Completeness in topological vector spaces and filters on N

TL;DR

This paper investigates how the concept of completeness in topological vector spaces varies with different filters on natural numbers, highlighting differences between metrizable and non-metrizable cases.

Contribution

It extends the understanding of completeness by analyzing its dependence on various filters, especially ultrafilters, in non-metrizable topological vector spaces.

Findings

In metrizable spaces, all types of completeness coincide.

In non-metrizable spaces, completeness can vary with different ultrafilters.

Completeness with respect to filters relates to f-statistical convergence in normed spaces.

Abstract

We study completeness of a topological vector space with respect to different filters on the set N of all naturals. 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 N, but incomplete with respect to another. Our study was motivated by [Aizpuru, List\'an-Garc\'ia and Rambla-Barreno; Quaest. Math., 2014] and [List\'an-Garc\'ia; 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.