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.

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