Majorizing Norms on Majorized Spaces

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.