$\pmb{D}$-boundedness and $\pmb{D}$-compactness in finite dimensional probabilistic normed spaces

TL;DR

This paper investigates the properties of D-boundedness and D-compactness in finite-dimensional probabilistic normed spaces, establishing the equivalence of probabilistic norms and exploring their implications.

Contribution

It introduces and studies the concepts of D-boundedness and D-compactness, providing new insights into probabilistic normed spaces.

Findings

Probabilistic norms are equivalent in finite-dimensional spaces.

D-compactness and D-boundedness are characterized in these spaces.

Theoretical foundations for probabilistic normed space topology are strengthened.

Abstract

In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion of $D$-compactness and $D$-boundedness in probabilistic normed spaces.