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Math Help - bounded lineal operators

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    mms
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    bounded lineal operators

    Let X,Y be two banach spaces and T_n :X \to Y be linear operators that converge punctually to T_{} :X \to Y Show that
    T_n  \to T uniformly in each compact subset of X
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    Quote Originally Posted by mms View Post
    Let X,Y be two banach spaces and T_n :X \to Y be linear operators that converge punctually to T_{} :X \to Y Show that
    T_n  \to T uniformly in each compact subset of X
    First, replacing T_n by T_n-T, we may assume that T=0, so that T_nx\to0 for all x in X.

    Next, it follows from the uniform boundedness principle that \{T_n\} is bounded, say \|T_n\|\leqslant M for all n.

    Now let \varepsilon>0. For each x in X, there exists N_x such that \|T_nx\|<\varepsilon/2 whenever n\geqslant N_x. Let U_x denote the open set \{y\in X:\|y-x\|<\frac\varepsilon{2M}\}. Then \|T_ny\|\leqslant \|T_nx\| + \|T_n(y-x)\|<\varepsilon for all y in U_x, whenever n\geqslant N_x. Such sets U_x form an open cover of a given compact subset of X, and by looking at a finite subcover you should be able to see that the pointwise convergence T_nx\to0 is uniform on that set.
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