(b) Prove that the unit ball of a normed linearspace is compact if and only if the normedlinear space is finite dimensional. Use thisto show that the identity map oninfinite dimensional normed space is notcompact.

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Answered: (b) Prove that the unit ball of a…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.2: Norms And Distance Functions

Problem 33EQ

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(b) Prove that the unit ball of a normed linear space is compact if and only if the normed linear space is finite dimensional. Use this to show that the identity map on an infinite dimensional normed space is not compact.