The fact that, for functions from a compact manifold with boundary $M$ toward $\mathbb{R}$ or $\mathbb{R}^n$, the Sobolev spaces associated are compactly embedded with Sobolev spaces of less regularity is well known (see for example: https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem). However I would like to know if there is an analog statement when the functions are for exemple of the type $M \xrightarrow{} N$ where $N$ is a compact différentiable subset of a Banach space.
Do you are aware of any reference that deal with this kind of generalisation ? Do you know what are the kind of hypothesis that we need to have on this target space $N$ in order to have compact sobolev embedding ? (I would expect $N$ compact to be enough but maybe we can ask less)