# 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