Journal of Convex Analysis 25 (2018), No. 1, 181--199
Copyright Heldermann Verlag 2018
On Countable Tightness and the Lindelöf Property in Non-Archimedean Banach Spaces
Jerzy Kakol
Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznan, Poland
kakol@amu.edu.pl
Albert Kubzdela
Institute of Civil Engineering, University of Technology, Ul. Piotrowo 5, 61-138 Poznan, Poland
albert.kubzdela@put.poznan.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
[Abstract-pdf]
Let $\mathbb{K}$ be a non-archimedean valued field and let $E$ be a non-archimedean 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 infinite-dimensional 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 non-archimedean 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 non-archimedean 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 non-archimedean counterparts from the ``classical'' corresponding theorems for Banach spaces over the real or complex field.
Keywords: Non-archimedean Banach spaces, weak topology, Lindel\"of property, countable tightness.
MSC: 46S10, 54D20
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