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?
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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 $\lim_{N\to +\infty}\sup_{x\in S}\sum_{k=N}^{+\infty}|x_k|^2=0$.
Davide Giraudo
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2Of note: A generalization of this holds for $\ell_p$-spaces, $1\le p<\infty$. – David Mitra May 02 '13 at 20:23
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Thank you very much. I am really surprised that this definition/characterization is nowhere to be found (I searched wikipedia and dozens of scripts) – CBenni May 02 '13 at 20:26
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3@CBenni see here the prooof of this characterization – Norbert May 02 '13 at 20:51