By Erwin Kreyszig
"Provides avenues for utilising practical research to the sensible research of average sciences in addition to arithmetic. comprises labored difficulties on Hilbert area conception and on Banach areas and emphasizes ideas, rules, tools and significant purposes of practical analysis."
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Additional resources for Introductory Functional Analysis with Applications
Hence x EM. This proves that M is closed because x EM was arbitrary. Conversely, let M be closed and (xn) Cauchy in M. 4-6(a), and x EM since M = M by assumption. Hence the arbitrary Cauchy sequence (xn) converges in M, which proves completeness of M. • This theorem is very useful, and we shall need it quite often. 5-3 in the next section includes the first application, which is typical. The last of our present three theorems shows the importance of convergence of sequences in connection with the continuity of a mapping.
4-8 Theorem (Continuous mapping). 4 31 Convergence, Cauchy Sequence, Completeness Xo E X if and only if Xn ------i> implies Xo Proof Assume T to be continuous at Xo; cf. Def. 3-3. Then for a given E > 0 there is a l) > 0 such that d(x, xo) < Let Xn ------i> implies l) d(Tx, Txo) < E. Xo. Then there is an N such that for all n > N we have Hence for all n > N, By definition this means that TX n ------i> Txo. Conversely, we assume that implies and prove that then T is continuous at Xo. Suppose this is false.
And if a sequence (x,,) satisfies the condition of the Cauchy criterion, we may call it a Cauchy sequence. Then the Cauchy criterion simply says that a sequence of real or complex numbers converges on R or in C if and only if it is a Cauchy sequence. This refers to the situation in R or C. Unfortunately, in more general spaces the situation may be more complicated, and there may be Cauchy sequences which do not converge. Such a space is then lacking a property which is so important that it deserves a name, namely, completeness.