Bounded linear operators in PN-spaces

TL;DR

This paper introduces a new formulation of probabilistic normed spaces and extends fundamental functional analysis theorems like open mapping and Banach-Steinhaus to PN-spaces, enhancing the theoretical framework of probabilistic analysis.

Contribution

It presents a novel formulation of PN-spaces and proves key theorems in this setting, expanding the mathematical foundation of probabilistic functional analysis.

Findings

Established a new formulation of PN-spaces

Proved open mapping theorem in PN-spaces

Extended Banach-Steinhaus theorem to PN-spaces

Abstract

In this paper, first we present a new useful way of formulating probabilistic normed spaces. Then by using this formulation and probabilistic normed space version of the Baire category theorem, we prove four important results of functional analysis, i.e. the open mapping, closed graph, principle of uniform boundedness and Banach-Steinhaus theorem in PN-spaces.