$\textbf{Definition}$ Let $ \Omega $ be open in $\mathbb{R}^n$ and $\alpha \in (0,1]$. \begin{align*} \textrm{For } \alpha\in(0,1],& \newline\\ &[u]_{\alpha,\Omega}:=\sup_{x\neq y}\frac{\vert u(x)-u(y) \vert}{\vert x-y \vert^{\alpha}} \quad \textrm{for } x,y\in \Omega\\ \newline &\Vert u \Vert_{C^0(\bar{\Omega})}:= \sup_{x\in \Omega} \vert u(x) \vert \\ \newline &\Vert u\Vert_{C^k(\bar {\Omega})} :=\sum_{0\leq \vert \beta \vert \leq k} \Vert D^{\beta}u \Vert _{C^0(\bar{\Omega})}\\ \newline &\Vert u\Vert_{C^{k,\alpha}(\bar {\Omega})}:= \Vert u \Vert_{C^k(\bar{\Omega})} + \sum_{\vert \beta \vert =k} [D^{\beta}u]_{\alpha, \Omega} \end{align*} $\textbf{Holder space} \quad C^{k,\alpha}(\bar{\Omega}):=\{u\in C^{k}(\bar{\Omega}); \Vert u \Vert_{C^{k,\alpha}}(\bar{\Omega})< \infty \}$
I want to know whether Holder space has finite bases or not. How to find bases of Holder space??...
Any help is appreciated... Thank you!