PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 3, March 2011, Pages 957–967 S 0002-9939(2010)10508-7 Article electronically published on July 28, 2010 p-CONVERGENT SEQUENCES AND BANACH SPACES IN WHICH p-COMPACT SETS ARE q-COMPACT C ´ ANDIDO PI ˜ NEIRO AND JUAN MANUEL DELGADO (Communicated by Nigel J. Kalton) Abstract. We introduce and investigate the notion of p-convergence in a Ba- nach 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 character- ize 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. 1. Introduction By a well known characterization due to Grothendieck [7] (see, e.g., [9, p. 30]), a subset A of a Banach space X is relatively compact if and only if there exists (x n ) in c 0 (X) (the space of norm-null sequences in X) such that A ⊂{ ∑ n a n x n : ∑ n |a n |≤ 1}. Since then, several authors have dealt with stronger forms of compactness, studying sets sitting inside the convex hulls of special types of null sequences. For instance, it was observed in [14] (see also [1]) that if one considers, instead of c 0 (X), the space of q-summable sequences ℓ q (X), for some fixed q ≥ 1, then this stronger form of compactness characterizes Reinov’s approximation property of order p, 0 <p< 1. This latter form of compactness was recently further strengthened by Sinha and Karn [15] as follows. Let 1 ≤ p ≤∞ and let p ′ be the conjugate index of p (i.e., 1/p +1/p ′ = 1). The p-convex hull of a sequence (x n ) ∈ ℓ p (X) is defined as p- co (x n )= { ∑ n a n x n : ∑ n |a n | p ′ ≤ 1} (sup |a n |≤ 1 if p = 1). A set A ⊂ X is said to be relatively p-compact if there exists (x n ) ∈ ℓ p (X) ((x n ) ∈ c 0 (X) if p = ∞) such that A ⊂ p- co (x n ). (Note that similar notions with (x n ) being a weakly p-summable sequence were already considered in [2, p. 51].) Some results concerning this type of relatively compact set have been set in [3]. The aim of this article is to deepen the study of the geometry of Banach spaces related to p-compact sets. In this way, the notion of p-convergent sequence is introduced in Section 2 and a Grothendieck-like result is obtained; namely, a subset Received by the editors January 21, 2010 and, in revised form, March 6, 2010 and March 22, 2010. 2010 Mathematics Subject Classification. Primary 46B50, 47B07; Secondary 47B10. Key words and phrases. p-compact set, p-convergent sequence, p-nuclear operator, p-summing operator, cotype. This research was supported by MTM2009-14483-C02-01 project (Spain). c 2010 American Mathematical Society Reverts to public domain 28 years from publication 957 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use