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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 {\it P. Kostyrko}, {\it T. Šalát} and {\it 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.

40J05Summability in abstract structures
40A35Ideal and statistical convergence
46S50Functional analysis in probabilistic metric linear spaces
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