Real and complex linear extensions for locally convex cones
type: Item
instance of: scholarly article
title: Real and complex linear extensions for locally convex cones
zbMATH Open document ID: 0903.46012
author: Walter Roth
published in: Journal of Functional Analysis
publication date: 1999-01-07T00:00:00Z
review text: Locally convex (preordered) cones were introduced by the author and \textit{K. Keimel} [``Ordered cones and approximation'', Lect. Notes Math. 1517 (1992; Zbl 0752.41033)] in order to have a unifying concept, in which Korovkin-type theorems can be treated, in their definition of a cone there is no cancellation law (i.e. \(a+c= b+c\) implies \(a=b\)), therefore there is no embedding into a real vector space in general. A locally convex cone \(P\) is called linear over \(\mathbb{K}\) \((\in\{\mathbb{R},\mathbb{C}\})\) if the operation of \(\mathbb{R}^+\) on \(P\) is extended to \(\mathbb{K}\), such that right-distributivity, associativity, and a weakened form of left-distributivity \(((\alpha+\beta)a\leq \alpha a+\beta a)\) is valid; if, furthermore, a certain condition concerning a modification of the given order holds, then \(P\) is called a locally convex cone over \(\mathbb{K}\). The first result of the paper shows, that every locally convex cone can be embedded into a locally convex cone over \(\mathbb{K}\). In the second part continuous linear functionals on these cones are investigated. After defining regular linear functionals a kind of integral representation theorem is proved. The theory is illustrated by the example formed by the set of non-empty closed convex sets of a locally convex ordered topological vector space.
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 52A01
zbMATH DE Number: 1117826
zbMATH Keywords: Korovkin-type theorems
zbMATH Keywords: right-distributivity
zbMATH Keywords: associativity
zbMATH Keywords: locally convex cone
zbMATH Keywords: regular linear functionals
zbMATH Keywords: integral representation
zbMATH Keywords: locally convex ordered topological vector space
MaRDI profile type: Publication
full work available at URL: jfan.1997.3142
OpenAlex ID: W2067602752
cites work: Hahn-Banach type theorems for hypolinear functionals
cites work: Q3935671
cites work: Ordered cones and approximation
cites work: Q4952824
cites work: On Positive Contractions In L p -Spaces
DOI: 10.1006/JFAN.1997.3142
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description: scientific article; zbMATH DE number 1117826