\(f\)-statistical convergence, completeness and \(f\)-cluster points (Q2629125)

An increasing continuous function \(f:\mathbb{R}^+\to\mathbb{R}^+\) is called a modulus function if \(f(0)=0\), \(f(t) > 0\) for \(t > 0\), and \(f(t+s)\leq f(t)+f(s)\) for every \(t,s \in \mathbb{R}^+\). Let \(f\) be an unbounded modulus function. A sequence \((x_n)\) in a normed space \(X\) is said to be \(f\)-statistically convergent to \(x \in X\) if, for every \(\varepsilon > 0\), \[ \lim_{n\to \infty} \frac{f(|\{k \in \mathbb N: k \leq n \, \mathrm{ and } \, \|x_k - x\| > \varepsilon\}|)}{f(n)} = 0. \] For \(f(t) = t\), the corresponding \(f\)-statistical convergence coincides with the ordinary statistical convergence widely studied in past decades. Statistical and \(f\)-statistical convergences are particular cases of filter convergence or, in other terminology, of ideal convergence. The author proves that for a normed space \(X\) its completeness with respect to \(f\)-statistical convergence is equivalent to the usual completeness. Relation of weak \(f\)-statistical convergence with weak convergence on a subset of maximal \(f\)-density is studied. The last subsection is devoted to \(f\)-statistical cluster points.