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Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(\Omega)$. See the wikipedia page:

http://en.wikipedia.org/wiki/H%C3%B6lder_condition

More precisely, I want to know the exact reference of the theory related to the following statement: [Claim] Given $\{f_n\}$ is a sequence of functions with $\|f_n\|_\beta <1$ for all $n$. Then, there exists a subsequence $\{n_k\}$ and $f\in C^\alpha$ such that, $\lim_{k\to \infty} \|f_{n_k} - f\|_\alpha = 0$. In the above, $\|\cdot\|_\alpha$ is Hölder-$\alpha$ norm.

However, I could not find a precise reference from some books on functional analysis.

1) Can anybody indicate a precise reference for this theorem?

2) If possible, I would like to know a reference on the similar result on parabolic Hölder space.

Thanks.