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A quantitative version of Krein's theorems for Fréchet spaces

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A quantitative version of Krein's theorems for Fréchet spaces

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dc.contributor.author Angosto Hernández, Carlos es_ES
dc.contributor.author Kakol, Jerzy es_ES
dc.contributor.author Kubzdela, Albert es_ES
dc.contributor.author López Pellicer, Manuel es_ES
dc.date.accessioned 2014-06-04T07:26:06Z
dc.date.issued 2013-05
dc.identifier.issn 0003-889X
dc.identifier.uri http://hdl.handle.net/10251/37911
dc.description.abstract For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being &#963;(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein¿s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E, the above function k(H) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds k(coH) < (2^(n+1) &#8722; 2)k(H) + 1/2^n for all n &#8712; N. Consequently, this yields also the following formula k(coH) &#8804; (k(H))^(1/2))(3-2(k(H)^(1/2))). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein¿s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet spaces. We also define and discuss two other measures of weak non-compactness lk(H) and k'(H) for a Fréchet space and provide two quantitative versions of Krein¿s theorem for both functions. es_ES
dc.description.sponsorship The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation. en_EN
dc.language Inglés es_ES
dc.publisher Springer Verlag (Germany) es_ES
dc.relation.ispartof Archiv der Mathematik es_ES
dc.rights Reserva de todos los derechos es_ES
dc.subject Krein's theorem es_ES
dc.subject Compactness es_ES
dc.subject Fréchet space es_ES
dc.subject Space of continuous functions es_ES
dc.subject.classification MATEMATICA APLICADA es_ES
dc.title A quantitative version of Krein's theorems for Fréchet spaces es_ES
dc.type Artículo es_ES
dc.embargo.lift 10000-01-01
dc.embargo.terms forever es_ES
dc.identifier.doi 10.1007/s00013-013-0513-4
dc.relation.projectID info:eu-repo/grantAgreement/MICINN//MTM2008-05396/ES/LA INTERACCION ENTRE TEORIA DE LA MEDIDA, TOPOLOGIA Y ANALISIS FUNCIONAL/
dc.relation.projectID info:eu-repo/grantAgreement/NCN//N N201 605340/
dc.relation.projectID info:eu-repo/grantAgreement/MICINN//MTM2010-12374-E/ES/ESTRATEGIAS PARA EL PROGRESO MATEMATICO EN ESPAÑA/
dc.rights.accessRights Abierto es_ES
dc.contributor.affiliation Universitat Politècnica de València. Departamento de Matemática Aplicada - Departament de Matemàtica Aplicada es_ES
dc.description.bibliographicCitation Angosto Hernández, C.; Kakol, J.; Kubzdela, A.; López Pellicer, M. (2013). A quantitative version of Krein's theorems for Fréchet spaces. Archiv der Mathematik. 101(1):65-77. https://doi.org/10.1007/s00013-013-0513-4 es_ES
dc.description.accrualMethod S es_ES
dc.relation.publisherversion http://link.springer.com/content/pdf/10.1007%2Fs00013-013-0513-4.pdf es_ES
dc.description.upvformatpinicio 65 es_ES
dc.description.upvformatpfin 77 es_ES
dc.type.version info:eu-repo/semantics/publishedVersion es_ES
dc.description.volume 101 es_ES
dc.description.issue 1 es_ES
dc.relation.senia 246301
dc.contributor.funder Ministerio de Ciencia e Innovación
dc.contributor.funder National Science Centre, Polonia
dc.description.references Angosto C., Cascales B.: Measures of weak noncompactness in Banach spaces. Topology Appl. 156, 1412–1421 (2009) es_ES
dc.description.references C. Angosto, Distance to spaces of functions, PhD thesis, Universidad de Murcia (2007). es_ES
dc.description.references C. Angosto and B. Cascales, A new look at compactness via distances to functions spaces, World Sc. Pub. Co. (2008). es_ES
dc.description.references Angosto C., Cascales B.: The quantitative difference between countable compactness and compactness. J. Math. Anal. Appl. 343, 479–491 (2008) es_ES
dc.description.references Angosto C., Cascales B., Namioka I.: Distances to spaces of Baire one functions. Math. Z. 263, 103–124 (2009) es_ES
dc.description.references C. Angosto, J. Ka̧kol, and M. López-Pellicer, A quantitative approach to weak compactness in Fréchet spaces and spaces C(X), J. Math. Anal. Appl. 403 (2013), 13–22. es_ES
dc.description.references Cascales B., Marciszesky W., Raja M.: Distance to spaces of continuous functions. Topology Appl. 153, 2303–2319 (2006) es_ES
dc.description.references M. Fabian et al. Functional Analysis and Infinite-dimensional geometry, CMS Books in Mathematics, Canadian Math. Soc., Springer (2001). es_ES
dc.description.references M. Fabian et al. A quantitative version of Krein’s theorem, Rev. Mat. Iberoam. 21 (2005), 237–248 es_ES
dc.description.references Granero A. S.: An extension of the Krein-Smulian Theorem. Rev. Mat. Iberoam. 22, 93–110 (2006) es_ES
dc.description.references Granero A. S., Hájek P., Montesinos V.: Santalucía, Convexity and ω*-compactness in Banach spaces. Math. Ann. 328, 625–631 (2004) es_ES
dc.description.references Grothendieck A.: Criteres de compacité dans les spaces fonctionnelles généraux. Amer. J. Math. 74, 168–186 (1952) es_ES
dc.description.references Khurana S. S.: Weakly compactly generated Fréchet spaces. Int. J. Math. Math. Sci. 2, 721–724 (1979) es_ES


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