Completeness in topological vector spaces and filters on $\mathbb N$

Abstract

We study completeness of a topological vector space with respect to different filters on $\mathbb N$. In the metrizable case all these kinds of completeness are the same, but in non-metrizable case the situation changes. For example, a space may be complete with respect to one ultrafilter on $\mathbb N$, but incomplete with respect to another. Our study was motivated by [Aizpuru, Listán-García and Rambla-Barreno; Quaest. Math., 2014] and [Listán-García; Bull. Belg. Math. Soc. Simon Stevin, 2016] where for normed spaces the equivalence of the ordinary completeness and completeness with respect to $f$-statistical convergence was established.