DOI: 10.68381/jca24037 ISSN: 0944-6532
On Topological Properties of the Weak Topology of a Banach Space
Saak Gabriyelyan, Jerzy Kąkol, Lyubomyr Zdomskyy
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
ℓ
1
, 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^*
c
s
∗
-character and is a k-space. As a corollary we obtain that B is Polish if and only if it has countable
cs^*
c
s
∗
-character and is Čech-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].