Let X be a Banach space and T = L(X, X) have ||T|| < 1. Define T° to be the identity map (that is,Tº(x) = = x, for all x € X). Let r = ||T||. Show that ||T|| ≤ r", for all n € N.

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Answered: Let X be a Banach space and T = L(X, X)…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.5: Permutations And Inverses

Problem 10E: 10. Let and be mappings from to. Prove that if is invertible, then is onto and is...

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Let X be a Banach space and T = L(X, X) have ||T|| < 1. Define Tº to be the identity map (that is, Tº(x) = x, for all x € X). Let r = ||T||. Show that ||T"|| ≤ r", for all n € N.