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2000 | 138 | 2 | 179-191
Tytuł artykułu

Valdivia compacta and equivalent norms

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EN
Abstrakty
EN
We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density $ℵ_1$ is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss' theorem.
Czasopismo
Rocznik
Tom
138
Numer
2
Strony
179-191
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-03-15
Twórcy
  • Department of Mathematical Analysisi, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, kalenda@karlin.mff.cuni.cz
Bibliografia
  • [AL] D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. 88 (1968), 35-46.
  • [AMN] S. Argyros, S. Mercourakis and S. Negrepontis, Functional-analytic properties of Corson-compact spaces, Studia Math. 89 (1988), 197-229.
  • [DG] R. Deville and G. Godefroy, Some applications of projective resolutions of identity, Proc. London Math. Soc. 67 (1993), 183-199.
  • [D] J. Diestel, Sequences and Series in Banach Spaces, Springer, Berlin, 1984.
  • [FGZ] M. Fabian, G. Godefroy and V. Zizler, A note on Asplund generated Banach spaces, Bull. Acad. Polon. Sci. 47 (1999), 221-230.
  • [HHZ] P. Habala, P. Hájek and V. Zizler, Introduction to Banach Spaces, lecture notes, Matfyzpress, Prague, 1996.
  • [K1] O. Kalenda, An example concerning Valdivia compact spaces, Serdica Math. J. 25 (1999), 131-140.
  • [K2] O. Kalenda, Continuous images and other topological properties of Valdivia compacta, Fund. Math. 162 (1999), 181-192.
  • [K3] O. Kalenda, A characterization of Valdivia compact spaces, Collect. Math. (to appear).
  • [K4] O. Kalenda, Valdivia compacta and subspaces of C(K) spaces, Extracta Math. (to appear).
  • [V1] M. Valdivia, Resolutions of the identity in certain Banach spaces, Collect. Math. 39 (1988), 127-140.
  • [V2] M. Valdivia, Projective resolutions of the identity in C(K) spaces, Arch. Math. (Basel) 54 (1990), 493-498.
  • [V3] M. Valdivia, Simultaneous resolutions of the identity operator in normed spaces, Collect. Math. 42 (1991), 265-285.
  • [V4] M. Valdivia, On certain compact topological spaces, Rev. Mat. Univ. Complut. Madrid 10 (1997), 81-84.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv138i2p179bwm
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