Holder Spaces

Let $\Omega \subset \mathbb{R}^n$ be an open set with $k \geq 0$ being an integer and $0 < \alpha \leq 1$. The space of Holder continuous functions $C^{0,\alpha}(\Omega)$ is defined as consisting of all those continuous functions $f \in C(\Omega)$ so that $$[f]_{C^{0,\alpha}(K)} := \sup_{x,y \in K, x\neq y} \left\{\frac{|u(x)-u(y)|}{|x-y|^\alpha}\right\} < \infty$$ for every compact $K \subset \Omega$. Similarly, the space $C^{0,\alpha}(\bar{\Omega})$ is the set $f\in C(\bar{\Omega})$ so that $[f]_{C^{0,\alpha}(\bar{\Omega})} < \infty$. This space is equipped with the norm $\|f\|_{C^{0,\alpha}(\bar{\Omega})} := \|f\|_{C^0(\bar{\Omega})} + [f]_{C^{0,\alpha}(\bar{\Omega})}$.

$(C^{0,\alpha}(\bar{\Omega}), \|\cdot\|_{C^{0,\alpha}(\bar{\Omega})})$ is a Banach space

Let $f_n$ be a cauchy sequence in $C^{0,\alpha}(\bar{\Omega})$. Therefore, for every $\epsilon > 0$ there exits $M$ such that for all $n,m > M$, we have $\|f_n-f_m\|_{C^{0,\alpha}(\bar{\Omega})} < \epsilon$. This gives that $[f_n-f_m]_{C^{0,\alpha}(\bar{\Omega})} < \epsilon$ and $\|f_n -f_m\|_{C^0(\bar{\Omega})} < \epsilon$. From the latter we get that $f_n$ converges (in the sup norm) to a bounded continuous function $f$ (on $\bar{\Omega}$). From the former we get that $[f_n-f]_{C^{0,\alpha}(\bar{\Omega})} < \epsilon$ by letting $m \to \infty$. Therefore $\|f_n-f\|_{C^{0,\alpha}(\bar{\Omega})} \to 0$ as $n \to \infty$. It remains to show that $f \in {C^{0,\alpha}(\bar{\Omega})}$ but this follows easily by $$[f]_{C^{0,\alpha}(\bar{\Omega})} = [f-f_n+f_n]_{C^{0,\alpha}(\bar{\Omega})} \leq [f-f_n]_{C^{0,\alpha}(\bar{\Omega})} + [f_n]_{C^{0,\alpha}(\bar{\Omega})} < \infty$$

Note that this result does not depend on $\alpha$.