TITLE: Proving that a Hölder space is a Banach space
QUESTION [14 upvotes]: I am trying to show that the Hölder space $C^{k,\gamma}(\bar{U})$ is a Banach space. To do this, I successfully proved that the mapping $\| \quad \| : C^{k,\gamma}(\bar{U}) \to [0,\infty)$ is a norm, by proving its properties. 
But how do I show that the sequence $\{u_k\}_{k=1}^\infty \subset C^{k,\gamma}(\bar{U})$ converges to $u \in C^{k,\gamma}(\bar{U})$, that is, how do I show that $$\lim_{k \to \infty}\|u_k-u\|=0,$$ which would mean the normed linear space is complete, and hence a Banach space?
Here are the following taken from PDE Evans, 2nd edition, page 255:

Definition. The Hölder space $$C^{k,\gamma}(\bar{U})$$ consists of all functions $u \in C^k(\bar{U})$ for which the norm $$\|u\|_{C^{k,y}(\bar{U})}:= \sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{0,\gamma}(\bar{U})}$$ is finite.

Also from page 254,

Definitions. (i) If $u : U \to \mathbb{R}$ is bounded and continuous, we write $$\|u\|_{C(\bar{U})}:=\sup_{x\in U}|u(x)|.$$
(ii) The $\gamma$th-Hölder seminorm of $u : U \to \mathbb{R}$ is $$[u]_{C^{0,\gamma}(\bar{U})}:=\sup_{\substack{x,y\in U \\ x \neq y}} \left\{\frac{|u(x)-u(y)|}{|x-y|^\gamma} \right\},$$ and the $\gamma$th-Hölder norm is $$\|u\|_{C^{0,\gamma}(\bar{U})}:=\|u\|_{C(\bar{U})}+[u]_{C^{0,\gamma}(\bar{U})}.$$

This is all I got so far:
\begin{align}
\|u_k-u\|_{C^{k,\gamma}(\bar{U})}&=\sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{0,\gamma}(\bar{U})} \\
&= \sum_{|\alpha|\le k} \sup_{x\in U} |u_k(x)-u(x)|+ \sum_{|\alpha|=k} \sup_{\substack{x,y\in U \\ x \neq y}} \left\{\frac{|[u_k(x)-u(x)]-[u_k(y)-u(y)]|}{|x-y|^\gamma} \right\}.
\end{align}
Now, where can I go from here, to show that $\lim_{k \to \infty}\|u_k-u\|_{C^{k,\gamma}(\bar{U})}=0$? The sequence is Cauchy, and I have to use that fact somehow.

REPLY [6 votes]: Since your comment to my first answer is tricky to read, let me restate it. The follow-up question is: how does
$$
\frac{|v(x)-v(y)|}{|x-y|^\gamma} \le M_\alpha < \infty \qquad independent\ of\ x\neq y \qquad\qquad\qquad(*)$$
for every multi-index $\alpha$ of order $k$ imply
$$
u \in C^{k,\gamma}(\bar{U})?
$$
The answer is as follows: in order to show $u \in C^{k,\gamma}(\bar{U}),$ we have to show that all the ingredients of the Hölder norm of $u$ can be calculated, and the value of the Hölder norm thus obtained is finite.
By construction of $u,$ we have
$$
\sum_{|\beta|\leq k} \|D^\beta u\|_{C(\overline U)} < \infty.
$$
Moreover, (*) above implies that for a fixed multi-index $\alpha$ of order $k,$ we have
$$
[D^\alpha u]_{C^{,\gamma}(\overline U)} \leq M_\alpha < \infty.
$$
From this we get
$$
\sum_{|\alpha| = k}[D^\alpha u]_{C^{,\gamma}(\overline U)} \leq \sum_{|\alpha| = k} M_\alpha < \infty.
$$
So we have all in all
$$
\|u\|_{C^{k,\gamma}(\bar{U})} = \sum_{|\beta|\leq k} \|D^\beta u\|_{C(\overline U)} + \sum_{|\alpha| = k}[D^\alpha u]_{C^{,\gamma}(\overline U)} < \infty
$$
and thus
$$
u \in C^{k,\gamma}(\bar{U}),
$$
as desired.