TITLE: Examples of compact sets that are infinite dimensional and not bounded
QUESTION [25 upvotes]: In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, then why do I sometimes see phrases like "for any compact subspaces of some vector space"?
Also, do compact sets in infinite dimensional Banach spaces have to be bounded?
Is there a general result that tells us what compact subsets of infinite dimensional vector spaces look like?
Thanks so much!

REPLY [2 votes]: To answer the first question, Does a compact subset have to be finite dimensional?:
Tychonoff's Theorem states that an arbitrary cartesian product of compact spaces is compact.
So for instance in infinite-dimensional separable Hilbert space, the cartesian product of the closed intervals [-1/n, 1/n], n = 1, 2, 3, ... is a compact subset.