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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

Jerzy Kakol
Faculty of Mathematics and Informatics, A. Mickiewicz University, Matejki 48-49, 60-769 Poznan, Poland

Lyubomyr Zdomskyy
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Str. 25, 1090 Vienna, Austria

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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