# 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

The unit ball in a Hilbert space is compact if and only if the Hilbert space is finite-dimensional.

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Didn't the OP mean weak compactness? I admit to being a bit confused by the phrasing of the question. – Harald Hanche-Olsen Oct 20 '12 at 22:06
The OP has already mentioned that the (closed) unit ball is always weakly compact. – Chris Eagle Oct 20 '12 at 22:06
Yeah, but I thought perhaps he meant, can you get that result without using the full power of Banach–Alaoglu? – Harald Hanche-Olsen Oct 20 '12 at 22:07
I think the word "condition" makes it fairly clear that he's not talking about weak compactness. – anonymous Oct 20 '12 at 22:25