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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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up vote 10 down vote accepted

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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@Jesse: I'm usually not that terse in answering, but I honestly don't think it makes much sense for me to repeat the results here because these notes are very well written. Note that there are sections with historical comments in the linked file where you can find a lot more information and references. In particular, there is also a result due to Phillips characterizing compactness for $L^{p}(X)$ for an arbitrary measure space, but it's much less useful than Kolmogorov-Riesz (and as I recall his paper is rather hard to read). Please ask if you have further questions. –  t.b. Apr 26 '11 at 9:50
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+1 for linking to the article by Hanche-Olsen and Holden –  Willie Wong Apr 26 '11 at 15:10
    
@Willie: Thanks, I find this article really neat. It is an art to write in such a way that one has the impression that the authors only point out the obvious... –  t.b. Apr 26 '11 at 15:36
    
For future reference: en.wikipedia.org/wiki/Fr%C3%A9chet-Kolmogorov_theorem –  Jesse Madnick Feb 6 '13 at 11:34
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