, W). Similarly, if (X, d_{X}) is a nonempty metric space, then UC_{b}(X, W) is a linear subspace of C_{b}(X, W).

Let A be an algebra over k. If X is a nonempty set, then the space c(X, A) of A-valued functions on X is an algebra over k 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 in X is the multiplicative identity element in c(X, A).

Let N be a a submultiplicative norm on A with respect to |·| on k. It is easy to see that ℓ^{∞}(X, A) is a subalgebra of c(X, A). Let ∥ · ∥_{∞} be the supremum norm on ℓ^{∞}(X, A) corresponding to N on A, as in Section 8. If f, g ∈ ℓ^{∞}(X, A), then

∥f g∥_{∞} = sup_{x∈X} N(f(x) g(x)) ≤ sup_{x∈X}(N(f(x)) N(g(x))) ≤ ∥f∥_{∞} ∥g∥_{∞}, (31.4)

so that ∥ · ∥_{∞} is submultiplicative on ℓ^{∞}(X, A).

If X is a nonempty topological space, then the space C(X, A) of continuous mappings from X into A is subalgebra of c(X, A). Similarly,

C_{b}(X, A) = C(X, A) ∩ ℓ^{∞}(X, A) (31.5)

is a subalgebra of ℓ^{∞}(X, A). The supremum norm of f ∈ C_{b}(X, A) may also be denoted ∥f∥_{sup}.

Let (X, d_{X}) be a nonempty metric space, and let f, g ∈ UC_{b}(X, A) be given. Observe that

f(x) g(x) − f(y) g(y) = (f(x) − f(y)) g(x) + f(y) (g(x) − g(y)) (31.6)

for every x, y ∈ X, so that

N(f(x) g(x) − f(y) g(y)) (31.7)

≤ N(f(x) − f(y)) N(g(x)) + N(f(y)) N(g(x) − g(y))
≤ N(f(x) − f(y)) ∥g∥_{sup} + ∥f∥_{sup} N(g(x) − g(y))

for every x, y ∈ X. Using this, it is easy to see that f g is uniformly continuous on X, so that UC_{b}(X, A) is a subalgebra of C_{b}(X, A). Similarly, if N is an ultranorm on A, then

N(f(x) g(x) − f(y) g(y)) (31.8)

≤ max(N(f(x) − f(y)) N(g(x)), N(f(y)) N(g(x) − g(y)))
≤ max(N(f(x) − f(y)) ∥g∥_{sup}, ∥f∥_{sup} N(g(x) − g(y)))

for every x, y ∈ X.

32 Bounded Lipschitz functions

Let k be a field with an absolute value function |·|, and let A be an algebra over k with a submultiplicative norm N with respect to | · | on k. Also let (X, d_{X}) be a nonempty metric space, and let

Lip_{b}(X, A) = Lip(X, A) ∩ ℓ^{∞}(X, A) (32.1)

be the space of A-valued functions on X that are bounded and Lipschitz. If f, g ∈ Lip_{b}(X, A), then one can check that f g is Lipschitz on X as well, using (31.7). To make this more precise, let ∥ · ∥_{Lip} be the seminorm on Lip(X, A) defined in Section 30, and let ∥ · ∥_{sup} be the supremum norm on C_{b}(X, A), as in the previous section. If f, g ∈ Lip_{b}(X, A), then it is easy to see that

(32.2) ∥f g∥_{Lip} ≤ ∥f∥_{Lip} ∥g∥_{sup} + ∥f∥_{sup} ∥g∥_{Lip},

using (31.7). If N is an ultranorm on A, then one can verify that

(32.3) ∥f g∥_{Lip} ≤ max(∥f∥_{Lip} ∥g∥_{sup}, ∥f∥_{sup} ∥g∥_{Lip}),

using (31.8). Thus Lip_{b}(X, A) is a subalgebra of UC_{b}(X, A).

If r is a nonnegative real number, then

∥f∥_{sup} + r ∥f∥_{Lip} (32.4)

and max(∥f∥_{sup}, r ∥f∥_{Lip}) (32.5)

define norms on Lip_{b}(X, A) with respect to | · | on k. One can check that (32.4) is submultiplicative on Lip_{b}(X, A) for every r ≥ 0, using (32.2). If N is an ultranorm on A, then (32.5) is submultiplicative on Lip_{b}(X, A) too, because of (32.3). In this case, (32.5) is an ultranorm on Lip_{b}(X, A), because ∥f∥_{sup} is an ultranorm on Lip_{b}(X, A), and ∥f∥_{Lip} is a semi-ultranorm on Lip(X, A), as in Section 30.

Note that (32.5) is less than or equal to (32.4) for every r ≥ 0, and that (32.4) is less than or equal to two times (32.5). Of course, (32.4) and (32.5) both increase monotonically in r. If 0 < r ≤ r′ < ∞, then the analogue of (32.4) with r replaced by r′ is bounded by r′/r times (32.4), and similarly for (32.5).

If A is complete with respect to the metric associated to N, and r > 0, then Lip_{b}(X, A) is complete with respect to the metric associated to (32.4) or (32.5). To see this, let {f_{j}}_{j=1}^{∞} be a Cauchy sequence in Lip_{b}(X, A) with respect to the metric associated to (32.4) or (32.5). In particular, {f_{j}}_{j=1}^{∞} is a Cauchy sequence with respect to the supremum metric, and hence converges uniformly to a bounded A-valued function f on X, as in the previous section. One can check that f is Lipschitz on X, because Cauchy sequences are bounded, so that ∥f_{j}∥_{Lip} is bounded. One can use the Cauchy condition in Lip_{b}(X, A) again to get that {f_{j}}_{j=1}^{∞} converges to f with respect to the Lipschitz seminorm under these conditions.

Let f ∈ Lip_{b}(X, A) be given, so that f(x)^{j} is an element of Lip_{b}(X, A) for each j ∈ Z_{+}. One can check that

∥f^{j}∥_{Lip} ≤ j ∥f∥_{sup}^{j−1} ∥f∥_{Lip} (32.6)

for every j ≥ 1, using (32.2) and induction. If N is an ultranorm on A, then

∥f^{j}∥_{Lip} ≤ ∥f∥_{sup}^{j−1} ∥f∥_{Lip} (32.7)

for every j ≥ 1, because of (32.3).

33 Polynomials

Let k be a field, and let a_{0}, …, a_{n} be n + 1 elements of k for some nonnegative integer n. Also let T be an indeterminate, so that

f(T) = ∑_{j=0}^{n} a_{j} T^{j} (33.1)

is a formal polynomial in T with coefficients in k. As in [1, 5], we use uppercase letters like T for indeterminates, and lower-case letters for elements of k, or elements of algebras over k. Let A be an algebra over k, and suppose that A has a multiplicative identity element e. If x ∈ A, then

f(x) = ∑_{j=0}^{n} a_{j} x^{j} (33.2)

defines an element of A, where x^{j} is interpreted as being equal to e when j = 0.

Let | · | be an absolute value function on k, and let N be a submultiplicative norm on A such that N(e) = 1. Observe that

N(f(x)) ≤ ∑_{j=0}^{n} |a_{j}| N(x^{j}) ≤ ∑_{j=0}^{n} |a_{j}| N(x)^{j} (33.3)

for every x ∈ A. If N is an ultranorm on A, then

N(f(x)) ≤ max_{0≤j≤n}(|a_{j}| N(x^{j})) ≤ max_{0≤j≤n}(|a_{j}| N(x)^{j}) (33.4)

for every x ∈ A. Of course, (33.2) defines a continuous mapping from A into itself, with respect to the topology determined on A by the metric associated to N.

Let r be a positive real number, and let

\overline{B}_{r} = \overline{B}(0, r) = \{x ∈ A : N(x) ≤ r\} (33.5)

be the closed ball in A centered at 0 with radius r with respect to the metric associated to N. We may consider \overline{B}_{r} as a metric space, using the restriction of the metric on A associated to N to \overline{B}_{r}. Let ∥ · ∥_{Lip(\overline{B}_{r},A)} be the seminorm defined on Lip(\overline{B_{r}}, A) as in Section 30. If j is a nonnegative integer, then the restriction of x^{j} to \overline{B}_{r} defines a Lipschitz function on \overline{B}_{r} with values in A, with

∥x^{j}∥_{Lip(\overline{B}_{r},A)} ≤ j r^{j−1}. (33.6)

More precisely, if j = 0, then x^{j} is the constant function equal to e, which is Lipschitz with constant 0. If j = 1, then x^{j} is the identity mapping, which is Lipschitz with constant 1. If j ≥ 2, then (33.6) follows from (32.6), because the supremum norm of x on \overline{B}_{r} is less than or equal to r. If N is an ultranorm on A, then we get that ∥x^{j}∥_{Lip(\overline{B}_{r},A)} ≤ r^{j−1} (33.7)

for every j ≥ 1, using (32.7).

Similarly, the restriction of (33.2) to x ∈ \overline{B}_{r} defines a Lipschitz function on \overline{B}_{r} with values in A. Using (33.6), we get that

∥f∥_{Lip(\overline{B}_{r},A)} ≤ ∑_{j=0}^{n} |a_{j}| ∥x^{j}∥_{Lip(\overline{B}_{r},A)} ≤ ∑_{j=1}^{n} j |a_{j}| r^{j−1}. (33.8)

If N is an ultranorm on A, then ∥·∥_{Lip(\overline{B}_{r},A)} is a semi-ultranorm on Lip(\overline{B}_{r}, A), as in Section 30. In this case, we have that

(33.9) ∥f∥_{Lip(\overline{B}_{r},A)} ≤ max_{0≤j≤n}(|a_{j}| ∥x^{j}∥_{Lip(\overline{B}_{r},A)}) ≤ max_{1≤j≤n}(|a_{j}| r^{j−1}),

using (33.7) in the second step. Of course, the right sides of (33.8) and (33.9) should be interpreted as being equal to 0 when n = 0.

34 Power series

Let k be a field, and let a_{0}, a_{1}, a_{2}, a_{3}, … be a sequence of elements of k. If T is an indeterminate, then

f(T) = ∑_{j=0}^{∞} a_{j} T^{j} (34.1)

is a formal power series in T with coefficients in k. We would like to consider associated functions on algebras over k, under suitable convegence conditions. Let | · | be an absolute value function on k, and let A be an algebra over k with a submultiplicative norm N with respect to | · | on k. Suppose that A has a multiplicative identity element e, with N(e) = 1, and that A is complete with respect to the metric associated to N.

Let r be a positive real number, and suppose for the moment that

∑_{j=0}^{∞} |a_{j}| r^{j} (34.2)

converges. If x ∈ A and N(x) ≤ r, then

∑_{j=0}^{∞} N(a_{j} x^{j}) ≤ ∑_{j=0}^{∞} |a_{j}| N(x)^{j} ≤ ∑_{j=0}^{∞} |a_{j}| r^{j}. (34.3)

Under these conditions, we put

f(x) = ∑_{j=0}^{∞} a_{j} x^{j}, (34.4)

where the convergence of the series on the right follows from the remarks in Section 10. We also have that

N(f(x)) ≤ ∑_{j=0}^{∞} N(a_{j} x^{j}) ≤ ∑_{j=0}^{∞} |a_{j}| r^{j} (34.5)

when N(x) ≤ r. Let \overline{B}_{r} be the closed ball in A centered at 0 with radius r, as in (33.5). In this situation, the partial sums of the series on the right side of (34.4) converge uniformly on \overline{B}_{r}, by the well-known criterion of Weierstrass. It follows that (34.4) is uniformly continuous on \overline{B}_{r}, because the partial sums are uniformly continuous on \overline{B}_{r}.

Let us suppose for the rest of the section that N is an ultranorm on A. Let r be a positive real number again, and suppose for the moment that

lim_{j→∞} |a_{j}| r^{j} = 0. (34.6)

If x ∈ A and N(x) ≤ r, then

N(a_{j} x^{j}) = |a_{j}| N(x^{j}) ≤ |a_{j}| N(x)^{j} ≤ |a_{j}| r^{j} (34.7)

for each j, so that

lim_{j→∞} N(a_{j} x^{j}) = 0. (34.8)

This implies that the series on the right side of (34.4) converges in A, as in Section 10. If f(x) is the value of the sum, as before, then

N(f(x)) ≤ max_{j≥0} N(a_{j} x^{j}) ≤ max_{j≥0}(|a_{j}| r^{j}). (34.9)

Suppose now that |a_{j}| r^{j} has a finite upper bound, instead of (34.6). If x ∈ A, then

(34.10)  N(a_{j} x^{j}) ≤ |a_{j}| N(x)^{j} = (|a_{j}| r^{j}) (N(x)/r)^{j}
≤ ( sup_{l≥0}(|a_{l}| r^{l})) (N(x)/r)^{j}

for each j. This implies that (34.8) holds when N(x) < r, so that the series on the right side of (34.4) converges in A, as in Section 10. If f(x) is the value of the sum again, then

N(f(x)) ≤ max_{j≥0} N(a_{j} x^{j}) ≤ sup_{j≥0}(|a_{j}| r_{j}). (34.11)

35 Lipschitz conditions

Let k be a field with an absolute value function | · |, and let A be an algebra over k with a submultiplicative norm N with respect to | · | on k. Suppose that A has a multiplicative identity element e, and that A is complete with respect to the metric associated to N. To be consistent with the previous sections, one might ask that N(e) = 1, but this will not really be needed in this section. This was also not needed for the Lipschitz conditions mentioned in Section 33. Let f(T) = ∑_{j=0}^{∞} a_{j} T^{j} be a formal power series in an indeterminate T with coefficients in k, and let a positive real number r be given. As in (33.5), we let \overline{B}_{r} be the closed ball in A centered at 0 with radius r with respect to N. Similarly, let B_{r} = B(0, 1) = {x ∈ A : N(x) < r} (35.1)

be the open ball in A centered at 0 with radius r with respect to N. Suppose for the moment that

∑_{j=1}^{∞} j |a_{j}| r^{j} (35.2)

converges, which implies that (34.2) converges. Thus the series on the right side of (34.4) converges absolutely for every x ∈ A with N(x) ≤ r, as before, so that (34.4) defines an A-valued function f on \overline{B}_{r}. In this case, one can check that f is Lipschitz on \overline{B}_{r}, with respect to the restriction of the metric on A associated to N to \overline{B}_{r}. More precisely, we have that

∥f∥_{Lip(\overline{B}_{r},A)} ≤ ∑_{j=1}^{∞} j |a_{j}| r^{j−1}, (35.3)

as in (33.8).

Let us now consider the case where N is an ultranorm on A. If (34.6) holds, then (34.8) holds for every x ∈ A such that N(x) ≤ r, which implies that the series on the right side of (34.4) converges in A, as before. It follows that (34.4) defines an A-valued function f on \overline{B}_{r} again, and one can verify that f is Lipschitz on \overline{B}_{r}. More precisely,

∥f∥_{Lip(\overline{B}_{r},A)} ≤ max_{j≥1}(|a_{j}| r^{j−1}), (35.4)

as in (33.9).

If |a_{j}| r^{j} has a finite upper bound, then (34.8) holds for every x ∈ A with N(x) < r, as in the previous section. Hence the series on the right side of (34.4) converges in A for every x ∈ A with N(x) < r, so that (34.4) defines an A-valued function f on B_{r}. As usual, one can check that f is Lipschitz on B_{r}, with respect to the restriction of the metric on A associated to N to B_{r}. We also have that

∥f∥_{Lip(B_{r},A)} ≤ sup_{j≥1}(|a_{j}| r^{j−1}), (35.5)

as in (33.9) again.

Let r_{0} be a positive real number strictly less than r, so that

\overline{B}_{r_{0}} ⊆ B_{r}. (35.6)

If |a_{j}| r^{j} has a finite upper bound, then

lim_{j→∞} |a_{j}| r_{0}^{j} = 0, (35.7)

because |a_{j}| r_{0}^{j} = |a_{j}| r^{j} (r_{0}/r)^{j} is bounded by a constant times (r_{0}/r)^{j}. If f is defined on B_{r} as in the preceding paragraph, then the restriction of f to \overline{B}_{r_{0}} can be treated as in the paragraph before that.

36 Some isometric mappings

Let k be a field with an ultrametric absolute value function | · |, and let A be an algebra over k with a submultiplicative ultranorm N with respect to | · | on k. To be consistent with the previous sections, one might ask A to have a multiplicative identity element, but this is not really needed here, because we shall consider power series with constant term equal to 0. Let g(T) be a formal power series in an indeterminate T with coefficients in k of the form

g(T) = ∑_{j=2}^{∞} a_{j} T^{j}, (36.1)

and put

f(T) = T + g(T) = T + ∑_{j=2}^{∞} a_{j} T^{j}. (36.2)

Suppose that A is complete with respect to the ultrametric associated to N, and let a positive real number r be given. Also let B_{r} and \overline{B}_{r} be the open and closed balls in A centered at 0 with radius r with respect to N, as in (35.1) and (33.5), respectively.

Suppose that

lim_{j→∞} |a_{j}| r^{j} = 0, (36.3)

and put

g(x) = ∑_{j=2}^{∞} a_{j} x^{j} (36.4)

for every x ∈ A with N(x) ≤ r, as in Section 34. This defines a Lipschitz mapping from \overline{B}_{r} into A, with

∥g∥_{Lip(\overline{B}_{r},A)} ≤ max_{j≥2}(|a_{j}| r^{j−1}), (36.5)

as in (35.4). In particular, if |a_{j}| r^{j−1} ≤ 1 (36.6)

for each j ≥ 2, then g is Lipschitz with constant 1 on \overline{B}_{r} with respect to the ultrametric associated to N on A. Similarly, put

f(x) = x + g(x) = x + ∑_{j=2}^{∞} a_{j} x^{j} (36.7)

for each x ∈ A with N(x) ≤ r. This defines a Lipschitz mapping from \overline{B}_{r} into A with constant 1 when (36.6) holds. This can be derived from (35.4), or from the analogous statements for g and the identity mapping on \overline{B}_{r}, using the fact that the Lipschitz seminorm is a semi-ultranorm in this case, as in Section 30. Note that N(f(x)) ≤ N(x) (36.8)

for every x ∈ A with N(x) ≤ r when (36.6) holds. This can be obtained from the fact that f is Lipschitz with constant 1 and f(0) = 0, or by estimating N(f(x)) as in the first step in (34.9). It follows that

f(\overline{B}_{r}) ⊆ \overline{B}_{r} (36.9)

when (36.6) holds.

Suppose that

max_{j≥2}(|a_{j}| r^{j−1}) < 1, (36.10)

in addition to (36.3), which also ensures that the maximum on the right side of (36.10) is attained. Let x, y ∈ A with N(x), N(y) ≤ r be given, and let us check that N(f(x) − f(y)) = N(x − y). (36.11)

The left side of (36.11) is less than or equal to the right side, because f is Lipschitz with constant 1 on \overline{B}_{r}, as in the preceding paragraph. Thus it suffices to verify that the right side of (36.11) is less than or equal to the left side. Of course, this is trivial when x = y, and so we may suppose that x ≠ y. Observe that x − y = f(x) − f(y) − (g(x) − g(y)), (36.12)

so that N(x − y) ≤ max(N(f(x) − f(y)), N(g(x) − g(y))), (36.13)

by the ultranorm version of the triangle inequality. We also have that

N(g(x) − g(y)) ≤ ( max_{j≥2}(|a_{j}| r^{j−1})) N(x − y), (36.14)

by (36.5). If x ≠ y, then (36.10) implies that the right side of (36.14) is strictly less that N(x − y). This means that

N(g(x) − g(y)) < N(x − y), (36.15) by (36.14). It follows that

N(x − y) ≤ N(f(x) − f(y)), (36.16)

by (36.13).

Suppose now that (36.6) holds for each j ≥ 2, but not necessarily (36.3) or (36.10). If x ∈ A and N(x) < r, then we can define g(x) and f(x) as elements of A as in (36.4) and (36.7), as mentioned in Section 34. In this situation, g is Lipschitz as a mapping from B_{r} into A, with

∥g∥_{Lip(B_{r},A)} ≤ sup_{j≥2}(|a_{j}| r^{j−1}) ≤ 1, (36.17)

as in (35.5). Similarly, f is Lipschitz as a mapping from B_{r} into A, with constant 1. This can be obtained from (35.5), or using (36.17) and the fact the identity mapping on B_{r} is Lipschitz with constant 1, as before. We also have that (36.8) holds for every x ∈ A with N(x) < r, for essentially the same reasons as before. This implies that f(B_{r}) ⊆ B_{r}. (36.18)

Let r_{0} be a positive real number with r_{0} < r, so that (35.7) holds. Observe that

max_{j≥2}(|a_{j}| r_{0}^{j−1}) ≤ (r_{0}/r) ( sup_{j≥2}(|a_{j}| r^{j−1}|)) ≤ r_{0}/r < 1, (36.19)

using (36.6) in the second step. Thus the earlier remarks can be applied to the restrictions of f and g to \overline{B}_{r_{0}}. In particular, (36.11) holds for every x, y ∈ A with N(x), N(y) ≤ r_{0}, as before. It follows that (36.11) holds for every x, y ∈ A with N(x), N(y) < 1, by choosing r_{0} so that N(x), N(y) ≤ r_{0}.

37 Some contractions

Let k be a field with an ultrametric absolute value function | · | again, and let A be an algebra over k with a submultiplicative ultranorm N with respect to | · | on k. Also let g(T) be a formal power series in an indeterminate T with coefficients in k of the form

g(T) = ∑_{j=2}^{∞} a_{j} T^{j}, (37.1)

and suppose that A is complete with respect to the ultrametric associated to N. Let r be a positive real number, and suppose that

lim_{j→∞} |a_{j}| r^{j} = 0. (37.2)

If x ∈ A and N(x) ≤ r, then put

g(x) = ∑_{j=2}^{∞} a_{j} x^{j}, (37.3)

as in Section 34. Note that A does not need to have a multiplicative identity element here, as in the previous section. Let \overline{B}_{r} be the closed ball in A centered at 0 with radius r, as in (33.5), so that g defines an A-valued function on \overline{B}_{r}. More precisely, g is Lipschitz with respect to the ultrametric associated to N on A and its restriction to \overline{B}_{r}, with

∥g∥_{Lip(\overline{B}_{r},A)} ≤ max_{j≥2}(|a_{j}| r^{j−1}), (37.4)

as in (35.4).

Let z_{0} be an element of A, and put

h_{0}(x) = z_{0} − g(x) (37.5)

for every x ∈ \overline{B}_{r}. This defines a Lipschitz mapping from \overline{B}_{r} into A, with

∥h_{0}∥_{Lip(\overline{B}_{r},A)} = ∥g∥_{Lip(\overline{B}_{r},A)}. (37.6)

If the right side of (37.4) is less than or equal to 1, and

N(z_{0}) ≤ r, (37.7)

then h_{0}(\overline{B}_{r}) ⊆ \overline{B}_{r}. (37.8)

More precisely, N(h_{0}(x)) ≤ r for every x ∈ \overline{B}_{r}, because h_{0}(0) = z_{0}, h_{0} is Lipschitz with constant 1 on \overline{B}_{r}, and N is an ultranorm on A. One can also verify this more directly from the definitions of g and h_{0}.

Suppose that (37.7) holds and

max_{j≥2}(|a_{j}| r^{j−1}) < 1, (37.9)

so that h_{0} is a Lipschitz mapping from \overline{B}_{r} into itself with constant strictly less than 1, by (37.4) and (37.6). In this case, the contraction mapping principle implies that there is an x_{0} ∈ \overline{B}_{r} such that

h_{0}(x_{0}) = x_{0}. (37.10)

This uses the fact that \overline{B}_{r} is complete as a metric space with respect to the restriction of the ultrametric on A associated to N, because A is complete, by hypothesis, and \overline{B}_{r} is a closed set in A. If f is the A-valued function defined on \overline{B}_{r} as in (36.7), then we get that

f(x_{0}) = x_{0} + g(x_{0}) = z_{0}, (37.11)

because of (37.5) and (37.10). It follows that

f(\overline{B}_{r}) = \overline{B}_{r} (37.12)

under these conditions, because we can take z_{0} to be any element of \overline{B}_{r}.

Suppose now that |a_{j}| r^{j−1} ≤ 1 (37.13)

for every j ≥ 2, without asking that (37.2) or (37.9) hold. If x ∈ A and N(x) < 1, then g(x) can be defined as an element of A as in (37.3), as mentioned in Section 34. Let B_{r} be the open ball in A centered at 0 with radius r, as in (35.1), so that g defines an A-valued function on B_{r}. In fact, g is Lipschitz with respect to the ultrametric associated to N on A and its restriction to B_{r}, with

∥g∥_{Lip(B_{r},A)} ≤ sup_{j≥2}(|a_{j}| r^{j−1}), (37.14)

as in (35.5). Let z_{0} ∈ A be given again, and let h_{0}(x) be defined for x ∈ B_{r} as in (37.5). Note that h_{0} is Lipschitz as a mapping from B_{r} into A, with

∥h_{0}∥_{Lip(B_{r},A)} = ∥g∥_{Lip(B_{r},A)}. (37.15)

If N(z_{0}) < 1, (37.16)

then h_{0}(B_{r}) ⊆ B_{r}. (37.17)

More precisely, one can check that N(h_{0}(x)) < r for every x ∈ B_{r}, because h_{0}(0) = z_{0}, h_{0} is Lipschitz with constant 1 on B_{r}, and N is an ultranorm on A. One can also get this more directly from the definitions of g and h_{0}, as before.

Remember that B_{r} is a closed set in A, because N is an ultranorm on A, as in Section 1. This implies that B_{r} is complete as a metric space with respect to the restriction of the ultrametric on A associated to N, because A is complete. If (37.16) holds, and the right side of (37.14) is strictly less than 1, then one can apply the contraction mapping principle to h_{0} on B_{r}, to get that there is an x_{0} ∈ B_{r} that satisfies (37.10). However, we can also get this using (37.13), without asking that the right side of (37.14) be less than 1, as follows. Let r_{0} be a positive real number such that r_{0} < r and

N(z_{0}) ≤ r_{0}. (37.18)

Note that (35.7) and (36.19) hold, because r_{0} < r, as before. Thus we can apply the earlier remarks to the restrictions of g and h_{0} to \overline{B}_{r_{0}}, to get that there is an x_{0} ∈ \overline{B}_{r_{0}} that satisfies (37.10). If f is the A-valued function defined on B_{r} as in (36.7), then (37.11) holds, as before, so that

f(B_{r}) = B_{r}. (37.19)

Part IV

The exponential function

38 The real and complex cases

Let k be a field of characteristic 0, so that there is a natural embedding of Q into k, and we shall simply think of Q as being a subfield of k. If T is an indeterminate, then

exp(T) = ∑_{j=0}^{∞}(1/j!) T^{j} (38.1)

may be considered as a formal power series in T with coefficients in k, which is the formal power series corresponding to the exponential function. Of course, j! is “j factorial”, the product of the positive integers from 1 to j, which is interpreted as being equal to 1 when j = 0. We shall consider the convergence of the corresponding power series in algebras over k in various situations.

Suppose first that r is a nonnegative real number, and put

exp(r) = ∑_{j=0}^{∞} r^{j}/j!, (38.2)

where the sum on the right is considered as an infinite series of nonnegative real numbers. It is well known and not difficult to show that this series converges, so that (38.2) is defined as a nonnegative real number. Note that exp(0) = 1, and that exp(r) increases monotonically in r, because r^{j} increases monotonically in r ≥ 0 for each j ≥ 0. We also have that exp(r) → +∞ as r → +∞, because of the analogous property of r^{j} when j ≥ 1.

Suppose now that k = R or C, equipped with the standard absolute value function. Let A be an algebra over k with a multiplicative identity element e and a submultiplicative norm N with respect to | · |, and suppose that A is complete with respect to the metric associated to N. It is also convenient to ask that N(e) = 1. If x ∈ A, then we would like to define

exp(x) = ∑_{j=0}^{∞}(1/j!) x^{j} (38.3)

as an element of A. Observe that

∑_{j=0}^{∞} N((1/j!) x^{j}) ≤ ∑_{j=0}^{∞}(1/j!) N(x)^{j} = exp(N(x)), (38.4)

so that the series on the right side of (38.3) converges absolutely. Hence this series converges in A, because A is complete with respect to the metric associated to N, as in Section 10. We also get that

N(exp(x)) ≤ ∑_{j=0}^{∞} N((1/j!) x^{j}) ≤ exp(N(x)). (38.5)

Let x, y be commuting elements of A, so that x y = y x. It is well known that

(x + y)^{l} = ∑_{j=0}^{l} \binom{l}{j} x^{j} y^{l−j} (38.6)

for each nonnegative integer j, where

\binom{l}{j} = \frac{l!}{j! (l − j)!} (38.7)

is the usual binomial coefficient. Thus

exp(x + y) = ∑_{l=0}^{∞}(1/l!) (x + y)^{l} = ∑_{l=0}^{∞} ( ∑_{j=0}^{l} \frac{1}{j! (l − j)!} x^{j} y^{l−j}). (38.8)

The sum on the right corresponds exactly to the Cauchy product of the series used to define exp(x) and exp(y). It follows that

(38.9) exp(x + y) = exp(x) exp(y)

under these conditions, as in Section 15.

Note that exp(0) = e (38.10)

in A. If x is any element of A, then x commutes with −x, so that

(38.11) exp(x) exp(−x) = exp(x − x) = e,

as in (38.9). Similarly, exp(−x) exp(x) = e. (38.12)

Thus exp(−x) is the multiplicative inverse of exp(x) in A.

39 Trivial absolute values on Q

Let k be a field of characteristic 0 again, so that Q may be considered as a subfield of k. Also let | · | be an absolute value function on k, and suppose that the restriction of | · | to Q is the trivial absolute value function on Q. In particular, this implies that | · | is non-archimedean on k, so that | · | is an ultrametric absolute value function on k, as in Section 4. Let A be an algebra over k with a multiplicative identity element e and a submultiplicative ultranorm N, and suppose that A is complete with respect to the ultrametric associated to N. As before, it is convenient to ask that N(e) = 1 too.

If x ∈ A and N(x) < 1, then we can define

exp(x) = ∑_{j=0}^{∞}(1/j!) x^{j} (39.1)

as an element of A, as in Section 34. More precisely, N((1/j!) x^{j}) = |1/j!| N(x^{j}) ≤ N(x)^{j} (39.2) for every nonnegative integer j, using the hypothesis that | · | is trivial on Q in the second step. This implies that

lim_{j→∞} N((1/j!) x^{j}) = 0 (39.3)

when N(x) < 1. Using (39.3), we get that the series on the right side of (39.1) converges in A, because N is an ultranorm on A, and A is complete with respect to the ultrametric associated to N, as in Section 10. We also have that

N(exp(x)) ≤ max_{j≥0} N((1/j!) x^{j}) = 1, (39.4)

using (39.2) and N(e) = 1 in the second step. Similarly,

N(exp(x) − e) = N( ∑_{j=1}^{∞}(1/j!) x^{j}) ≤ max_{j≥1} N((1/j!) x^{j}) ≤ N(x), (39.5)

using (39.2) in the second step again. The ultranorm version of the triangle inequality implies that (39.6) 1 = N(e) ≤ max(N(e − exp(x)), N(exp(x))). It follows that N(exp(x)) ≥ 1, because the right side of (39.5) is strictly less than 1, by hypothesis. Combining this with (39.4), we obtain that N(exp(x)) = 1 (39.7) for every x ∈ A with N(x) < 1.

Let x, y be commuting elements of A with N(x), N(y) < 1. Thus N(x + y) ≤ max(N(x), N(y)) < 1, (39.8) by the ultranorm version of the triangle inequality. Under these conditions, we have that (39.9) exp(x + y) = exp(x) exp(y), where each of the exponentials is defined as an element of A as in the preceding paragraph. More precisely, the series defining exp(x + y) corresponds to the Cauchy product of the series defining exp(x) and exp(y), as in (38.8). This implies (39.9), as in Section 15 again. In particular, x automatically commutes with −x, so that

(39.10) exp(x) exp(−x) = exp(−x) exp(x) = e,

as in the previous section. Thus exp(−x) is the multiplicative inverse of exp(x) in A for every x ∈ A with N(x) < 1.

If x, y ∈ A and N(x), N(y) < 1, then

N(exp(x) − exp(y)) = N(x − y). (39.11)

This basically corresponds to (36.11), in the situation described in the paragraphs after the one containing (36.11) in Section 36, with r = 1. More precisely, the exponential function is Lipschitz with constant 1 as a mapping from the open unit ball in A into A, with respect to the ultrametric associated to N and its restriction to the open unit ball, as in (35.5). This implies that the left side of (39.11) is less than or equal to the right side of (39.11), and the opposite inequality is discussed in Section 36. Of course, the constant terms in the definitions of exp(x) and exp(y) cancel each other in the left side of (39.11), and they were not included in Section 36.

Observe that the exponential function maps

B_{1} = B(0, 1) = {x ∈ A : N(x) < 1} (39.12)

into B(e, 1) = {z ∈ A : N(z − e) < 1}, (39.13) by (39.5). In fact, exp(B(0, 1)) = B(e, 1), (39.14) essentially by (37.19), with r = 1. This takes the constant term into account for the exponential function, which was taken to be 0 in Section 37.

40 Some preliminary facts

Let p be a prime number, and let | · |_{p} be the p-adic absolute value function on Q, as in Section 2. In order to deal with the exponential function in p-adic situations, we need to estimate

|1/l!|_{p} = 1/|l!|_{p} (40.1)

for nonnegative integers l, as in [1, 5]. Equivalently, we want to estimate the total number of factors of p in l!. This is the same as the sum of the factors of p in the positive integers less than or equal to l. In particular, there are no factors of p in l! when l < p, so that (40.1) is equal to 1.

Let [r] denote the integer part of a nonnegative real number r, which is the largest integer less than or equal to r. If l is a nonnegative integer, then the total number of factors of p in l! can be given by

∑_{j=1}^{∞}[\frac{l}{p^{j}} ]. (40.2)

Of course, [l/p^{j}] = 0 when l < p^{j}, so that all but finitely many terms in the sum are equal to 0. The first term [l/p] is the same as the number of positive integers less than or equal to l that are multiples of p. These are the positive integers less than or equal to l that lead to factors of p in l!. However, these positive integers less than or equal to l may have more than one factor of p, which lead to additional factors of p in l!. The second term [l/p^{2}] in (40.2) is the number of positive integers less than or equal to l that are multiples of p^{2}. Each of these positive integers less than or equal to l lead to at least one additional factor of p in l!. If j is any positive integer, then [l/p^{j}] is the number of positive integers less than or equal to l that are multiples of p^{j}, each of which leads to at least j factors of p in l!. In this way, the sum over j is the total number number of factors of p in l!.

Observe that

∑_{j=1}^{∞}[\frac{l}{p^{j}} ] < ∑_{j=1}^{∞} \frac{l}{p^{j}} (40.3)

for each positive integer l, because l/p^{j} > 0 for every j, but [l/p^{j}] = 0 when l < p^{j}. Of course, we can sum the geometric series, to get that

∑_{j=1}^{∞} \frac{l}{p^{j}} = (l/p) ∑_{j=0}^{∞} p^{−j} = (l/p) (1 − (1/p))^{−1} = \frac{l}{p − 1}. (40.4)

It follows that |l!|_{p} > p^{−l/(p−1)} (40.5)

for every l ≥ 1, so that 1/|l!|_{p} < p^{l/(p−1)}. (40.6)

If n is a positive integer, then we have that

∑_{j=1}^{∞}[\frac{p^{n}}{p^{j}} ] = ∑_{j=1}^{n} p^{n−j} = ∑_{j=0}^{n−1} p^{j} = \frac{p^{n} − 1}{p − 1}. (40.7)

This also works when n = 0, with the two sums in the middle interpreted as being equal to 0.

Using (40.3) and (40.4), we get that

(p − 1) ∑_{j=1}^{∞}[\frac{l}{p^{j}} ] < l (40.8)

for every l ≥ 1. This implies that

(p − 1) ∑_{j=1}^{∞}[\frac{l}{p^{j}} ] ≤ l − 1, (40.9)

because the left side is an integer. Thus

∑_{j=1}^{∞}[\frac{l}{p^{j}} ] ≤ \frac{l − 1}{p − 1} (40.10)

for every l ≥ 1, so that |l!|_{p} ≥ p^{−(l−1)/(p−1)}, (40.11)

and hence 1/|l!|_{p} ≤ p^{(l−1)/(p−1)}. (40.12)

41 The p-adic case

Let k be a field of characteristic 0 again, and let |·| be an absolute value function on k. As before, we can think of Q as being a subfield of k, so that the restriction of | · | to Q defines an absolute value function on Q. Let p be a prime number, and suppose that the restriction of | · | to Q is the same as the p-adic absolute value function | · |_{p} on Q. This implies that | · | is non-archimedean on k, so that | · | is an ultrametric absolute value function on k, as in Section 4. Let A be an algebra over k with a multiplicative identity element e and a submultiplicative ultranorm N, and suppose that A is complete with respect to the ultrametric associated to N. As usual, it is convenient to also ask that N(e) = 1. If k is not already complete with respect to the metric associated to | · |, then one can pass to a completion, which contains Q_{p} as a subfield in this situation. One can consider A as an algebra over the completion of k, because A is complete, and in particular one can consider A as an algebra over Q_{p}.

If x ∈ A, then we would like to put

exp(x) = ∑_{j=0}^{∞}(1/j!) x^{j} (41.1)

under suitable conditions on N(x). Observe that

N((1/j!) x^{j}) = |1/j!|_{p} N(x^{j}) ≤ (1/|j!|_{p}) N(x)^{j}
(41.2)  ≤ p^{j/(p−1)} N(x)^{j} = (p^{1/(p−1)} N(x))^{j}

for every nonnegative integer j, using (40.6) in the first step on the second line. If N(x) < p^{−1/(p−1)}, (41.3)

then the right side of (41.2) tends to 0 as j → ∞. This implies that the series on the right side of (41.1) converges in A, as in Section 10. Thus exp(x) may be defined as an element of A when (41.3) holds, in which case we have that

N(exp(x)) ≤ max_{j≥0} N((1/j!) x^{j}) = 1, (41.4)

using (41.2) and N(e) = 1 in the second step.

Similarly, if x ∈ A satisfies (41.3), then

(41.5) N(exp(x) − e) ≤ max_{j≥1} N((1/j!) x^{j}) ≤ max_{j≥1}((1/|j!|_{p}) N(x)^{j}), using the first line in (41.2) in the second step. If j is a positive integer, then

(1/|j!|_{p}) N(x)^{j−1} ≤ p^{(j−1)/(p−1)} N(x)^{j−1} = (p^{1/(p−1)} N(x))^{j−1}, (41.6)

using (40.12) in the first step. Combining this with (41.5), we get that

N(exp(x) − e) ≤ N(x), (41.7)

because of (41.3). It follows that

(41.8)  1 = N(e) ≤ max(N(exp(x) − e), N(exp(x)))
≤ max(N(x), N(exp(x))),

using the ultranorm version of the triangle inequality in the first step. Note that N(x) < 1, by (41.3). Thus (41.8) implies that N(exp(x)) ≥ 1. Hence

N(exp(x)) = 1 (41.9)

when x ∈ A satisfies (41.3), by (41.4).

Let x, y be commuting elements of A satisfying (41.3) and

N(y) < p^{−1/(p−1)}. (41.10)

Note that N(x + y) ≤ max(N(x), N(y)) < p^{−1/(p−1)} (41.11)

too, by the ultranorm version of the triangle inequality. Under these conditions, we have that (41.12) exp(x + y) = exp(x) exp(y),

where each of the exponentials is defined as an element of A, as before. Indeed, the series defining exp(x + y) corresponds to the Cauchy products of the series defining exp(x) and exp(y), as in (38.8). This implies (41.12), as in Section 10. In particular, x automatically commutes with −x, so that

(41.13) exp(x) exp(−x) = exp(−x) exp(x) = e,

as before. This implies that exp(−x) is the multiplicative inverse of exp(x) in A when x ∈ A satisfies (41.3).

As usual, B(x, r) denotes the open ball in A centered at x ∈ A with radius r > 0 with respect to the ultrametric associated to N. Thus the exponential function defines an A-valued function on B(0, p^{−1/(p−1)}). If j is a positive integer, then |1/j!|_{p} (p^{−1/(p−1)})^{j−1} ≤ 1, (41.14)

as in (40.12). This implies that the exponential function is Lipschitz with constant 1 on B(0, p^{−1/(p−1)}), as in (35.5). More precisely, the exponential function corresponds to taking a_{j} = 1/j! in (35.5), and we take r = p^{−1/(p−1)}.

If x, y ∈ B(0, p^{−1/(p−1)}), then

N(exp(x) − exp(y)) = N(x − y). (41.15)

Of course, the left side of (41.15) is less than or equal to the right side, because the exponential function is Lipschitz with constant 1 on B(0, p^{−1/(p−1)}), as in the previous paragraph. The fact that equality holds in (41.15) basically corresponds to (36.11), in the situation described in the paragraphs after the one containing (36.11) in Section 36, with r = p^{−1/(p−1)}. Note that (36.6) corresponds to (41.14) here. As before, the constant terms in the definitions of exp(x) and exp(y) cancel out in the left side of (41.15), and they were not included in Section 36.

The exponential function maps B(0, p^{−1/(p−1)}) into B(e, p^{−1/(p−1)}), because of (41.7). We actually have that

exp(B(0, p^{−1/(p−1)})) = B(e, p^{−1/(p−1)}), (41.16)

as in (37.19), with r = p^{−1/(p−1)}. This takes the constant term into account for the exponential function, as before, while the constant term was taken to be 0 in Section 37. The condition (37.13) corresponds to (41.14) here, as in the preceding paragraph.

References

[1] J. Cassels, Local Fields, Cambridge University Press, 1986.

[2] R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lecture Notes in Mathematics 242, Springer-Verlag, 1971.

[3] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bulletin of the American Mathematical Society 83 (1977), 569– 645.

[4] A. Escassut, Ultrametric Banach Algebras, World Scientific, 2003.

[5] F. Gouvêa, p-Adic Numbers: An Introduction, Springer-Verlag, 1997.

[6] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer-Verlag, 2001.

[7] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, 1969.

[8] W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.

[9] W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, 1987.

[10] W. Rudin, Functional Analysis, 2nd edition, McGraw-Hill, 1991.

[11] L. Steen and J. Seebach, Jr., Counterexamples in Topology, Dover, 1995.

[12] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

Index

absolute convergence, 17 absolute value functions, 5 adjoint mappings, 32 algebras, 22 archimedean absolute value functions,

8

Banach algebras, 22 Banach spaces, 19, 20 BL(V), 23 BL(V, W), 19 bounded functions, 13, 55 bounded linear functionals, 27 bounded linear mappings, 18 bounded sets, 55

C, 5 c(X, V), 12 C(X, Y), 54 c_{0}(X, V), 13 c_{00}(X, V), 12 C_{b}(X, Y), 55 Cauchy products, 23 Cauchy–Schwarz inequality, 31 closed balls, 4 complex-linear mappings, 30 conjugate-linear mappings, 30 convergence of infinite series, 16

discrete absolute value functions, 8 discrete metric, 3 dual norms, 28 dual spaces, 28

equivalent absolute value functions, 6 exponential function, 67

Fatou’s lemma, 15 formal polynomials, 58 formal power series, 59

Hilbert spaces, 31

identity matrix, 34 inner products, 30 involutions, 36, 37 isometric linear mappings, 33

ℓ^{1}(X, V), 15 ℓ^{∞}(X, V), 13 linear functionals, 27 Lip(X, Y), 54 Lip_{b}(X, A), 57 Lipschitz mappings, 52, 54

M_{n}(R), 34

matrix multiplication, 34, 38, 44, 46 metrics, 3 multiplicative seminorms, 22

N′, 28

non-archimedean absolute value func-

tions, 8 norms, 9

open balls, 4 operator norm, 19 orthogonal linear mappings, 33 Ostrowski’s theorems, 6, 9

p-adic absolute value function, 6

p-adic numbers, 8

Q, 6 Q_{p}, 8

R, 5 R_{+}, 8 real-linear mappings, 30

semi-ultranorms, 9

seminorms, 9

submultiplicative seminorms, 22 summable functions, 14, 15 supports of functions, 12, 38, 40 supremum metric, 55

transpose matrix, 36 trivial absolute value function, 6 trivial ultranorm, 10

UC(X, Y), 54 UC_{b}(X, Y), 55 ultrametric absolute value functions, 6 ultrametrics, 3 ultranorms, 10 unitary mappings, 33

V′, 28 vanishing at infinity, 13

Z_{+}, 8