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It is well know that every bounded set in a banach space is relatively weakly compact if and only if the banach space space is reflexive.

Do they mean its norm closure or weak closure is compact in the weak topology ?

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    $\begingroup$ Well, ist the unit sphere (not ball) in an infinite dimensional Hilbert space weakly compact or not? Is the unit sphere norm closed? What does this tell you? $\endgroup$ – PhoemueX Jul 12 at 16:33
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    $\begingroup$ Relatively weakly compact means relatively compact in the weak topology: the weak closure is weakly compact. $\endgroup$ – David C. Ullrich Jul 12 at 17:25

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