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

Abstract

A convex (or concave) real-valued function, ƒ, 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 ƒ, 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.