A note on Mackey topologies on Banach spaces
[EN] There is a maybe unexpected connection between three apparently unrelated notions concerning a given w*-dense subspace Y of the dual X* of a Banach space X: (i) The norming character of Y, (ii) the fact that (Y, w*) has the Mazur property, and (iii) the completeness of the Mackey topology mu(X,...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2017 |
| País: | España |
| Institución: | Universitat Politècnica de València (UPV) |
| Repositorio: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Idioma: | inglés |
| OAI Identifier: | oai:riunet.upv.es:10251/107361 |
| Acceso en línea: | https://riunet.upv.es/handle/10251/107361 |
| Access Level: | acceso abierto |
| Palabra clave: | Banach space Mackey topology Completeness Norming subspace Mazur space Weak compactness MATEMATICA APLICADA |
| Sumario: | [EN] There is a maybe unexpected connection between three apparently unrelated notions concerning a given w*-dense subspace Y of the dual X* of a Banach space X: (i) The norming character of Y, (ii) the fact that (Y, w*) has the Mazur property, and (iii) the completeness of the Mackey topology mu(X, Y), i.e., the topology on X of the uniform convergence on the family of all absolutely convex w*-compact subsets of Y. To clarify these connections is the purpose of this note. The starting point was a question raised by M. Kunze and W. Arendt and the answer provided by J. Bonet and B. Cascales. We fully characterize mu(X, Y)-completeness or its failure in the case of Banach spaces X with a w*-angelic dual unit ball in particular, separable Banach spaces or, more generally, wealdy compactly generated ones-by using the norming or, alternatively, the Mazur character of Y. We characterize the class of spaces where the original Kunze-Arendt question has always a positive answer. Some other applications are also provided. (C) 2016 Elsevier Inc. All rights reserved. |
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