| 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:
|
http://hdl.handle.net/10338.dmlcz/133414 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| . |
