For X and Y normed linear spaces, let {Tn} be a sequence in L(X, Y) such that Tỵ →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 {T} to T in L(X,Y). T≤M, \n € N, where M = sup{||T₁||, ||T₂||, ||TN-1||, 1 + ||T||}, for some N & N. (b) for all n € N, ||Tn(Un) – T(u)||x ≤ ||Tn|| · ||un – u||x + ||Tn − T|| · ||u||x. Use (a) and (b) to show that Tn (un) → T(u) in Y.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
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.
(a) Let ε = 1 in the definition of convergence of {T} to T in L(X, Y).
||T|| ≤M, \n N, where M = sup{||T₁||, ||T₂||, ..
||TN-1||, 1 + ||T||}, for some NE N.
(b) for all n € N, ||Tn(Un) — T(u)||y ≤ ||Tn|| · ||Un − u||x + ||Tn − T|| · ||u|| x.
Use (a) and (b) to show that Tn(un) → T(u) in Y.
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. (a) Let ε = 1 in the definition of convergence of {T} to T in L(X, Y). ||T|| ≤M, \n N, where M = sup{||T₁||, ||T₂||, .. ||TN-1||, 1 + ||T||}, for some NE N. (b) for all n € N, ||Tn(Un) — T(u)||y ≤ ||Tn|| · ||Un − u||x + ||Tn − T|| · ||u|| x. Use (a) and (b) to show that Tn(un) → T(u) in Y.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,