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 $\ell_1$, while $E$ has the Pytkeev property if and only if it is finite-dimensional. We extend Schl\"uchtermann and Wheeler's result by showing that $B$ is a (separable) metrizable space if and only if it has countable $cs^\ast$-character and is a $k$-space. As a corollary we obtain that $B$ is Polish if and only if it has countable $cs^\ast$-character and is \v{C}ech-complete, that supplements a result of Edgar and Wheeler.