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Abstract

$L$-norms and $M$-norms on Banach lattices, unit-norms and base norms on ordered vector spaces are well known. In this paper $m$- and $m_\le$-norms are introduced on ordered normed spaces. They generalize the notions of the $M$-norm and the order-unit norm, possess also some interesting properties and can be characterized by means of the constants of reproducibility of cones. In particular, the dual norm of an ordered Banach space with a closed cone turns out to be additive on the dual cone if and only if the norm of the Banach space is an $m_\le$-norm and, on the other hand, the norm of an ordered normed space with a reproducing cone is an $L$-norm if and only if the dual norm is an $m_\le$-norm. Conditions are given for the operator norm to be an $m_\le$- or an $L$-norm.