
Journal of Convex Analysis 25 (2018), No. 1, [final page numbers not yet available] 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, 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. [ Fulltextpdf (389 KB)] for subscribers only. 