Mackey continuity of convex functions on dual Banach spaces: a review
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...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Extremadura
2020-12-01
|
| Series: | Extracta Mathematicae |
| Subjects: | |
| Online Access: | https://publicaciones.unex.es/index.php/EM/article/view/361 |
| Summary: | 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.
|
|---|---|
| ISSN: | 0213-8743 2605-5686 |