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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 e = 1 in the definition of convergence of {Tn} to T in L(X,Y). ||Tn|| < M, Vn e N, where M = sup{||T||, ||T2||, . . , [[TN-1||, 1 + ||T||}, for some N e N. (b) for all n E N, ||Tn(Un) – T(u)||y < ||Tn|| |n – u||x + |Tn – T|| · ||u||x . Use (a) and (b) to show that Tn(Un) → T(u) in Y.

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 e = 1 in the definition of convergence of {Tn} to T in L(X,Y). ||Tn|| < M, Vn e N, where M = sup{||T||, ||T2||, . . , [[TN-1||, 1 + ||T||}, for some N e N. (b) for all n E N, ||Tn(Un) – T(u)||y < ||Tn|| |n – u||x + |Tn – T|| · ||u||x . Use (a) and (b) to show that Tn(Un) → T(u) in Y.

Advanced Engineering Mathematics
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 {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.
Transcribed Image Text: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.
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 {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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