# Banach-Alaoglu theorem

If we have a Hilbert space $H$, (so it is reflexive) then by Banach-Alaoglu's theorem, the closed unit ball $B\subset H$ is weakly-compact. My question is,

Is there any corollary or similar theorem or conditions that gives compactness? I mean, some ingredient or condition that shows that the unit ball of a Hilbert space is compact?

Thank you very much for your help! :)

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I think you should be able to prove weak compactness by expressing everything in terms of an orthonormal basis. But you can't easily get away from using the Tychonov theorem in some fashion. –  Harald Hanche-Olsen Oct 20 '12 at 22:06