Journal of Convex Analysis 24 (2017), No. 2, 571--586
Copyright Heldermann Verlag 2017
On Topological Properties of the Weak Topology of a Banach Space
Saak Gabriyelyan
Department of Mathematics, Ben-Gurion University of the Negev, P. O. 653, Beer-Sheva, Israel
saak@math.bgu.ac.il
Jerzy Kakol
Faculty of Mathematics and Informatics, A. Mickiewicz University, Matejki 48-49, 60-769 Poznan, Poland
kakol@amu.edu.pl
Lyubomyr Zdomskyy
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Str. 25, 1090 Vienna, Austria
lyubomyr.zdomskyy@univie.ac.at
Being motivated by the famous Kaplansky theorem we study various sequential properties of a Banach space E and its closed unit ball B, both endowed with the weak topology of E. We show that B has the Pytkeev property if and only if E in the norm topology contains no isomorphic copy of l1, while E has the Pytkeev property if and only if it is finite-dimensional. We extend a result of G. Schlüchtermann and R. F. Wheeler [The Mackey dual of a Banach space, Noti de Matematica XI (1991) 273--287] by showing that B is a (separable) metrizable space if and only if it has countable cs*-character and is a k-space. As a corollary we obtain that B is Polish if and only if it has countable cs*-character and is Cech-complete, that supplements a result of G. A. Edgar and R. F. Wheeler [Topological properties of Banach spaces, Pacific J. Math. 115 (1984) 317--350].
Keywords: Weak topology, Banach space, aleph-space, k-space, cs*-character.
MSC: 46A03, 54E18; 54C35, 54E20
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