Majorizing Norms on Majorized Spaces

TL;DR

This paper introduces a new class of norms for ordered vector spaces, explores their completeness properties, and generalizes key results in Banach lattice theory related to principal ideals and L-norms.

Contribution

It defines a novel norm for majorized spaces, establishes conditions for completeness, and extends classical Banach lattice results to broader contexts.

Findings

New norm generalizes order unit and base norms

Provides sufficient and necessary conditions for completeness

Generalizes results on AM-spaces and L-norms in Banach lattices

Abstract

We introduce a new type of norm for ordered vector spaces majorized by a proper (convex) cone that generalizes the notions of order unit norm and base norm. Then we give sufficient conditions to ensure its completeness. In the case of normed Riesz spaces, we also present necessary completeness conditions and describe the completed norm. Finally, we define a particular family of well-behaving sets. Spaces majorized by the cone generated by these sets display surprisingly interesting structural properties. In particular, we generalize two famous standard results in Banach lattice theory. Namely, every principal ideal in a Banach lattice is an AM-space and every Banach lattice having a generating cone with a compact base admits an equivalent L-norm.