Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications

This article deals with smooth removability of compact sets in infinite-dimensional Banach spaces. The main result states that ifX is an infinite-dimensional Banach space which has a not necessarily equivalent Cp-smooth norm and K is a compact subset of X, then X and X r K are Cp diffeomorphic. The...

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Detalles Bibliográficos
Autores: Azagra Rueda, Daniel, Dobrowolski, Tadeusz
Tipo de recurso: artículo
Fecha de publicación:1998
País:España
Institución:Universidad Complutense de Madrid (UCM)
Repositorio:Docta Complutense
Idioma:inglés
OAI Identifier:oai:docta.ucm.es:20.500.14352/57112
Acceso en línea:https://hdl.handle.net/20.500.14352/57112
Access Level:acceso abierto
Palabra clave:517.98
Cp diffeomorphisms
Cp smooth norm
Complete smooth classification of the convex bodies of every Banach space
Garay’s phenomena for ODE’s in Banach spaces
Existence of periodic diffeomorphisms without fixed points
Análisis funcional y teoría de operadores
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spelling Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applicationsAzagra Rueda, DanielDobrowolski, Tadeusz517.98Cp diffeomorphismsCp smooth normComplete smooth classification of the convex bodies of every Banach spaceGaray’s phenomena for ODE’s in Banach spacesExistence of periodic diffeomorphisms without fixed pointsAnálisis funcional y teoría de operadoresThis article deals with smooth removability of compact sets in infinite-dimensional Banach spaces. The main result states that ifX is an infinite-dimensional Banach space which has a not necessarily equivalent Cp-smooth norm and K is a compact subset of X, then X and X r K are Cp diffeomorphic. The proof relies on the construction of a “deleting path” through a nontrivial refinement of Bessaga’s incomplete-norm technique. However, norms are not at present available and the construction requires the use of asymmetric functionals. The noncompleteness of such functionals relies in turn on James’ theorem on existence of linear functionals which do not attain their norm on every nonreflexive space. Applications are given which show that several important theorems on finite-dimensional spaces completely fail in the infinite-dimensional case: for instance, on any Banach space isomorphic to its Cartesian square and for any natural number n _ 2 there exists a C1-diffeomorphism of pure period n with no fixed point. This work opens the way to several interesting open questions on nonseparable Banach spaces: Does every Banach space with a C1 smooth norm admit a nonequivalent C1-smooth norm? In which Banach spaces is every compact subset the set where a certain C1 real-valued function vanishes?SpringerUniversidad Complutense de Madrid19981998-01-0119981998-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/57112reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/571122026-06-02T12:44:21Z
dc.title.none.fl_str_mv Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
title Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
spellingShingle Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
Azagra Rueda, Daniel
517.98
Cp diffeomorphisms
Cp smooth norm
Complete smooth classification of the convex bodies of every Banach space
Garay’s phenomena for ODE’s in Banach spaces
Existence of periodic diffeomorphisms without fixed points
Análisis funcional y teoría de operadores
title_short Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
title_full Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
title_fullStr Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
title_full_unstemmed Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
title_sort Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
dc.creator.none.fl_str_mv Azagra Rueda, Daniel
Dobrowolski, Tadeusz
author Azagra Rueda, Daniel
author_facet Azagra Rueda, Daniel
Dobrowolski, Tadeusz
author_role author
author2 Dobrowolski, Tadeusz
author2_role author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 517.98
Cp diffeomorphisms
Cp smooth norm
Complete smooth classification of the convex bodies of every Banach space
Garay’s phenomena for ODE’s in Banach spaces
Existence of periodic diffeomorphisms without fixed points
Análisis funcional y teoría de operadores
topic 517.98
Cp diffeomorphisms
Cp smooth norm
Complete smooth classification of the convex bodies of every Banach space
Garay’s phenomena for ODE’s in Banach spaces
Existence of periodic diffeomorphisms without fixed points
Análisis funcional y teoría de operadores
description This article deals with smooth removability of compact sets in infinite-dimensional Banach spaces. The main result states that ifX is an infinite-dimensional Banach space which has a not necessarily equivalent Cp-smooth norm and K is a compact subset of X, then X and X r K are Cp diffeomorphic. The proof relies on the construction of a “deleting path” through a nontrivial refinement of Bessaga’s incomplete-norm technique. However, norms are not at present available and the construction requires the use of asymmetric functionals. The noncompleteness of such functionals relies in turn on James’ theorem on existence of linear functionals which do not attain their norm on every nonreflexive space. Applications are given which show that several important theorems on finite-dimensional spaces completely fail in the infinite-dimensional case: for instance, on any Banach space isomorphic to its Cartesian square and for any natural number n _ 2 there exists a C1-diffeomorphism of pure period n with no fixed point. This work opens the way to several interesting open questions on nonseparable Banach spaces: Does every Banach space with a C1 smooth norm admit a nonequivalent C1-smooth norm? In which Banach spaces is every compact subset the set where a certain C1 real-valued function vanishes?
publishDate 1998
dc.date.none.fl_str_mv 1998
1998-01-01
1998
1998-01-01
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/20.500.14352/57112
url https://hdl.handle.net/20.500.14352/57112
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 Springer
publisher.none.fl_str_mv Springer
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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