# Article

 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: [Ba] Banakh I.: On Banach spaces possessing an $\varepsilon$-net without weak limit points.Math. Methods and Phys. Mech. Fields 43 3 (2000), 40--43. MR 1968634 Reference: [BPZ] Banakh T., Plichko A., Zagorodnyuk A.: Zeros of continuous quadratic functionals on non-separable Banach spaces.Colloq. Math. 100 (2004), 141--147. MR 2079354, 10.4064/cm100-1-13 Reference: [BFT] Bourgain J., Fremlin D., Talagrand M.: Pointwise compact sets of Baire-measurable functions.Amer. J. Math. 100 4 (1978), 845--886. Zbl 0413.54016, MR 0509077, 10.2307/2373913 Reference: [CG] Castillo J., González M.: Three-space problems in Banach space theory.Lecture Notes in Mathematics, 1667, Springer, Berlin, 1997. MR 1482801 Reference: [Dis] Diestel J.: Sequences and Series in Banach Spaces.Springer, New York, 1984. MR 0737004 Reference: [En] Engelking R.: General Topology.PWN, Warsaw, 1977. Zbl 0684.54001, MR 0500780 Reference: [Fab] Fabian M.: Gateaux Differentiability of Convex Functions and Topology.John Wiley & Sons, Inc., New York, 1997. Zbl 0883.46011, MR 1461271 Reference: [HHZ] Habala P., Hájek P., Zizler V.: Introduction to Banach spaces.Matfyzpress, Praha, 1996. Reference: [OR] Odell E., Rosenthal H.P.: A double-dual characterization of separable Banach spaces containing $l^{1}$.Israel J. Math. 20 3--4 (1975), 375--384. MR 0377482, 10.1007/BF02760341 .

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