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
ℓ
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
∗
-character and is a
k
-space. As a corollary we obtain that
B
is Polish if and only if it has countable
c
s
∗
-character and is Čech-complete, that supplements a result of Edgar and Wheeler.