  In this work we study the Riemann-Liouville fractional integral of order
$\alpha\in(0,1/p)$ as an operator from $L^p(I;X)$ into $L^{q}(I;X)$, with
$1\leq q\leq p/(1-p\alpha)$, whether $I=[t_0,t_1]$ or $I=[t_0,\infty)$ and $X$
is a Banach space. Our main result give necessary and sufficient conditions to
ensure the compactness of the Riemann-Liouville fractional integral from
$L^p(t_0,t_1;X)$ into $L^{q}(t_0,t_1;X)$, when $1\leq q< p/(1-p\alpha)$.
