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.
Best Answer
One of the simplest consequences is that continuous function doesn't attain their minimum on the ball, fact that is always true in finite dimensional spaces for Weierstrass theorem. Consider for example the integral functional $\int_0^1|\cdot|:(C([0,1]),\Vert\cdot\Vert_{\infty})\to\mathbb{R}$, whose infimum over the ball is zero but zero is never attained on the ball.
Another interesting thing is that is easy to find an infinity of elements that lie on the ball which are $\epsilon$-separated. For an example consider the characteristics functions $\{f_n=\chi_{[n,n+1]}(x)\}_{n\in\mathbb{N}}\subset L^\infty(\mathbb{R})$. One can easily check that they belong to the unit ball of $L^\infty$ and that are 2-separated, i.e. $\Vert f_n-f_m\Vert_\infty=2$ for each $n\ne m$.