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Some of my favorite theorems in analysis are those which classify the (pre-)compact subsets of a particular space. For example:

Can we give a similar description of the compact subsets of $L^p$?


Notes: I realize that I'm being a little vague in two senses.

First, $L^p$ of what? Frankly, I don't know. I would be very interested in seeing a theorem about $L^p[0,1]$, or more generally about $L^p(X)$ where $X$ is locally compact Hausdorff. I just want to know what's out there.

Second, what kind of "classification" am I looking for? Well, hopefully one that is similar in spirit to the above three examples, and in some sense specific to $L^p$. For instance, saying that a set is compact if and only if it is complete and totally bounded does not really count (since that is true in any metric space).

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    Relevant: http://en.wikipedia.org/wiki/Fr%C3%A9chet-Kolmogorov_theorem2011-11-26

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Look at the nice notes by Hanche-Olsen and Holden on the Kolmogorov-Riesz compactness theorem characterizing norm compactness in $L^p(\mathbb{R}^n)$ for $1 \leq p \lt \infty$ in terms of uniform $p$-integrability and $p$-tightness, see Theorem 5 on page 3 for the precise statement of the result. This is very much in the spirit of the theorems you mention. Don't miss the historical notes in section 4, where you can find a host of references to further related results and applications. These notes and the references therein should answer your question entirely.

I'd also like to point you to the quite closely related Dunford-Pettis theorem on weak compactness in $L^1$.

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    For future reference: http://en.wikipedia.org/wiki/Fr%C3%A9chet-Kolmogorov_theorem2013-02-06