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,...

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Autores: Guirao Sánchez, Antonio José|||0000-0002-1031-3954, Montesinos Santalucia, Vicente, Zizler, Vaclav
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
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spelling A note on Mackey topologies on Banach spacesGuirao Sánchez, Antonio José|||0000-0002-1031-3954Montesinos Santalucia, VicenteZizler, VaclavBanach spaceMackey topologyCompletenessNorming subspaceMazur spaceWeak compactnessMATEMATICA APLICADA[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.Supported in part by MICINN (Project MTM2014-57838-C2-2-P).ElsevierEscuela Técnica Superior de Ingeniería de TelecomunicaciónDepartamento de Matemática AplicadaInstituto Universitario de Matemática Pura y AplicadaMinisterio de Economía y CompetitividadFundación Séneca-Agencia de Ciencia y Tecnología de la Región de MurciaRepositorio Institucional de la Universitat Politècnica de València Riunet20172017-01-01journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfapplication/pdfhttps://riunet.upv.es/handle/10251/107361reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valénciainstname:Universitat Politècnica de València (UPV)InglésengFundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia https://doi.org/10.13039/100007801 19368%2FPI%2F14Ministerio de Economía y Competitividad http://dx.doi.org/10.13039/501100003329 MTM2014-57838-C2-1-P LA INTERACCION ENTRE GEOMETRIA Y TOPOLOGIA EN ESPACIOS DE BANACH. APLICACIONESopen accesshttp://purl.org/coar/access_right/c_abf2Reconocimiento - No comercial - Sin obra derivada (by-nc-nd) http://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:riunet.upv.es:10251/1073612026-06-13T07:49:27Z
dc.title.none.fl_str_mv A note on Mackey topologies on Banach spaces
title A note on Mackey topologies on Banach spaces
spellingShingle A note on Mackey topologies on Banach spaces
Guirao Sánchez, Antonio José|||0000-0002-1031-3954
Banach space
Mackey topology
Completeness
Norming subspace
Mazur space
Weak compactness
MATEMATICA APLICADA
title_short A note on Mackey topologies on Banach spaces
title_full A note on Mackey topologies on Banach spaces
title_fullStr A note on Mackey topologies on Banach spaces
title_full_unstemmed A note on Mackey topologies on Banach spaces
title_sort A note on Mackey topologies on Banach spaces
dc.creator.none.fl_str_mv Guirao Sánchez, Antonio José|||0000-0002-1031-3954
Montesinos Santalucia, Vicente
Zizler, Vaclav
author Guirao Sánchez, Antonio José|||0000-0002-1031-3954
author_facet Guirao Sánchez, Antonio José|||0000-0002-1031-3954
Montesinos Santalucia, Vicente
Zizler, Vaclav
author_role author
author2 Montesinos Santalucia, Vicente
Zizler, Vaclav
author2_role author
author
dc.contributor.none.fl_str_mv Escuela Técnica Superior de Ingeniería de Telecomunicación
Departamento de Matemática Aplicada
Instituto Universitario de Matemática Pura y Aplicada
Ministerio de Economía y Competitividad
Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia
Repositorio Institucional de la Universitat Politècnica de València Riunet
dc.subject.none.fl_str_mv Banach space
Mackey topology
Completeness
Norming subspace
Mazur space
Weak compactness
MATEMATICA APLICADA
topic Banach space
Mackey topology
Completeness
Norming subspace
Mazur space
Weak compactness
MATEMATICA APLICADA
description [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.
publishDate 2017
dc.date.none.fl_str_mv 2017
2017-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://riunet.upv.es/handle/10251/107361
url https://riunet.upv.es/handle/10251/107361
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia https://doi.org/10.13039/100007801 19368%2FPI%2F14
Ministerio de Economía y Competitividad http://dx.doi.org/10.13039/501100003329 MTM2014-57838-C2-1-P LA INTERACCION ENTRE GEOMETRIA Y TOPOLOGIA EN ESPACIOS DE BANACH. APLICACIONES
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reconocimiento - No comercial - Sin obra derivada (by-nc-nd)
http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reconocimiento - No comercial - Sin obra derivada (by-nc-nd)
http://creativecommons.org/licenses/by-nc-nd/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
instname:Universitat Politècnica de València (UPV)
instname_str Universitat Politècnica de València (UPV)
reponame_str RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
collection RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
repository.name.fl_str_mv
repository.mail.fl_str_mv
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