Compactness in vector-valued Banach function spaces

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 $1\le p<\mathrm{\infty }$, and extend the result to vector-valued Banach function spaces $E_X$, where $E$ is a Banach function space with order continuous norm.