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The weak topology on q ‐convex Banach function spaces
Author(s) -
Agud L.,
Calabuig J. M.,
Sánchez Pérez E. A.
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201000030
Subject(s) - mathematics , convexity , banach space , topology (electrical circuits) , weak topology (polar topology) , unit sphere , reflexive space , convex function , regular polygon , pure mathematics , function space , discrete mathematics , topological space , combinatorics , general topology , functional analysis , interpolation space , extension topology , geometry , biochemistry , chemistry , financial economics , economics , gene
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).