Here is a beautiful little theorem. The unit ball in an infinite dimensional Hilbert space is not compact. The proof is quite simple. So the unit ball . Recall that since this is a Hilbert space, we have an inner product defining this norm .
Since our space is infinite dimensional, we can choose to be linearly independent inductively. Basic application of Gram-Schmidt gets us this set to be orthonormal, in particular, each one has norm 1 and so is in the unit ball.
Now look at the distance between any two
Thus all are apart and hence there is no subsequence that converges. Since sequential compact and compact are the same here, we are done. (Note the inner product is not symmetric, but I knew the middle terms would be zero, so I went ahead and abused that instead of writing two zeros).
I know the result to be true in a Banach space as well, but I don’t see a quick fix without the inner product…
A lot more of these things will start popping up with my analysis prelim two weeks from today.