##
**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 |

### Keywords:

\(t\)-norm; probabilistic normed space; \(I\)-convergence; \(I\)-limit points; \(I\)-cluster points### Citations:

Zbl 1021.40001
PDFBibTeX
XMLCite

\textit{M. Mursaleen} and \textit{S. A. Mohiuddine}, Math. Slovaca 62, No. 1, 49--62 (2012; Zbl 1274.40034)

Full Text:
DOI

### References:

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.