On the Wikipedia page on interpolation space, it is written that the space $C^\theta([0, 1])$ is the (real) interpolation of $C^0([0, 1])$ and $C^1([0, 1])$, where $C^\theta([0, 1])$ denotes the space of $\theta$-H\"older continuous functions on $[0, 1]$.

However, I was unable to find the detailed proof of this fact (all books I checked treat $L^p$-spaces and their derived spaces). Do you have a reference for this?

To be more precise, the result I would be interested in is the following:

Given $\Omega$ a (non-empty) open subset of $\mathbb{R}^n$, is it true that $C^{k, \alpha}(\Omega)$ is the (real) interpolation space of $C^0(\Omega)$ and $C^l(\Omega)$ for any $l > k+\alpha$ and (possibly) $\alpha \in (0, 1)$?