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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 78E

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Question

For X and Y normed linear spaces, let {Tn} 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.
(a) Let ɛ = 1 in the definition of convergence of {Tn} to T in L(X,Y).
||Tn|| < M, Vn E N, where M = sup{||T1||, ||T2||,
||TN-1||, 1+ ||T||}, for some N E N.
....,
(b) for all n E N, ||Tn(un) – T(u)||y < ||Tn|| ||Uun – u||x + ||Tn – T|| - ||u||x.
Use (a) and (b) to show that Tn(Um)
→ T(u) in Y.

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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 {n} be a sequence in X such that un → u in X.
Let ε = 1 in the definition of convergence of {T} to T in L(X, Y). Show that for all n ≤ N,
||Tn(Un) – T(u)||y ≤ ||Tn|| · ||Un − U||x + ||Tn − T|| · ||u||x.

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