Is the fact that a unit ball in an infinite-dimensional (Banach) space is a noncompact topological space...?
If it is, how would one go about proving it...?
I found a proof in Kreyszig.
Every Banach space is a metric space, and in a compact subset of a metric space, every sequence has a convergent subsequence.
Kreyszig assumes that the closed unit ball of an infinite-dimensional Banach space is compact. He then uses Riesz's Lemma (which isn't the Riesz Representation Theorem) to construct a sequence that doesn't have convergent subsequence.
Conclusion: the closed unit ball of a Banach space is compact if and only if the Banach space is finite-dimensional.