
Journal of Convex Analysis 24 (2017), No. 2, 571586 Copyright Heldermann Verlag 2017 On Topological Properties of the Weak Topology of a Banach Space Saak Gabriyelyan Department of Mathematics, BenGurion University of the Negev, P. O. 653, BeerSheva, Israel saak@math.bgu.ac.il Jerzy Kakol Faculty of Mathematics and Informatics, A. Mickiewicz University, Matejki 4849, 60769 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 l_{1}, while E has the Pytkeev property if and only if it is finitedimensional. We extend a result of G. Schlüchtermann and R. F. Wheeler [The Mackey dual of a Banach space, Noti de Matematica XI (1991) 273287] by showing that B is a (separable) metrizable space if and only if it has countable cs*character and is a kspace. As a corollary we obtain that B is Polish if and only if it has countable cs*character and is Cechcomplete, that supplements a result of G. A. Edgar and R. F. Wheeler [Topological properties of Banach spaces, Pacific J. Math. 115 (1984) 317350]. Keywords: Weak topology, Banach space, alephspace, kspace, cs*character. MSC: 46A03, 54E18; 54C35, 54E20 [ Fulltextpdf (188 KB)] for subscribers only. 