Normal cones and Thompson metric

Abstract

The aim of this paper is to study the basic properties of the Thompson metric dTin the general case of a linear space X ordered by a cone K. We show that dThas monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of dT; and some results concerning the topology of dT; including a brief study of the dT-convergence of monotone sequences. It is shown that most results are true without any assumption of an Archimedean-type property for K. One considers various completeness properties and one studies the relations between them. Since dTis defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. Thompson metric dTand order-unit (semi)norms |·|u are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although dTand |·|uare only topologically (and not metrically) equivalent on Ku, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.