6.6 TheoremLet X and Y be normed spaces. For F E BL(X,Y), define||F|| = sup{||F(x)|| : x € X, ||1|| ≤ 1}.Then || || is a norm on BL(X,Y), called the operator norm. Forall z E X, we have||F(x)|| ≤ ||F|| ||1||.In fact,|||F|| = inf{a ≥ 0 : ||F(1)|| ≤ a||1|| for all x € X}.RequustexplainI how theyare sameAlso, if X {0}, then||F|| = sup{|| F(x)|| : x € X, ||x|| = 1} = sup{||F(x)|| : x € X, ||z|| < 1}.

There are definitions of norms on bounded linear operators, they all are equivalent.

Answered: Let X and Y be normed spaces. For F €…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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