Asymmetric norms, cones and partial orders
type: Item
instance of: scholarly article
title: Asymmetric norms, cones and partial orders
zbMATH Open document ID: 1344.46003
author: Jurie Conradie
published in: Topology and its Applications
publication date: 2015-08-20T00:00:00Z
review text: This interesting paper is devoted to the study of the relations among asymmetric norms and orders in linear spaces. Although some of the main examples of asymmetric norms are defined by means of orderings using the lattice structure of some classical Banach spaces, no systematic study of asymmetric linear lattices was done before. The author is able to show how this topic can be understood in a unified framework, introducing the main mathematical tools that allow to do this. It must be said that the main known applications of the theory of asymmetric normed spaces -- complexity analysis of algorithms, optimization --, use lattice-based asymmetric norms. The author defines three asymmetric norms relating cones, ordering and topology, and studies the relations between them. An order provides a cone, and the cone together with a norm provides asymmetric norms that are closely related to the normed lattice structure; the reciprocal relations are essentially true and are also analyzed. Some applications and nice examples are given. The last part of the paper is devoted to the ordering associated to what is called Yudin cone, as a part of a specific study of finite-dimensional spaces with lattice asymmetric norms.
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46B40
zbMATH DE Number: 6474003
zbMATH Keywords: asymmetric norms
zbMATH Keywords: cone
zbMATH Keywords: partially ordered vector space
zbMATH Keywords: vector lattice
MaRDI profile type: Publication
full work available at URL: j.topol.2015.06.007
OpenAlex ID: W767580019
cites work: Compactness in asymmetric normed spaces
cites work: Q5293839
cites work: One-parameter semigroups of positive operators
cites work: Functional Analysis in Asymmetric Normed Spaces
cites work: Quasi-uniform structures in linear lattices
cites work: Q5182141
cites work: Compactness and finite dimension in asymmetric normed linear spaces
cites work: Q3146072
cites work: Q4459334
DOI: 10.1016/J.TOPOL.2015.06.007
description: scientific article; zbMATH DE number 6474003