Compactness and D-boundedness in Menger's 2-probabilistic normed spaces

TL;DR

This paper explores convexity, boundedness, and compactness concepts in Menger's 2-probabilistic normed spaces, extending the understanding of their structural properties and introducing the notion of D-boundedness.

Contribution

It introduces and analyzes convex series closedness, convex series compactness, and D-boundedness in Menger's 2-probabilistic normed spaces, expanding existing theoretical frameworks.

Findings

Established relationships between convex series closedness and compactness.

Defined and discussed D-boundedness in Menger's 2-probabilistic normed spaces.

Extended concepts to Menger's generalized 2-probabilistic normed spaces.

Abstract

The idea of convex sets and various related results in 2-Probabilistic normed spaces were established in [HR]. In this paper, We obtain the concepts of convex series closedness, convex series compactness, boundedness and their interrelationships in Menger's 2-probabilistic normed space. Finally, the idea of $ \mathcal{D}- $ Boundedness in Menger's 2-probabilistic normed spaces and Menger's Generalized 2-Probabilistic Normed spaces are discussed.