# Reference request: Arzela-Ascoli theorem for smooth Hölder norms

Could anyone suggest a textbook account of the Arzela-Ascoli theorem for $C^{k,\alpha}$ norms?

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Not sure I understand. Could you provide the precise statement that you want a reference for? –  Deane Yang Jan 12 '11 at 23:26
There are various versions of Arzela-Ascoli, but typically one starts with a sequence of functions that has a uniform bound on $C^{k,\alpha}$ norm, an one want to extract a subsequence converging to in $C^{m,\beta}$ norm where $m+\beta<k+\alpha$, and one also gets information about regularity of the limit. There are also some subtleties when $\alpha = 0$. I found that I do not have a sufficiently firm grip on these matters. –  Igor Belegradek Jan 12 '11 at 23:49
Maybe a book that explains Schauder estimates for elliptic PDE's? Gilbarg and Trudinger? Or the book by L. C. Evans? –  Deane Yang Jan 13 '11 at 1:53
Deane, I don't have the books handy, and won't get to them before Friday (due to snowstorm) but what I see in google.books while searching them isn't promising. Anyway, I agree that these should be the first books to look at, thanks. –  Igor Belegradek Jan 13 '11 at 2:10
The standard version of Arzela-Ascoli theorem says an equicontinuous family of functions is compact in the $C^0$ topology. I believe that it is straightforward to show that a uniform $C^\alpha$ bound implies equicontinuity. I also believe it is reasonably straightforward extend this argument to the general case. But I haven't worked out the details. –  Deane Yang Jan 13 '11 at 4:25