DOI: 10.68381/jca25011 ISSN: 0944-6532
On Countable Tightness and the Lindelöf Property in Non-Archimedean Banach Spaces
Jerzy Kąkol, Albert Kubzdela, Cristina Perez-Garcia
Let
\mathbb{K}
K
be a non-archimedean valued field and let
E
E
be a non-archimedean Banach space over
\mathbb{K}
K
. By
E_{w}
E
w
we denote the space
E
E
equipped with its weak topology and by
E_{w^{\ast }}^{\ast }
E
w
∗
∗
the dual space
E^{\ast }
E
∗
equipped with its weak
^{\ast }
∗
topology. Several results about countable tightness and the Lindelöf property for
E_{w}
E
w
and
E_{w^{\ast }}^{\ast }
E
w
∗
∗
are provided. A key point is to prove that for a large class of infinite-dimensional polar Banach spaces
E
E
, countable tightness of
E_{w}
E
w
or
E_{w^{\ast }}^{\ast }
E
w
∗
∗
implies separability of
\mathbb{K}
K
. As a consequence we obtain the following two characterizations of the field
\mathbb{K}
K
: (a) A non-archimedean valued field
\mathbb{K}
K
is locally compact if and only if for every Banach space
E
E
over
\mathbb{K}
K
the space
E_{w}
E
w
has countable tightness if and only if for every Banach space
E
E
over
\mathbb{K}
K
the space
E^{\ast }_{w^{\ast } }
E
w
∗
∗
has the Lindelöf property. (b) A non-archimedean valued separable field
\mathbb{K}
K
is spherically complete if and only if every Banach space
E
E
over
\mathbb{K}
K
for which
E_{w}
E
w
has the Lindelöf property must be separable if and only if every Banach space
E
E
over
\mathbb{K}
K
for which
E^{\ast }_{w^{\ast }}
E
w
∗
∗
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.