FA23

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Let $X\,$ and $Y\,$ be Banach spaces and $T_n\isin B(X,Y)\,$ . Show that the following are equivalent:

(a) $\left\{||T_n||\right\}_{n=1}^\infty\,$ is bounded;

(b) $\left\{||T_nx||\right\}_{n=1}^\infty\,$ is bounded for each $x\isin X\,$ ;

(c) $\left\{g\left(T_nx\right)||\right\}_{n=1}^\infty\,$ is bounded for all $x\isin X\,$ and all $g \isin Y'\,$ .

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