Mackey continuity of convex functions on dual Banach spaces: a review

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AJ Wrobel

Abstract

A convex (or concave) real-valued function, f , on a dual Banach space P is continuous for the Mackey topology m (P, P ) if (and only if) it is Mackey continuous on bounded subsets of P . Equivalence of Mackey continuity to sequential Mackey continuity follows when P is strongly weakly compactly generated, e.g., when P = L1(T ), where T is a set that carries a sigma-finite measure σ. This result of Delbaen, Orihuela and Owari extends their earlier work on the case that P is either L (T ) or a dual Orlicz space. An earlier result of this kind is recalled also: it derives Mackey continuity from bounded Mackey continuity for a nondecreasing concave function, F , that is defined and finite only on the nonnegative cone L+. Applied to a linear f , the Delbaen-Orihuela-Owari result shows that the convex bounded Mackey topology is identical to the Mackey topology, i.e., cbm (P, P ) = m (P, P ); here, this is shown to follow also from Grothendieck’s Completeness Theorem. As for the bounded Mackey topology, bm (P, P ), it is conjectured here not to be a vector topology, or equivalently to be strictly stronger than m (P, P ), except when P is reflexive.

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How to Cite
Wrobel, A. J. (2020). Mackey continuity of convex functions on dual Banach spaces: a review. Extracta Mathematicae, 35(2), 185-195. Retrieved from https://publicaciones.unex.es/index.php/EM/article/view/2605-5686.35.2.185
Section
Banach Spaces and Operator Theory

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