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.
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.
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e4c2197-3e48-421d-8cec-615cc195654a%2F28955a7f-ad19-4ef7-95ac-a623f05eed27%2Fpsz0v5f_processed.png&w=3840&q=75)
Transcribed Image Text: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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