On topological properties of the weak topology of a Banach space

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 ${\ell }_1$, while $E$ has the Pytkeev property if and only if it is finite-dimensional. We extend Schlüchtermann and Wheeler's result by showing that $B$ is a (separable) metrizable space if and only if it has countable $c{s}^{\ast }$-character and is a $k$-space. As a corollary we obtain that $B$ is Polish if and only if it has countable $c{s}^{\ast }$-character and is Čech-complete, that supplements a result of Edgar and Wheeler.

Comments and remarks are welcome

Cited by in corpus (2)

• The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces
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