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.