Abstract

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\ell_1, 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∗cs^*-character and is a k-space. As a corollary we obtain that B is Polish if and only if it has countable cs∗cs^*-character and is Čech-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].

Author information

Contact details are reproduced from the original publication and may be historical.

Saak Gabriyelyan

Department of Mathematics, Ben-Gurion University of the Negev, P. O. 653, Beer-Sheva, Israel

saak@math.bgu.ac.il

Jerzy Kąkol

Faculty of Mathematics and Informatics, A. Mickiewicz University, Matejki 48-49, 60-769 Poznań, 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

Keywords

  • Weak topology
  • Banach space
  • aleph-space
  • k-space
  • cs*-character

Mathematics Subject Classification

46A03, 54E18; 54C35, 54E20

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Suggested citation

S. Gabriyelyan, J. Kąkol, L. Zdomskyy. “On Topological Properties of the Weak Topology of a Banach Space.” Journal of Convex Analysis 24 (2017), No. 2, 571–586. https://doi.org/10.68381/jca24037

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