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Math Help - Convergence of Sequences of Operator and Functionals

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    Convergence of Sequences of Operator and Functionals

    Let {T}_{n}\to T, where {T}_{n}\in B(X,Y).. show that for every \epsilon > 0 and every closed ball  K\subset X there is an N such that [tex] \left\|T_nx-Tx\right\| < \epsilon for all n > N and  x\in K
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by geezeela View Post
    Let {T}_{n}\to T, where {T}_{n}\in B(X,Y).. show that for every \epsilon > 0 and every closed ball  K\subset X there is an N such that [tex] \left\|T_nx-Tx\right\| < \epsilon for all n > N and  x\in K
    I just did this somewhere. Where's the effort man? How about using the fact that \|T_n(x)-T(x)\|\leqslant \|T_n-T\|_\text{op}\|x\|.
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    Quote Originally Posted by Drexel28 View Post
    I just did this somewhere. Where's the effort man? How about using the fact that \|T_n(x)-T(x)\|\leqslant \|T_n-T\|_\text{op}\|x\|.
    what does the {op} stands for?..
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    Quote Originally Posted by Drexel28 View Post
    I just did this somewhere. Where's the effort man? How about using the fact that \|T_n(x)-T(x)\|\leqslant \|T_n-T\|_\text{op}\|x\|.
    What does that op stands for??
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by geezeela View Post
    What does that op stands for??
    Operator norm? Isn't that the norm you're putting on \mathcal{B}(X,Y).
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