Squaring, we have for m, r> Nand j = 1,· .. , n and This shows that for each fixed j, (1 ~j~ n), the sequence (~?l, ~~2l, ... ) is a Cauchy sequence of real numbers. It converges by Theorem 1.4-4, say, ~~m) ~ ~i as m ~ 00. Using these n limits, we define x = (~l> ... , ~n). Clearly, x ERn. From (1), with r ~ 00,

Show that all complex m x n matrices A = (ajk) with fixed m and n constitute an mn-dimensional vector space Z. Show that all norms on Z are equivalent. What would be the analogues of II· III> I . 112 and I . 1100 in Prob. 8, Sec. 2.2, for the present space Z? Metric Space 2 Further Examples of Metric Spaces 9 Open Set, Closed Set, Neighborhood 17 Convergence, Cauchy Sequence, Completeness 25 Examples. Completeness Proofs 32 Completion of Metric Spaces 41 Theorem (Uniform convergence). Convergence Xm ~ x in the space C[a, b] is uniform convergence, that is, (Xm) converges uniformly on [a, b] to x. Hence the metric on C[a, b] describes uniform convergence on [a, b] and, for this reason, is sometimes called the uniform metric. To gain a good understanding of completeness and related concepts, let us finally look at some This space is not complete. For instance, if [a, b] = [0, 1], the sequence in 1.5-9 is also Cauchy in the present space X; this is almost obvious from Fig. 10, Sec. 1.5, and results formally by integration because for n > m we obtain

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cannot be obtained from a norm. This may immediately be seen from the following lemma which states two basic properties of a metric d obtained from a norm. The first property, as expressed by (9a), is called the translation invariance of d. 2.2-9 Lemma (Translation invariance). . on a normed space X satisfies Topics In Complex Function Theory. Volume III ―Abelian Functions & Modular Functions of Several Variables J. J. Stoker

Problems 1. Show that C c [00 is a vector subspace of [00 (cf. 1.5-3) and so is space of all sequences of scalars converging to zero. 2. Show that Co in Prob. 1 is a closed subspace of [00, so that by 1.5-2 and 1.4-7. The book is suitable for a one-semester course meeting five hours per week or for a two-semester course meeting three hours per week. The book can also be utilized for shorter courses. In fact, chapters can be omitted without destroying the continuity or making the rest of the book a torso where L I~il = 1, so that not all ~i can be zero. Since {XI, ..• ,xn } is a linearly independent set, we thus have Y-F O. On the other hand, Yn,m ~ Yimplies IIYn,ml1 ~ IIYII, by the continuity of the norm. Since IIYml1 ~ 0 by assumption and (Yn,m) is a subsequence of (Ym), we must have IIYn,ml1 ~ O. Hence IIYII = 0, so that Y= 0 by (N2) in Sec. 2.2. This contradicts Y-F 0, and the lemma is proved. • / The reader will notice that in these cases (Examples 1.5-1 to 1.5-5) we get help from the completeness of the real line or the complex plane (Theorem 1.4-4). This is typical. Examples 1.5-1 Completeness of R n and C n • space C n are complete. (Cf. 1.1-5.) We shall now introduce two more concepts, which are related. Let M be a subset of a metric space X. Then a point Xo of X (which mayor may not be a point of M) is called an accumulation point of M (or limit point of M) if every neighborhood of Xo contains at least one point Y E M distinct from Xo. The set consisting of the points of M and the accumulation points of M is called the closure of M and is denoted byFor every x E M there is a sequence (x..) in M such that x; cf. 1.4-6(a). Since M is compact, x E M Hence M is closed The concept of convergence of a series can be used to define a "basis" as follows. If a normed space X contains a sequence (en) with the property that for every X E X there is a unique sequence of scalars (un) such that (as For example, IP in 2.2-3 has a Schauder basis, namely (en), where en = (8,.j), that is, en is the sequence whose nth term is 1 and all other