Hostname: page-component-5c6d5d7d68-tdptf Total loading time: 0 Render date: 2024-08-07T07:27:17.291Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly compact sets and smooth norms in Banach spaces

Published online by Cambridge University Press:  17 April 2009

Marián Fabian ,
Vicente Montesinos  and
Václav Zizler
Marián Fabian
Affiliation:
Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 11567, Prague 1, Czech Repulic e-mail: Fabian@math.cas.cz
Vicente Montesinos
Affiliation:
Departmento de Matemática Aplicada, E.T.S.I. Telecomunicación, Universidad Politécnica de ValenciaC/Vera, S/n. 46071, Valencia, Spain e-mail: vmontesinos@mat.upv.es
Václav Zizler
Affiliation:
Department of Mathematical Sciences, University of Alberta, 632 Central Academic Building, Edmonton, Alberta T6G 2G1, Canada e-mail: vzizler@math.ualberta.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Two smoothness characterisations of weakly compact sets in Banach spaces are given. One that involves pointwise lower semicontinuous norms and one that involves projectional resolutions of identity.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 2 , April 2002 , pp. 223 - 230
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Borwein, J.M., ‘Weak local supportability and application to approximation’ Pacific J. Math 82 (1979), 323–338.Google Scholar
[2]Deville, R. and Godefroy, G., ‘Some applications of projective resolutions of identity’, Bull. London Math. Soc. 67 (1993), 183–199.CrossRefGoogle Scholar
[3]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64 (Longman Scientific and Technical, Harlow, 1993).Google Scholar
[4]Fabian, M., Gâteaux differentiability of convex functions and topology. Weak Asplund spaces (John Wiley and Sons, New York, 1997).Google Scholar
[5]Fabian, M., Godefroy, G. and Zizler, V., ‘The structure of uniformly Gâteaux smooth Banach spaces’, Israel J. Math. 124 (2001), 243–252.Google Scholar
[6]Fabian, M., Habala, P., Hájek, P., Pelant, J., Montesinos, V. and Zizler, V., Functional analysis and infinite dimensional geometry, CMS Books in Mathematics 8 (Springer-Verlag, New York, 2001).Google Scholar
[7]Fabian, M., Hájek, P. and Zizler, V., ‘A note on lattice renormings’, Comment. Math. Univ. Carolin. 38 (1997), 263–272.Google Scholar
[8]Fabian, M., Montesinos, V. and Zizler, V., ‘Pointwise lower semicontinuous smooth norms’ Arch. Math. (to appear).Google Scholar
[9]Hájek, R., ‘Dual renormings of Banach spaces’, Comment. Math. Univ. Carolin. 37 (1996), 241–253.Google Scholar
[10]Haydon, R., ‘Trees in renorming theory’, Proc. London Math. Soc. 78 (1999), 541–584.CrossRefGoogle Scholar
[11]John, K. and Zizler, V., ‘Smoothness and its equivalents in weakly compactly generated Banach spaces’, J. Funct. Anal. 15 (1974), 161–166.Google Scholar
[12]Orihuela, J., Schachermayerand, W.Valdivia, M., ‘Every Radon-Nikodým Corson Compact is Eberlein compact’, Studia Math. 98 (1991), 157–174.CrossRefGoogle Scholar
[13]Rosenthal, H.P., ‘The heredity problem for weakly compactly generated Banach spaces’, Compisito. Math. 28 (1974), 83–111.Google Scholar
[14]Stegall, Ch., ‘More facts about conjugate Banach spaces with the Radon-Nikodým property II’, Acta Univ. Carolin. Math. Phys. 32 (1991), 47–57.Google Scholar