For X and Y normed linear spaces, let {T} be a sequence in L(X, Y) such that Tn →T in L(X, Y)and let {un} be a sequence in X such that un → u in X.Prove that for all n ≤ N, ||Tn(un) − T(u)||y ≤ ||Tn|| · ||un − u||x + ||Tn — T||· ||u||x.

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Answered: For X and Y normed linear spaces, let…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 74E

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For X and Y normed linear spaces, let {T} be a sequence in L(X, Y) such that Tn →T in L(X, Y) and let {un} be a sequence in X such that un → u in X. Prove that for all n ≤ N, ||Tn(un) − T(u)||y ≤ ||Tn|| · ||un − u||x + ||Tn — T||· ||u||x.