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2014 | OriginalPaper | Buchkapitel

Normal Cones and Thompson Metric

verfasst von : Ştefan Cobzaş, Mircea-Dan Rus

Erschienen in: Topics in Mathematical Analysis and Applications

Verlag: Springer International Publishing

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Abstract

The aim of this paper is to study the basic properties of the Thompson metric d _T in the general case of a linear space X ordered by a cone K. We show that d _T has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of d _T, and some results concerning the topology of d _T, including a brief study of the d _T-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 d _T is 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 d _T and order-unit (semi)norms | ⋅ | _u are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although d _T and | ⋅ | _u are only topologically (and not metrically) equivalent on K _u, 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.

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Vorheriges Kapitel Extension Operators that Preserve Geometric and Analytic Properties of Biholomorphic Mappings

Nächstes Kapitel Functional Operators and Approximate Solutions of Functional Equations

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Titel: Normal Cones and Thompson Metric
verfasst von: Ştefan Cobzaş
Mircea-Dan Rus
Verlag: Springer International Publishing
Buch: Topics in Mathematical Analysis and Applications
Print ISBN: 978-3-319-06553-3

Electronic ISBN: 978-3-319-06554-0

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-06554-0_9

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