$f-$statistical convergence, completeness and $f-$cluster points

Abstract

We study f −statistical convergence, which is a generalization of the classical statistical convergence. In terms of it, we give a characterization of completeness in a normed space. We also introduce ' f −statistical cluster points', which is a richer concept than the classic one. Namely, each (usual) limit point of a sequence is an f −statistical cluster point for some f .

On statistical limit points with respect to power series methods and modulus functions

“…By using the modulus functions, Aizpuru et al [1] introduced the concept of f -statistical convergence which depends on the other new concept of f -density of subsets of natural numbers (where f is a modulus function). Listán-García [13] gave the definition of f -statistical limit points and cluster points with respect to a modulus function f and proved some relations including the properties of the sets of f -statistical limit points and f -cluster points.…”

Completeness in topological vector spaces and filters on N

“…Let us say that a normed space X is f -complete, if every f -statistically Cauchy sequence (x n ) ⊂ X is f -statistically convergent. Our research is motivated by [12,Theorem 2.4] (see also [1,Theorem 3.3]): Let X be a normed space. The following are equivalent: (1) X is complete; (2) X is f -complete for every unbounded modulus f ;…”

“…In the next section we recall, for the reader's convenience, the definitions and basic facts about filters and filter convergence in topological spaces. After that, in the section "Completeness, sequential completeness, and completeness over a filter on N", we recall the basic facts about Cauchy filters and completeness in TVS, introduce formally the completeness over a filter on N, list some features of this new property, and deduce, for general filters on N and a metrizable TVS, the validity of equivalences like those in [12,Theorem 2.4]. After that we pass to the general non-metrizable case (Section "Various types of completeness and classes of filters and spaces").…”

“…After that we pass to the general non-metrizable case (Section "Various types of completeness and classes of filters and spaces"). We discuss the relationship between completeness, sequential completeness, and completeness with respect to various filters on N (subsection "Countable completeness"), demonstrate a non-metrisable version of [12,Theorem 2.4] in locally convex spaces under the additional boundedness condition for the Cauchy sequences in question (subsection "Completeness and boundedness"), give an example of a sequentially complete space which is not complete with respect to ANY free ultrafilter on N, and give an example of a space which is complete with respect to a free ultrafilter on N but is not complete with respect to some other ultrafilter (subsection "Completeness and ultrafilters"). We conclude the paper with some open questions.…”

“…Our study was motivated by [Aizpuru, Listán-García and Rambla-Barreno; Quaest. Math., 2014] and [Listán-García; Bull. Belg.…”