On Topological Properties of the Weak Topology of a Banach Space

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].