When is a subset of $\ell^2$ compact?

Question

I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà–Ascoli theorem that provides a useful tool, but I dont know how you can transfer it to $\ell^2$? Are there any similar theorems?

Answer 1

There is the following characterization:

$S\subset \ell^2$ is compact if and only the following conditions are satisfies:

• $S$ is closed;
• $S$ is bounded, i.e. $\sup_{x\in S}\lVert x\rVert_{\ell^2}<\infty$;
• we have $\lim_{N\to +\infty}\sup_{x\in S}\sum_{k=N}^{+\infty}|x_k|^2=0$.