I am writing a paper on the compactness of closed balls in Banach spaces, with particular attention paid to the following theorem
Let $V$ be a Banach space over $\mathbb R$ or $\mathbb C$. The closed unit ball in $V$ is compact if and only if $V$ is finite-dimensional.
I am looking for some consequences or applications of this theorem (probably mostly related to the part which asserts that: if $V$ is infinite-dimensional, then the closed unit ball is not compact). I do prove the immediate corollary of this theorem, which is basically replacing "the closed unit ball" with "the closed ball of radius $r>0$ around $x_0\in V$" in the statement of the theorem. I have also been looking at the notion of weak convergence, and how this can allow for compactness (in the weak sense) in infinite-dimensional spaces. Other than those two, I am looking for some other applications of this theorem. In particular, are there any specific interesting examples one can look at that follow from this theorem?
Any feedback is appreciated.