
Journal of Convex Analysis 25 (2018), No. 1, 181199 Copyright Heldermann Verlag 2018 On Countable Tightness and the Lindelöf Property in NonArchimedean Banach Spaces Jerzy Kakol Faculty of Mathematics and Informatics, A. Mickiewicz University, 61614 Poznan, Poland kakol@amu.edu.pl Albert Kubzdela Institute of Civil Engineering, University of Technology, Ul. Piotrowo 5, 61138 Poznan, Poland albert.kubzdela@put.poznan.pl Cristina PerezGarcia Dept. of Mathematics, Faculty of Sciences, Universidad de Cantabria, Avda. de los Castros s/n, 39071 Santander, Spain perezmc@unican.es [Abstractpdf] Let $\mathbb{K}$ be a nonarchimedean valued field and let $E$ be a nonarchimedean Banach space over $\mathbb{K}$. By $E_{w}$ we denote the space $E$ equipped with its weak topology and by $E_{w^{\ast }}^{\ast }$ the dual space $E^{\ast }$ equipped with its weak$^{\ast }$ topology. Several results about countable tightness and the Lindel\"{o}f property for $E_{w}$ and $E_{w^{\ast }}^{\ast }$ are provided. A key point is to prove that for a large class of infinitedimensional polar Banach spaces $E$, countable tightness of $E_{w}$ or $E_{w^{\ast }}^{\ast }$ implies separability of $% \mathbb{K}$. As a consequence we obtain the following two characterizations of the field $\mathbb{K}$:\par \medskip (a) A nonarchimedean valued field $\mathbb{K}$ is locally compact if and only if for every Banach space $E$ over $\mathbb{K}$ the space $E_{w}$ has countable tightness if and only if for every Banach space $E$ over $\mathbb{K% }$ the space $E^{\ast }_{w^{\ast } }$ has the Lindel\"{o}f property.\par \medskip (b) A nonarchimedean valued separable field $\mathbb{K}$ is spherically complete if and only if every Banach space $E$ over $\mathbb{K}$ for which $% E_{w}$ has the Lindel\"{o}f property must be separable if and only if every Banach space $E$ over $\mathbb{K}$ for which $E^{\ast }_{w^{\ast }}$ has countable tightness must be separable.\par \medskip Both results show how essentially different are nonarchimedean counterparts from the ``classical'' corresponding theorems for Banach spaces over the real or complex field. Keywords: Nonarchimedean Banach spaces, weak topology, Lindel\"of property, countable tightness. MSC: 46S10, 54D20 [ Fulltextpdf (163 KB)] for subscribers only. 