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Title: On $r$-reflexive Banach spaces (English)
Author: Banakh, Iryna
Author: Banakh, Taras
Author: Riss, Elena
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 50
Issue: 1
Year: 2009
Pages: 61-74
Summary lang: English
Category: math
Summary: A Banach space $X$ is called {\it $r$-reflexive\/} if for any cover $\Cal U$ of $X$ by weakly open sets there is a finite subfamily $\Cal V\subset\Cal U$ covering some ball of radius 1 centered at a point $x$ with $\|x\|\leq r$. We prove that an infinite-dimensional separable Banach space $X$ is $\infty$-reflexive ($r$-reflexive for some $r\in \Bbb N$) if and only if each $\varepsilon $-net for $X$ has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of $X$. We show that the quasireflexive James space $J$ is $r$-reflexive for no $r\in \Bbb N$. We do not know if each $\infty$-reflexive Banach space is reflexive, but we prove that each separable $\infty$-reflexive Banach space $X$ has Asplund dual. As a by-product of the proof we obtain a covering characterization of the Asplund property of Banach spaces. (English)
Keyword: reflexive Banach space
Keyword: $r$-reflexive Banach space
Keyword: Asplund Banach space
MSC: 46A25
MSC: 46B10
MSC: 46B22
idZBL: Zbl 1212.46022
idMR: MR2562803
Date available: 2009-08-18T12:22:58Z
Last updated: 2013-09-22
Stable URL:
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