On Countable Tightness and the Lindelöf Property in Non-Archimedean Banach Spaces

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 Lindelöf 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 $\mathbb{K}$: (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 Lindelöf property. (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 Lindelöf 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. Both results show how essentially different are non-archimedean counterparts from the ``classical'' corresponding theorems for Banach spaces over the real or complex field.