Importance of compactness of unit ball in normed vector spaces

Question

We know that for a normed vector space $(X, ||\cdot ||)$: X is finite dimensional if and only if the unit ball $K = \left\{x \in X: ||x|| \leq 1 \right\}$ is compact. How is this used in practice? Do we actually use this to prove that a space is finite dimensional? Other than that, are there examples of other theorems where this is used in the proofs?

Answer 1

I don't know of any result of the form

Theorem: Let $(V,\|\cdot\|)$ be [$\ldots$]. Then $V$ is finite / infinite dimensional.

Proof: let us show that the unit ball is compact / non-compact. $[\ldots]$

At least, I don't know of any such results where no simpler proof exists. However, the equivalence

$(V,\|\cdot\|)$ is finite dimensional $\iff$ the unit ball is compact.

is of crucial theoretical importance. Note that the fact that the unit ball is compact is equivalent to the fact that closed bounded domains are compact, or to the fact that bounded sequences admit limit points. Thus, the above result tells you that one cannot expect to use compactness arguments in infinite dimensional normed vector spaces to show existence of solutions to some minimising problems or some PDEs. Below is a hand-wavy illustration.

Assume $(V,\|\cdot\|)$ is finite dimensional, and let $F \colon V \to [0,+\infty)$ be some continuous functional. Assume that $F(u) \to \infty$ whenever $\|u\| \to \infty$. Then there exists $u_0 \in V$ such that $F(u_0) = \min_{u\in V} F(v)$. Indeed, one easily builds a sequence $u_k$ such that $F(u_k) \leqslant \inf F + \frac{1}{k}$, then shows that $\{u_k\}$ must be bounded, and exhibits a converging subsequence thanks to a compactness argument. The limit $u_0$ then solves the problem.

Now, assume that $(V,\|\cdot\|)$ is not finite dimensional. Then one can still create a "minimizing" subsequence $\{u_k\}$ such that $F(u_k) \to \inf F$. However, absolutely nothing can ensure the existence of a converging subsequence, and hence, you cannot expect to exhibit a solution to the problem this way.

This example is not artificial: some PDEs are naturally the Euler-Lagrange equations of some suitable functionals as above, and solving those said PDEs boils down to finding minimizers of the functionals. Since function spaces, such as $\mathcal{C}^{\infty}(X)$, $\mathcal{C}^{k,\alpha}(X)$, $\Bbb C[X]$, or $H^p(X)$, are infinite dimensional, you can't expect to use a normed topology to solve many problems in functional analysis, at least not in this way. One has to build appropriate tools to make the above heuristic work. Some of these tools are completeness, compact embeddings of Sobolev spaces, etc. One can also abandon the normed topology and look for a "better" topology that reflects the properties of the problem, such as weak topology, weak-* topology, and so on.