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On ideal convergence in probabilistic normed spaces. (English) Zbl 1274.40034

Summary: An interesting generalization of statistical convergence is \(I\)-convergence which was introduced by P. Kostyrko, T. Šalát and W. Wilczyński [Real Anal. Exch. 26, 669–685 (2001; Zbl 1021.40001)]. In this paper, we define and study the concept of \(I\)-convergence, \(I^{*}\)-convergence, \(I\)-limit points and \(I\)-cluster points in a probabilistic normed space. We discuss the relationship between \(I\)-convergence and \(I^{*}\)-convergence, i.e., we show that \(I^{*}\)-convergence implies \(I\)-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, \(I\)-convergence does not imply \(I^{*}\)-convergence in probabilistic normed spaces.

MSC:

40J05 Summability in abstract structures
40A35 Ideal and statistical convergence
46S50 Functional analysis in probabilistic metric linear spaces

Citations:

Zbl 1021.40001
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