Abstract
We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelöf number in the weak topology and other related questions.
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References
D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Annals of Mathematics (2) 88 (1968), 35–46.
S. Argyros and S. Mercourakis, On weakly Lindelöf Banach spaces, Rocky Mountains Journal of Mathematics 23 (1993), 395–446.
S. Argyros, S. Mercourakis, and S. Negrepontis, Functional-analytic properties of Corson-compact spaces, Studia Mathematica 89 (1988), 197–229.
A. D. Arvanitakis, Some remarks on Radon-Nikodým compact spaces, Fundamenta Mathematicae 172 (2002), 41–60.
A. Avilés, Radon-Nikodým compact spaces of low weight and Banach spaces, Studia Mathematica 166 (2005), 71–82.
M. Bell and W. Marciszewski, Function spaces on τ-Corson compacta and tightness of polyadic spaces, Czechoslovak Mathematical Journal 54(129) (2004), 899–914.
Y. Benyamini, M. E. Rudin, and M. Wage, Continuous images of weakly compact subsets of Banach spaces, Pacific Journal of Mathematics 70 (1977), 309–324.
M. R. Burke and M. Magidor, Shelah’s pcf theory and its applications, Annals of Pure and Applied Logic 50 (1990), 207–254.
B. Cascales, M. Muñoz, and J. Orihuela, Index of \(\mathcal{K}\)-determination of topological spaces, Preprint (2005).
H. H. Corson, The weak topology of a Banach space, Transactions of the American Mathematical Society 101 (1961), 1–15.
M. Fabian, Gâteaux Differentiability of Convex Functions and Topology, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1997, Weak Asplund spaces, A Wiley-Interscience Publication.
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Vol. 156, Springer-Verlag, New York, 1995.
M. Muñoz, Índice de \(\mathcal{K}\)-determinación de espacios topológicos σ-fragmentabilidad de aplicaciones, Doctoral dissertation, University of Murcia, 2004.
I. Namioka, On generalizations of Radon-Nikodým compact spaces, Topology Proceedings 26(2) (2001–2002), 741–750.
R. Pol, A function space C(X) which is weakly Lindelöf but not weakly compactly generated, Studia Mathematica 64 (1979), 279–285.
R. Pol, On weak and pointwise topology of function spaces, University of Warsaw preprint No. 184, Warsaw, 1984.
H. P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compositio Mathematica 28 (1974), 83–111.
M. Talagrand, Espaces de Banach faiblement \(\mathcal{K}\)-analytiques, Annals of Mathematics (2) 110 (1979), 407–438.
M. Talagrand, A new countably determined Banach space, Israel Journal of Mathematics 47 (1984), 75–80.
E. K. van Douwen, The integers and topology, in Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 111–167.
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This research was partially supported by the grant BFM2002-01719 of MCyT (Spain) and a FPU grant of MEC (Spain).
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Avilés, A. The number of weakly compact sets which generate a Banach space. Isr. J. Math. 159, 189–204 (2007). https://doi.org/10.1007/s11856-007-0042-6
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DOI: https://doi.org/10.1007/s11856-007-0042-6