𝑝-convergent sequences and Banach spaces in which 𝑝-compact sets are 𝑞-compact

Piñeiro, Cándido; Delgado, Juan

doi:10.1090/S0002-9939-2010-10508-7

$p$-convergent sequences and Banach spaces in which $p$-compact sets are $q$-compact
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by Cándido Piñeiro and Juan Manuel Delgado PDF

Proc. Amer. Math. Soc. 139 (2011), 957-967 Request permission

Abstract:

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 ($1\leq q<p$). 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.

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