A dual representation theorem on the conditional Orlicz space generated from a random normed module

TL;DR

This paper introduces a new dual representation theorem for conditional Orlicz spaces derived from random normed modules, extending existing results in the field of functional analysis.

Contribution

It defines the notion of a random Orlicz function, constructs the associated conditional Orlicz space, and proves a dual representation theorem that generalizes prior work.

Findings

Denseness of the Orlicz heart in the space with respect to the $(\,varepsilon, \,lambda)$-topology.

Establishment of a dual representation theorem for the conditional Orlicz space.

Extension and improvement of known results in the theory of random normed modules.

Abstract

In this paper, we first introduce the notion of a random Orlicz function, and further present the conditional Orlicz space generated from a random normed module. Second, we prove the denseness of the Orlicz heart of a random normed module $E$ in $E$ with respect to the $(\varepsilon, \lambda)$-topology. Finally, based on the above work, we establish a dual representation theorem on the conditional Orlicz space generated from a random normed module, which extends and improves some known results.