-Convergent sequences and Banach spaces in which -compact sets are -compact

We introduce and investigate the notion of p-convergence in a Banach space. Among others, a Grothendieck-like result is obtained; namely, a subset of a Banach space is relatively p-compact if and only if it is contained in the closed convex hull of a p-null sequence. We give a description of the topological dual of the space of all p-null sequences which is used to characterize the Banach spaces enjoying the property that every relatively p-compact subset is relatively q-compact (1 <= q < p). As an application, Banach spaces satisfying that every relatively p-compact set lies inside the range of a vector measure of bounded variation are characterized.