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Could I have a hint please on how to prove the following proposition:

Let $X$ and $Y$ be two normed space and $T$ be a linear operator from $X$ into $Y$. The operator $T$ is continuous if the set $T(K)$ is a (weakly) compact subset of $Y$ whenever $K$ is a (weakly) compact subset of $X$.

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What have you tried so far? –  Rasmus Jul 29 '12 at 13:31
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What is a glance proof? Please tell us a little bit about what you know about (weakly) compact sets and what efforts you made in order to establish these statements. –  t.b. Jul 30 '12 at 10:00
    
I suspect "glance proof" is intended to mean "please give me a hint". –  Matt N. Jul 30 '12 at 10:09
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