On r -reflexive Banach spaces

Iryna Banakh; Taras O. Banakh; Elena Riss

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 1, page 61-74
  • ISSN: 0010-2628

Abstract

top
A Banach space X is called r -reflexive if for any cover 𝒰 of X by weakly open sets there is a finite subfamily 𝒱 ⊂ 𝒰 covering some ball of radius 1 centered at a point x with ∥ x ∥ ≤ r . We prove that an infinite-dimensional separable Banach space X is ∞ -reflexive ( r -reflexive for some r ∈ ℕ ) if and only if each ε -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 ∈ ℕ . We do not know if each ∞ -reflexive Banach space is reflexive, but we prove that each separable ∞ -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.

How to cite

top

Banakh, Iryna, Banakh, Taras O., and Riss, Elena. "On $r$-reflexive Banach spaces." Commentationes Mathematicae Universitatis Carolinae 50.1 (2009): 61-74. <http://eudml.org/doc/32480>.

@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 -

References

top
  1. Banakh I., On Banach spaces possessing an ε -net without weak limit points, Math. Methods and Phys. Mech. Fields 43 3 (2000), 40--43. (2000) MR1968634
  2. Banakh T., Plichko A., Zagorodnyuk A., 10.4064/cm100-1-13, Colloq. Math. 100 (2004), 141--147. (2004) MR2079354DOI10.4064/cm100-1-13
  3. Bourgain J., Fremlin D., Talagrand M., 10.2307/2373913, Amer. J. Math. 100 4 (1978), 845--886. (1978) Zbl0413.54016MR0509077DOI10.2307/2373913
  4. Castillo J., González M., Three-space problems in Banach space theory, Lecture Notes in Mathematics, 1667, Springer, Berlin, 1997. MR1482801
  5. Diestel J., Sequences and Series in Banach Spaces, Springer, New York, 1984. MR0737004
  6. Engelking R., General Topology, PWN, Warsaw, 1977. Zbl0684.54001MR0500780
  7. Fabian M., Gateaux Differentiability of Convex Functions and Topology, John Wiley & Sons, Inc., New York, 1997. Zbl0883.46011MR1461271
  8. Habala P., Hájek P., Zizler V., Introduction to Banach spaces, Matfyzpress, Praha, 1996. 
  9. Odell E., Rosenthal H.P., 10.1007/BF02760341, Israel J. Math. 20 3--4 (1975), 375--384. (1975) MR0377482DOI10.1007/BF02760341

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.