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In $\mathbb{R}^n$ we know (Heine-Borel Theorem) that a set is compact if and only if it is closed and bounded.

In $C(X)$ for a compact metric space $X$, we know (corollary of Ascoli-Arzela Theorem) that a set is compact if and only if it is closed, bounded, and equicontinuous.

I am looking for as many examples as I can of other spaces where the extra condition for compactness is known.

Also, I am looking for as many examples as I can of (important) spaces where the extra condition is not currently known.

I am planning on doing some research (under a professor) and I thought this topic was particularly interesting, so I would very much appreciate some examples to start off with, just so I can get a feel for the problem.

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For Banach spaces, there is a big table in Dunford & Schwartz characterizing compactness of subsets in many different spaces.

Zev is right that "complete + totally bounded" is the general formulation. For subset in Banach space, complete holds if and only if closed. So the question is to characterize total boundedness.

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  • $\begingroup$ Thanks, GEdgar! What is the book by Dunford & Schwarz? $\endgroup$
    – Tim
    Nov 25, 2011 at 22:28
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    $\begingroup$ Tim: Just Google "Dunford and Schwarz" (or, better, "Dunford and Schwartz") and you'll figure this out yourself. $\endgroup$
    – KCd
    Nov 25, 2011 at 22:34
  • $\begingroup$ @KCD: Thanks! Found it. $\endgroup$
    – Tim
    Nov 25, 2011 at 22:39

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