Could anyone suggest a textbook account of the Arzela-Ascoli theorem for $C^{k,\alpha}$ norms?
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asked Jan 12, 2011 at 21:45
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$\begingroup$ Not sure I understand. Could you provide the precise statement that you want a reference for? $\endgroup$– Deane YangCommented Jan 12, 2011 at 23:26
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$\begingroup$ 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. $\endgroup$– Igor BelegradekCommented Jan 12, 2011 at 23:49
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$\begingroup$ Maybe a book that explains Schauder estimates for elliptic PDE's? Gilbarg and Trudinger? Or the book by L. C. Evans? $\endgroup$– Deane YangCommented Jan 13, 2011 at 1:53
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$\begingroup$ 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. $\endgroup$– Igor BelegradekCommented Jan 13, 2011 at 2:10
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1$\begingroup$ Sure, everything should be obtainable by straighforward iteration of the $C^0$ Arzela-Ascoli. I was just hoping to find some place where it is written. $\endgroup$– Igor BelegradekCommented Jan 13, 2011 at 12:51
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