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Let $\Omega\subset\mathbb R^n$ be compact and $C^{0,\alpha}(\Omega)$ the space of all $\alpha$-Hölder-continuous functions. Define $||u||_{C^{0,\alpha}(\Omega)}:=||u||_{\sup}+\sup\limits_{{x,y\in \Omega\space\&\space x\ne y}}\frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and consider $(C^{0,\alpha}(\Omega),||u||_{C^{0,\alpha}(\Omega)})$ and $\alpha\in]0,1]$ .

How can you prove that for any sequence in bounded closed set of $(C^{0,\alpha}(\Omega),||u||_{C^{0,\alpha}(\Omega)})$ there exists a convergent subsequence (concerning the uniform norm) and it limes is in $(C^{0,\alpha}(\Omega))$?

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    @Sheldor : Its very useful to learn Arzela Ascoli theorem.2012-06-15

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Hints:

1) Using estimations of norms prove that if set $F$ is bounded in $(C^{0,\alpha}(\Omega),\Vert\cdot\Vert_{C^{0,\alpha}(\Omega)})$ then it is bounded in $(C^{0}(\Omega),\Vert\cdot\Vert_{C^{0}(\Omega)})$

2) If $F$ is bounded in $(C^{0,\alpha}(\Omega),\Vert\cdot\Vert_{C^{0,\alpha}(\Omega)})$ then $ \exists C>0\quad\forall u\in F\quad \forall x,y\in\Omega\quad |u(x)-u(y)|\leq C|x-y|^\alpha $

3) Prove that 2) implies equicontinuity

4) From 1) and 4) you see that $F$ is relatively compact in $(C^{0}(\Omega),\Vert\cdot\Vert_{C^{0}(\Omega)})$.

5) If you get to this paragraph it is remains to prove uniform convergence. Using estimations of norms prove that if sequence $\{u_n:n\in\mathbb{N}\}$ converges in $(C^{0,\alpha}(\Omega),\Vert\cdot\Vert_{C^{0,\alpha}(\Omega)})$, then it converges in $(C^{0}(\Omega),\Vert\cdot\Vert_{C^{0}(\Omega)})$.

6) From statement of paragraph 2) prove that the limit function is $\alpha$-Hölder-continuous.

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    @sheldoor, I don't think so. Gener$a$lly sp$a$eking the family of uniformly continuous functions is not uniformly $b$ounded - $c$onsider $c$onst$a$$n$ts.2012-06-16