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FLAT SETS, p-GENERATING AND FIXING c0 IN THE NONSEPARABLE SETTING

  • M. FABIAN (a1), A. GONZÁLEZ (a2) and V. ZIZLER (a3)
Abstract

We define asymptotically p-flat and innerly asymptotically p-flat sets in Banach spaces in terms of uniform weak* Kadec–Klee asymptotic smoothness, and use these concepts to characterize weakly compactly generated (Asplund) spaces that are c0(ω1)-generated or p(ω1)-generated, where p∈(1,). In particular, we show that every subspace of c0(ω1) is c0(ω1)-generated and every subspace of p(ω1) is p(ω1)-generated for every p∈(1,). As a byproduct of the technology of projectional resolutions of the identity we get an alternative proof of Rosenthal’s theorem on fixing c0(ω1).

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Copyright
Corresponding author
For correspondence; e-mail: fabian@math.cas.cz
Footnotes
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The first author was supported by grants AVOZ 101 905 03 and IAA 100 190 610 and the Universidad Politécnica de Valencia. The second author was supported in a Grant CONACYT of the Mexican Government. The third author was supported by grants AVOZ 101 905 03 and GAČR 201/07/0394.

Footnotes
References
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[1]Diestel, J., Sequence and Series in Banach Spaces, Graduate Texts in Mathematics, 92 (Springer, New York, 1984).
[2]Fabian, M., Differentiability of Convex Functions and Topology-Weak Asplund Spaces (John Wiley and Sons, New York, 1997).
[3]Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J. and Zizler, V., Functional Analysis and Infinite Dimensional Geometry, Canadian Mathematical Society Books in Mathematics, 8 (Springer, New York, 2001).
[4]Fabian, M., Godefroy, G., Hájek, P., Zizler, V., Fabian, M., Godefroy, G., Hájek, P. and Zizler, V., ‘Hilbert-generated spaces’, J. Funct. Anal. 200 (2003), 301323.
[5]Fabian, M., Godefroy, G., Montesinos, V. and Zizler, V., ‘Inner characterization of weakly compactly generated Banach spaces and their relatives’, J. Math. Anal. Appl. 297 (2004), 419455.
[6]Fabian, M., Montesinos, V. and Zizler, V., ‘Weak compactness and sigma-Asplund generated Banach spaces’, Studia Math. 181 (2007), 125152.
[7]Godefroy, G., Kalton, N. and Lancien, G., ‘Subspaces of c 0(ℕ) and Lipschitz isomorphisms’, Geom. Funct. Anal. 10 (2000), 798820.
[8]Hájek, P., Montesinos, V., Vanderwerff, J. and Zizler, V., Biorthogonal Systems in Banach spaces, Canadian Mathematical Society Books in Mathematics (Canadian Mathematical Society, Springer Verlag, 2007).
[9]John, K. and Zizler, V., ‘Some notes on Markushevich bases in weakly compactly generated Banach spaces’, Compos. Math. 35 (1977), 113123.
[10]Lancien, G., ‘On uniformly convex and uniformly Kadec–Klee renormings’, Serdica Math. J. 21 (1995), 118.
[11]Milman, V. D., ‘Geometric theory of Banach spaces II. Geometry of the unit ball’ [in Russian], Uspekhi Mat. Nauk. 26(6 (162)) (1971), 73–149; Engl. Transl. Russian Math. Surveys 26 (1971), 6, 79–163..
[12]Rosenthal, H. P., ‘On relatively disjoint families of measures, with some applications to Banach space theory’, Studia Math. 37 (1970), 1330.
[13]Rosenthal, H. P., ‘The heredity problem for weakly compactly generated Banach spaces’, Compositio Math. 28 (1974), 83111.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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