Some results on the Orlicz space generated from a random normed module

TL;DR

This paper introduces and analyzes an Orlicz space derived from a random normed module, establishing duality and convexity properties that link the space's structure to the underlying module.

Contribution

It provides the first dual space representation theorem for the Orlicz space from a random normed module and relates convexity properties of the space to those of the module.

Findings

Dual space of the Orlicz heart is identified with the Orlicz space from the conjugate space.

Equivalence of strict convexity between the Orlicz space and the underlying module.

Equivalence of uniform convexity between the Orlicz space and the underlying module.

Abstract

Noting the important role the abstract $L^p$ space has played in the development of random normed modules, in this paper we introduce and study the Orlicz space generated from a random normed module. First, we give a basic dual space representation theorem which identify the dual of the Orlicz heart of a random normed module with the Orlicz space generated from the random conjugate space. Then, we establish the respective equivalence relations of the strict convexity and uniform convexity of this abstract Orlicz space to the random strict convexity and random uniform convexity of the underlying random normed module. These results demonstrate that it is possible to use the Orlicz space theory in the further development of random nomed modules.