Banach-Alaoglu theorem

Question

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! :)

Answer 1

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