Let X be a separable normed vector space and f : X → R be uniformly continuous and bounded. Show: For any sequence (xn) in X, there exists a subsequence (xn;) and a continuous function f : X → R such that f (an, + x) → f(x), j → 0, x €X, and sup f(xn, + x) – f(x)| →0, j → , rɛD for any compact subset of D of X.

Let X be a separable normed vector space and f : X → R be uniformly continuous and bounded. Show: For any sequence (xn) in X, there exists a subsequence (xn;) and a continuous function f : X → R such that f (an, + x) → f(x), j → 0, x €X, and sup f(xn, + x) – f(x)| →0, j → , rɛD for any compact subset of D of X.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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