Let $Z$ be a Banach-space and $A \subset Z$ compact and $X$ be another Banach-space. How can I now show that
$M \subset C(A;X)$ is pre-compact in $C(A;X)$ if and only if $M$ is equicontinuous and for all $t \in A$ the set $M(t)=\{v(t) \in X : v \in M \}$ is pre-compact in $X$.
How to begin?