  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.
