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On a three-space property for Lindelöf Sigma-spaces, (WCG)-spaces and the Sobczyk property

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On a three-space property for Lindelöf Sigma-spaces, (WCG)-spaces and the Sobczyk property

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dc.contributor.author Ferrer, Jesús es_ES
dc.contributor.author Kakol, Jerzy es_ES
dc.contributor.author López Pellicer, Manuel es_ES
dc.contributor.author Wójtowicz, Marek es_ES
dc.date.accessioned 2016-11-14T11:06:42Z
dc.date.available 2016-11-14T11:06:42Z
dc.date.issued 2011
dc.identifier.issn 0208-6573
dc.identifier.uri http://hdl.handle.net/10251/73925
dc.description.abstract [EN] Corson's example shows that there exists a Banach space EE which is not weakly normal but EE contains a closed subspace isomorphic to the Banach space C[0,1]C[0,1] and such that the quotient space E/C[0,1]E/C[0,1] is isomorphic to the weakly compactly generated Banach space c0[0,1]c0[0,1]. This applies to show the following two results: (i) The Lindelöf property is not a three-space property. (ii) The Lindelöf Σ-property is not a three-space property. In this note using the lifting property developed by Susanne Dierolf we present a very simple argument providing also (ii), see Theorem 1. This argument used in the proof applies also to show that under Continuum Hypothesis every infinite-dimensional topological vector space EE which contains a dense hyperplane admits a stronger vector topology υυ with the same topological dual and such that (E,υ)(E,υ) contains a dense non-Baire hyperplane. This partially answers a question of Saxon concerning Arias de Reyna-Valdivia-Saxon theorem. A Banach space EE has the Sobczyk Property if it contains an isomorphic copy of c0c0 and every such a copy is complemented in EE. The classical Sobczyk's theorem says that every separable Banach space has this property. We give an example of a C(K)C(K)-space EE and its subspace YY isometric to c0c0 such that E/YE/Y is isomorphic to c0(Γ)c0(Γ), with card(Γ)=2ℵ0card(Γ)=2ℵ0, yet YY is uncomplemented in EE. This complements Corson's example and shows that the Sobczyk Property (as well as the (WCG)-property, and the Separable Complementation Property) is not a~three-space property. In the last part we recall some facts (partially with a simpler presentation) concerning K-analytic, Lindelöf ΣΣ and analytic locally convex spaces. Additionally, a few remarks concerning weakly K-analytic spaces are include es_ES
dc.description.sponsorship The first author has been partially supported by MEC and FEDER Project MTM2008-03211. The research for the second named author was (partially) supported by Ministry of Science and Higher Education, Poland, grant no. NN201 2740 33, and for the second and third named author by the project MTM2008 - 01502 of the Spanish Ministry of Science and Innovation.
dc.language Inglés es_ES
dc.publisher Adam Mickiewicz University The Faculty of Mathematics and Computer Science
dc.relation MEC/MTM2008-03211 es_ES
dc.relation MICINN/MTM2008 - 01502 es_ES
dc.relation.ispartof Functiones et Approximatio, Commentarii mathematici es_ES
dc.rights Reserva de todos los derechos es_ES
dc.subject Lindelöf Σ-spaces es_ES
dc.subject WCG-spaces es_ES
dc.subject Banach spaces es_ES
dc.subject.classification MATEMATICA APLICADA es_ES
dc.title On a three-space property for Lindelöf Sigma-spaces, (WCG)-spaces and the Sobczyk property es_ES
dc.type Artículo es_ES
dc.identifier.doi 10.7169/facm/1308749133
dc.rights.accessRights Cerrado es_ES
dc.contributor.affiliation Universitat Politècnica de València. Escuela Técnica Superior de Ingeniería Agronómica y del Medio Natural - Escola Tècnica Superior d'Enginyeria Agronòmica i del Medi Natural es_ES
dc.description.bibliographicCitation Ferrer, J.; Kakol, J.; López Pellicer, M.; Wójtowicz, M. (2011). On a three-space property for Lindelöf Sigma-spaces, (WCG)-spaces and the Sobczyk property. Functiones et Approximatio, Commentarii mathematici. 44(2):289-306. doi:10.7169/facm/1308749133 es_ES
dc.description.accrualMethod S es_ES
dc.relation.publisherversion https://dx.doi.org/10.7169/facm/1308749133 es_ES
dc.description.upvformatpinicio 289 es_ES
dc.description.upvformatpfin 306 es_ES
dc.type.version info:eu-repo/semantics/publishedVersion es_ES
dc.description.volume 44 es_ES
dc.description.issue 2 es_ES
dc.relation.senia 212534 es_ES
dc.contributor.funder Ministerio de Educación y Ciencia
dc.contributor.funder Ministerio de Ciencia e Innovación


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