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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.



Is there a reference for compact imbedding theory of Hölder space?
For non-integer values of $\alpha$, the space $C^\alpha$ has a nice characterisation in terms of wavelet coefficients, see "[Wavelets and Operators](http://books.google.hu/books?id=y5L5HVlh3ngC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false)" by Yves Meyer. With that characterisation (essentially a weighted $\ell^\infty$ bound on the wavelet coefficients), the compactness statement boils down to the (trivial) statement that the set of sequences bounded by some fixed sequence $\{a\_n\}$ converging to $0$ is compact in $\ell^\infty$. For parabolic Hölder spaces (or any non-Euclidean scaling for that matter), Meyer's characterisation and therefore the compactness of the embedding still works, provided that one considers a suitably scaled tensor product wavelet basis.


