@MattN Probably the easiest way of saving your example in the real case would be to take the balls of radius $\sqrt{2}$ around $\pm e_n$ where $e_n = (0,\ldots,0,1,0,\ldots)$.
@MattN yes, and open sets $U$ in $B$ are of the form $V \cap B$, so if you insist on an open cover in the surrounding space, just replace the relatively open sets $U$ by sets $V$ such that $U = B \cap V$.
@MattN seen the comments to my answer? Here's a good exercise: Prove that the unit ball in an infinite dimensional normed space is never compact using Riesz's lemma.
