The weak topology on q‐convex Banach function spaces

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 τq 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 τw⊆τq⊆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).