Abstract

Let K\mathbb{K} be a non-archimedean valued field and let EE be a non-archimedean Banach space over K\mathbb{K}. By EwE_{w} we denote the space EE equipped with its weak topology and by Ew∗∗E_{w^{\ast }}^{\ast } the dual space E∗E^{\ast } equipped with its weak∗^{\ast } topology. Several results about countable tightness and the Lindelöf property for EwE_{w} and Ew∗∗E_{w^{\ast }}^{\ast } are provided. A key point is to prove that for a large class of infinite-dimensional polar Banach spaces EE, countable tightness of EwE_{w} or Ew∗∗E_{w^{\ast }}^{\ast } implies separability of K\mathbb{K}. As a consequence we obtain the following two characterizations of the field K\mathbb{K}: (a) A non-archimedean valued field K\mathbb{K} is locally compact if and only if for every Banach space EE over K\mathbb{K} the space EwE_{w} has countable tightness if and only if for every Banach space EE over K\mathbb{K} the space Ew∗∗E^{\ast }_{w^{\ast } } has the Lindelöf property. (b) A non-archimedean valued separable field K\mathbb{K} is spherically complete if and only if every Banach space EE over K\mathbb{K} for which EwE_{w} has the Lindelöf property must be separable if and only if every Banach space EE over K\mathbb{K} for which Ew∗∗E^{\ast }_{w^{\ast }} has countable tightness must be separable. Both results show how essentially different are non-archimedean counterparts from the ``classical'' corresponding theorems for Banach spaces over the real or complex field.

Author information

Contact details are reproduced from the original publication and may be historical.

Jerzy Kąkol

Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland
and: Institute of Mathematics, Czech Academy of Sciences, Praha, Czechia

kakol@amu.edu.pl

Cristina Perez-Garcia

Dept. of Mathematics, Faculty of Sciences, Universidad de Cantabria, Avda. de los Castros s/n, 39071 Santander, Spain

perezmc@unican.es

Keywords

  • Non-archimedean Banach spaces
  • weak topology
  • Lindelöf property
  • countable tightness

Mathematics Subject Classification

46S10, 54D20

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Suggested citation

J. Kąkol, A. Kubzdela, C. Perez-Garcia. “On Countable Tightness and the Lindelöf Property in Non-Archimedean Banach Spaces.” Journal of Convex Analysis 25 (2018), No. 1, 181–199. https://doi.org/10.68381/jca25011

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.