Does Arzela-Ascoli hold in $C_0(X)$ (vanishing at infinity)?... namely a subset of $C_0(X)$ is relatively compact iff it is equicontinuous and bounded. In particular, if $C_0^1(X)$ is the Banach subspace of $C_0(X)$ of differentiable functions such that the derivative is in $C_0(X)$, it would imply that the closed unit ball of $C_0^1(X)$ is relatively compact in $C_0(X)$, and therefore that $C_0^1(X)$ is compactly embedded in $C_0(X)$.
I have seen in some class notes that it is true that Arzela-Ascoli holds in $C_0(X)$... but I have seen nowhere that $C_0^1(X)$ is compactly embedded in $C_0(X)$. Nevertheless, it seems that we can proove this by using the Alexandroff compactification of $X$ and defining $f(\infty):=0$.