Let $X\subset\mathbb R^n$, let $m\in\mathbb N$, and let $0<\gamma\le 1$. For a map $f:X\to\mathbb R^m$, define

\begin{align*} \|f\|_{\infty}:=\sup_{x\in X}|f(x)| \end{align*}

and

\begin{align*} [f]_{C^{0,\gamma}(X)}:=\sup_{\substack{x, y\in X, x\ne y}}\frac{|f(x)-f(y)|}{|x-y|^\gamma}, \end{align*}

with the convention that the supremum over the empty set is $0$. Let $C^{0,\gamma}(X;\mathbb R^m)$ denote the [vector space](/page/Vector%20Space) of bounded maps $f:X\to\mathbb R^m$ with $[f]_{C^{0,\gamma}(X)}<\infty$. Then $C^{0,\gamma}(X;\mathbb R^m)$ is a [Banach space](/page/Banach%20Space) with respect to the norm

\begin{align*} \|f\|_{C^{0,\gamma}(X)}:=\|f\|_{\infty}+[f]_{C^{0,\gamma}(X)}. \end{align*}

Let $U\subset\mathbb R^n$ be bounded and open, let $k\in\mathbb N\cup\{0\}$, and let $0<\gamma\le 1$. Interpreting $C^{k,\gamma}(\overline U;\mathbb R^m)$ as the space of maps $f:U\to\mathbb R^m$ whose derivatives $D^\alpha f$ for $\alpha\in\mathbb N_0^n$ with $|\alpha|\le k$ extend continuously to maps $\overline U\to\mathbb R^m$, and whose extended derivatives of order exactly $k$ have finite $C^{0,\gamma}(\overline U)$ seminorm, the space $C^{k,\gamma}(\overline U;\mathbb R^m)$ is a Banach space with respect to the norm

\begin{align*} \|f\|_{C^{k,\gamma}(\overline U)}:=\sum_{|\alpha|\le k}\|D^\alpha f\|_{\infty}+\sum_{|\alpha|=k}[D^\alpha f]_{C^{0,\gamma}(\overline U)}. \end{align*}