On $r$-reflexive Banach spaces

@article{Banakh2009,
abstract = {A Banach space $X$ is called $r$-reflexive if for any cover $\mathcal \{U\}$ of $X$ by weakly open sets there is a finite subfamily $\mathcal \{V\}\subset \mathcal \{U\}$ covering some ball of radius 1 centered at a point $x$ with $\Vert x\Vert \le r$. We prove that an infinite-dimensional separable Banach space $X$ is $\infty $-reflexive ($r$-reflexive for some $r\in \mathbb \{N\}$) 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 $r\in \mathbb \{N\}$. 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.},
author = {Banakh, Iryna, Banakh, Taras O., Riss, Elena},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {reflexive Banach space; $r$-reflexive Banach space; Asplund Banach space; reflexive Banach space; -reflexive Banach space; net; weak accumulation point; weakly convergent sequence; James' space},
language = {eng},
number = {1},
pages = {61-74},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On $r$-reflexive Banach spaces},
url = {http://eudml.org/doc/32480},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Banakh, Iryna
AU - Banakh, Taras O.
AU - Riss, Elena
TI - On $r$-reflexive Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 1
SP - 61
EP - 74
AB - A Banach space $X$ is called $r$-reflexive if for any cover $\mathcal {U}$ of $X$ by weakly open sets there is a finite subfamily $\mathcal {V}\subset \mathcal {U}$ covering some ball of radius 1 centered at a point $x$ with $\Vert x\Vert \le r$. We prove that an infinite-dimensional separable Banach space $X$ is $\infty $-reflexive ($r$-reflexive for some $r\in \mathbb {N}$) 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 $r\in \mathbb {N}$. 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.
LA - eng
KW - reflexive Banach space; $r$-reflexive Banach space; Asplund Banach space; reflexive Banach space; -reflexive Banach space; net; weak accumulation point; weakly convergent sequence; James' space
UR - http://eudml.org/doc/32480
ER -