# Is C^{k+1}(X) compactly contained in C^{k}(X) for a closed manifold X?

Hi all,

I apologize if this question is too low level for mathoverflow. I'm happy to move it to math.stackexchange if so.

Let $X$ be a closed manifold, let $k$ be a nonnegative integer and let $C^k(X)$ denote the space of $k$-times continuously differentiable functions equipped with the $C^k$ norm.

Is $C^{k+1}(X)$ compactly contained in $C^k(X)$? Does this follow from Arzela-Ascoli?

Thanks.

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Yes. Please move it to MSE. –  timur Sep 1 '12 at 1:23
Thanks! Will do. Do you have a reference for this? –  trex Sep 1 '12 at 2:26

## 1 Answer

The answer depends on what you mean by a closed manifold and a compact operator. If the manifold is compact without boundary then the spaces involved are Banach spaces and the answer is yes and this is indeed proved using the Arzela-Ascoli theorem, after a standard localisation argument.

If the manifold is without boundary but not compact (think real line), then the spaces are Frechet spaces and the answer depends on what you mean by a compact operator on a locally convex space. If you define these as taking bounded sets into relatively compact ones, then the answer is yes, but if you mean those which take a neighbourhood of zero to a relativeley compact set (the usual definition), then the answer is no.

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I don't think the name "closed manifold" is ambiguous at all, in all the paper and books I read it always meant "compact without boundary". Wikipedia also agrees with me: en.wikipedia.org/wiki/Closed_manifold –  YangMills Sep 1 '12 at 18:29
I certainly meant compact without boundary. –  trex Sep 2 '12 at 1:37