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 unu in X. Let = 1 in the definition of convergence of {T} to T in L(X, Y). Show that ||T|| ≤M, VN, where M = sup{||T₁||, ||T2||, ..., ||TN-1||, 1 + ||T||}, for some NE N.
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 unu in X. Let = 1 in the definition of convergence of {T} to T in L(X, Y). Show that ||T|| ≤M, VN, where M = sup{||T₁||, ||T2||, ..., ||TN-1||, 1 + ||T||}, for some NE N.
Chapter6: Exponential And Logarithmic Functions
Section6.2: Graphs Of Exponential Functions
Problem 52SE: Prove the conjecture made in the previous exercise.
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Transcribed Image Text:For X and Y normed linear spaces, let {T} be a sequence in L(X, Y) such that T →T in L(X, Y)
and let {n} be a sequence in X such that un → u in X.
Vn EN.
Let = 1 in the definition of convergence of {T} to T in L(X, Y). Show that ||Tn|| ≤ M,
where M = sup{||T₁||, ||T₂||, ..., ||TN-1||, 1 + ||T||}, for some N € N.
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