ON a certain class of norms in semimodular spaces and their monotonicity properties

TL;DR

This paper introduces a new class of s-norms in semimodular spaces, generalizing existing norms like Orlicz-Amemiya and Luxemburg norms, and studies their monotonicity and continuity properties.

Contribution

It defines a novel class of s-norms on semimodular spaces that extend classical norms and analyzes their key functional properties.

Findings

Established conditions for order continuity of the new s-norms

Proved the Fatou property for the constructed semimodular spaces

Analyzed various monotonicity properties of these spaces

Abstract

Let X be a linear space over K, K=R or K=C and let for n>1 \rho_i be s-convex semimodular defined on X for any i\in{1,...,n-1}. Put \rho=\max_{1\leq i \leq n-1}\{\rho_i\} and X_{\rho}= { x \in X: \rho(dx) < \infty for some d > 0 }. In this paper we define a new class of s-norms (norms if s=1) on X_{\rho}. In particular, our defintion generalizes in a natural way the Orlicz-Amemiya and Luxemburg norms defined for s-convex semimodulars. Then, we investigate order continuous, the Fatou Property and various monotonicity properties of semimodular spaces equipped with these s-norms.