  We give a new proof of a recent characterization by Diaz and Mayoral of
compactness in the Lebesgue-Bochner spaces $L_X^p$, where $X$ is a Banach space
and $1\le p<\infty$, and extend the result to vector-valued Banach function
spaces $E_X$, where $E$ is a Banach function space with order continuous norm.
