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 Schlüchtermann and Wheeler's result from [27] 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 Edgar and Wheeler [8].