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

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 October 2009

M. FABIAN ,
A. GONZÁLEZ  and
V. ZIZLER
M. FABIAN*
Affiliation:
Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic (email: fabian@math.cas.cz)
A. GONZÁLEZ
Affiliation:
Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, C/Vera, s/n. 46022 Valencia, Spain (email: algoncor@doctor.upv.es)
V. ZIZLER
Affiliation:
Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic (email: zizler@math.cas.cz)
*
✉For correspondence; e-mail: fabian@math.cas.cz
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Abstract

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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 ℓp(ω1)-generated, where p∈(1,∞). In particular, we show that every subspace of c0(ω1) is c0(ω1)-generated and every subspace of ℓp(ω1) is ℓp(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).

Keywords

Lipschitz-weak*-Kadets–Klee normc0(Γ)-generated spaceℓp(Γ)-generated spaceweakly compactly generated spaceasymptotically p-flat setinnerly asymptotically p-flat set

MSC classification

Secondary: 46B26: Nonseparable Banach spaces 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 2 , October 2009 , pp. 197 - 210
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

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.

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