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The weak topology on q-convex Banach function spaces

✍ Scribed by L. Agud; J. M. Calabuig; E. A. Sánchez Pérez

Publisher
John Wiley and Sons
Year
2011
Tongue
English
Weight
198 KB
Volume
285
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let X(μ) be a Banach function space. In this paper we introduce two new geometric notions, q‐convexity and weak q‐convexity associated to a subset S of the unit ball of the dual of X(μ), 1 ≤ q < ∞. We prove that in the general case both notions are not equivalent and we study the relation between them, showing that they can be used for describing the weak topology in these spaces. We define the canonical q‐concave weak topology τ~q~ on X(μ)—a topology generated by q‐concave seminorms—for obtaining our main result: A σ‐order continuous Banach function space X(μ) is q‐convex if and only if the following topological inclusions τ~w~⊆τ~q~⊆τ~‖ · ‖~ hold. As an application, in the last section we prove a suitable Maurey‐Rosenthal type factorization theorem for operators from a Banach function space X(μ) into a Banach space that holds under weaker assumptions on the q‐convexity requirements for X(μ).

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