Let X and Y be normed spaces and F:X→Y(b)be linear. Prove that F is continuous if andonly if every cauchy sequence { x„ } in X,the sequence { F (x„) } is cauchy in Y. Showthat this is not true for non-linearcontinuous map.

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Answered: O Let X and Y be normed spaces and F…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 27E: 27. Let , where and are nonempty. Prove that has the property that for every subset of if and...

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O Let X and Y be normed spaces and F :X→Y be linear. Prove that F is continuous if and only if every cauchy sequence{ x„ } in X, the sequence { F (x„) } is cauchy in Y. Show that this is not true for non-linear