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

Abstract

Let X be a linear space over K, K=R or K=C and let for n>1 ρ_i be s-convex semimodular defined on X for any i\in{1,...,n-1}. Put ρ=\max_{1\leq i \leq n-1}\{ρ_i\} and X_ρ= { x \in X: ρ(dx) < \infty for some d > 0 }. In this paper we define a new class of s-norms (norms if s=1) on X_ρ. 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.

28 pages