The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces

We prove the following new characterization of Cp Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz) bump function if and only if it has another Cp smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the...

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Detalles Bibliográficos
Autores: Azagra Rueda, Daniel, Jiménez Sevilla, María Del Mar
Tipo de recurso: artículo
Fecha de publicación:2001
País:España
Institución:Universidad Complutense de Madrid (UCM)
Repositorio:Docta Complutense
Idioma:inglés
OAI Identifier:oai:docta.ucm.es:20.500.14352/57129
Acceso en línea:https://hdl.handle.net/20.500.14352/57129
Access Level:acceso abierto
Palabra clave:517.98
Negligibility
Rolle theorem
Smooth norm
Brouwer fixed point theorem
Bump
Análisis funcional y teoría de operadores
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repository_id_str
spelling The Failure of Rolle's Theorem in Infinite-Dimensional Banach SpacesAzagra Rueda, DanielJiménez Sevilla, María Del Mar517.98NegligibilityRolle theoremSmooth normBrouwer fixed point theoremBumpAnálisis funcional y teoría de operadoresWe prove the following new characterization of Cp Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz) bump function if and only if it has another Cp smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The ``twisted tube'' method we use in the proof is interesting in itself, as it provides other useful characterizations of Cp smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces.ElsevierUniversidad Complutense de Madrid20012001-05-1020012001-05-10journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/57129reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/571292026-06-02T12:44:21Z
dc.title.none.fl_str_mv The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
title The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
spellingShingle The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
Azagra Rueda, Daniel
517.98
Negligibility
Rolle theorem
Smooth norm
Brouwer fixed point theorem
Bump
Análisis funcional y teoría de operadores
title_short The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
title_full The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
title_fullStr The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
title_full_unstemmed The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
title_sort The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
dc.creator.none.fl_str_mv Azagra Rueda, Daniel
Jiménez Sevilla, María Del Mar
author Azagra Rueda, Daniel
author_facet Azagra Rueda, Daniel
Jiménez Sevilla, María Del Mar
author_role author
author2 Jiménez Sevilla, María Del Mar
author2_role author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 517.98
Negligibility
Rolle theorem
Smooth norm
Brouwer fixed point theorem
Bump
Análisis funcional y teoría de operadores
topic 517.98
Negligibility
Rolle theorem
Smooth norm
Brouwer fixed point theorem
Bump
Análisis funcional y teoría de operadores
description We prove the following new characterization of Cp Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz) bump function if and only if it has another Cp smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The ``twisted tube'' method we use in the proof is interesting in itself, as it provides other useful characterizations of Cp smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces.
publishDate 2001
dc.date.none.fl_str_mv 2001
2001-05-10
2001
2001-05-10
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/20.500.14352/57129
url https://hdl.handle.net/20.500.14352/57129
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 15,300719

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