## 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 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].

Original language | American English |
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Pages (from-to) | 571-586 |

Number of pages | 16 |

Journal | Journal of Convex Analysis |

Volume | 24 |

Issue number | 2 |

State | Published - 1 Jan 2017 |

## Keywords

- Banach space
- Cs∗-character
- K-space
- Weak topology
- X-space

## All Science Journal Classification (ASJC) codes

- Analysis
- General Mathematics