On the locally biorder-convex space and super efficiency
type: Item
instance of: scholarly article
title: On the locally biorder-convex space and super efficiency
zbMATH Open document ID: 0931.46007
author: Qiusheng Qiu
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1999-11-17T00:00:00Z
review text: A topological vector space is said to be ordered if a convex cone of positive elements is specified. Standard references for such spaces are the books by \textit{H. H. Schaefer} [``Topological vector spaces'', Macmillan (1966; Zbl 0141.30503)] and \textit{G. Jameson} [``Ordered linear spaces'', Lect. Notes Math. 141 (1970; Zbl 0196.13401)]. Key notions in the subject are order intervals, i.e. \([a,b]:=\{x:a\leq x\leq b\}=(a+K)\cap(b-K)\), normal cones and the decomposition property. A basic theorem connects the normality of \(K\) with the decomposition property for the dual space. In this paper the author considers a bi-ordered space for which two convex cones, \(C_1\) and \(C_2\) are specified. These are used to generalize the notion of order interval to: \[ [a,b]^{C_1}_{C_2}:=\{x:a\leq_{C_1}x\leq_{C_2}b\}=(a+C_1)\cap(b-C_2). \] This leads to a generalization of normality as an interesting relationship between \(C_1\) and \(C_2\), and a generalization of the theorem mentioned above. The paper is concisely written with the interplay between geometry and analysis clearly presented.
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 1283805
zbMATH Keywords: ordered space
zbMATH Keywords: bi-ordered space
zbMATH Keywords: positive decomposition
zbMATH Keywords: normal cone
zbMATH Keywords: order intervals
zbMATH Keywords: decomposition property
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1283805