It says in my lecture notes that the relatively compact sets of $C^k(K)$ are the sets $S$ for which the set $\{\delta^\alpha f|f\in S|\alpha|\le k\}$ is equicontinuous and bounded in $C^0(K)$. I may have lacked concentration then, but assuming I took them correctly I can't seem to agree.
I would have said that using the mean value theorem, it suffices that only the set $\{\delta^\alpha f|f\in s|\alpha|=1\}$ be bounded in $C^0$ to ensure that S be equicontinuous and obviously bounded.
Am I wrong about this?
It's important here that they're talking about sets which are compact in $C^k$, not in $C^0$. Because of the properties of the $C^k$ norm, relative compactness of $S$ in $C^k$ is equivalent to relative compactness of $S \cup \{ f' : f \in S \} \cup \dots \cup \{ f^{(k)} : f \in S\}$ in $C^0$. This is essentially because $f_n \to f$ in $C^k$ is equivalent to $f_n \to f$ in $C^0$, $f'_n \to f'$ in $C^0$, ... and $f^{(k)}_n \to f^{(k)}$ in $C^0$.