Geodesic
On Topological Properties of the Weak Topology of a Banach Space
Journal of convex analysis, Tome 24 (2017) no. 2, pp. 571-586
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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].
Classification : 46A03, 54E18, 54C35, 54E20
Mots-clés : Weak topology, Banach space, aleph-space, k-space, cs*-character 
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     author = {S. Gabriyelyan and J. Kakol and L. Zdomskyy},
     title = {On {Topological} {Properties} of the {Weak} {Topology} of a {Banach} {Space}},
     journal = {Journal of convex analysis},
     pages = {571--586},
     year = {2017},
     volume = {24},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2017_24_2_JCA_2017_24_2_a13/}
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S. Gabriyelyan; J. Kakol; L. Zdomskyy. On Topological Properties of the Weak Topology of a Banach Space. Journal of convex analysis, Tome 24 (2017) no. 2, pp. 571-586. http://geodesic.mathdoc.fr/item/JCA_2017_24_2_JCA_2017_24_2_a13/