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…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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.