James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump function sand to n-homogeneous polynomials on Banach spaces, and their geometrical proper ties are thus worth studying. In this paper we deal wit the question whether James' theorem on the...

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Detalhes bibliográficos
Autores: Azagra Rueda, Daniel, Deville, Robert
Formato: artículo
Fecha de publicación:2001
País:España
Recursos:Universidad Complutense de Madrid (UCM)
Repositorio:Docta Complutense
Idioma:inglés
OAI Identifier:oai:docta.ucm.es:20.500.14352/57130
Acesso em linha:https://hdl.handle.net/20.500.14352/57130
Access Level:acceso abierto
Palavra-chave:517.98
Starlike body
Convex body
James' theorem
Characterization of reflexivity
Análisis funcional y teoría de operadores
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spelling James' Theorem Fails for Starlike BodiesAzagra Rueda, DanielDeville, Robert517.98Starlike bodyConvex bodyJames' theoremCharacterization of reflexivityAnálisis funcional y teoría de operadoresStarlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump function sand to n-homogeneous polynomials on Banach spaces, and their geometrical proper ties are thus worth studying. In this paper we deal wit the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C-1 Lipschitz bump function if and only if there exists another C-1 smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.ElsevierUniversidad Complutense de Madrid20012001-03-1020012001-03-10journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/57130reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/571302026-06-02T12:44:21Z
dc.title.none.fl_str_mv James' Theorem Fails for Starlike Bodies
title James' Theorem Fails for Starlike Bodies
spellingShingle James' Theorem Fails for Starlike Bodies
Azagra Rueda, Daniel
517.98
Starlike body
Convex body
James' theorem
Characterization of reflexivity
Análisis funcional y teoría de operadores
title_short James' Theorem Fails for Starlike Bodies
title_full James' Theorem Fails for Starlike Bodies
title_fullStr James' Theorem Fails for Starlike Bodies
title_full_unstemmed James' Theorem Fails for Starlike Bodies
title_sort James' Theorem Fails for Starlike Bodies
dc.creator.none.fl_str_mv Azagra Rueda, Daniel
Deville, Robert
author Azagra Rueda, Daniel
author_facet Azagra Rueda, Daniel
Deville, Robert
author_role author
author2 Deville, Robert
author2_role author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 517.98
Starlike body
Convex body
James' theorem
Characterization of reflexivity
Análisis funcional y teoría de operadores
topic 517.98
Starlike body
Convex body
James' theorem
Characterization of reflexivity
Análisis funcional y teoría de operadores
description Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump function sand to n-homogeneous polynomials on Banach spaces, and their geometrical proper ties are thus worth studying. In this paper we deal wit the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C-1 Lipschitz bump function if and only if there exists another C-1 smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.
publishDate 2001
dc.date.none.fl_str_mv 2001
2001-03-10
2001
2001-03-10
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/57130
url https://hdl.handle.net/20.500.14352/57130
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
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
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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