Lattice-Subspaces and Positive Bases in Function Spaces

Polyrakis, IA

dc.contributor.author	Polyrakis, IA	en
dc.date.accessioned	2014-03-01T01:19:06Z
dc.date.available	2014-03-01T01:19:06Z
dc.date.issued	2003	en
dc.identifier.issn	1385-1292	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/15379
dc.subject	banach lattice	en
dc.subject	Convex Hull	en
dc.subject	Convex Set	en
dc.subject	Function Space	en
dc.subject	linear functionals	en
dc.subject	Value Function	en
dc.subject.classification	Mathematics	en
dc.title	Lattice-Subspaces and Positive Bases in Function Spaces	en
heal.type	journalArticle	en
heal.identifier.primary	10.1023/A:1026274014666	en
heal.identifier.secondary	http://dx.doi.org/10.1023/A:1026274014666	en
heal.language	English	en
heal.publicationDate	2003	en
heal.abstract	Let x(1),..., x(n) be linearly independent, positive elements of the space R-Omega of the real valued functions defined on a set Omega and let X be the vector subspace of R-Omega generated by the functions xi. We study the problem: Does a finite-dimensional minimal lattice-subspace ( or equivalently a finite-dimensional minimal subspace with a positive basis) of R-Omega which contains X exist? To this end we define the function beta(t) = 1/z(t) (x(1)(t), x(2)( t),..., x(n)(t)), where z( t) = x(1)( t)+ x(2)(t)+...+ x(n)(t), which we call basic function and takes values in the simplex Delta(n) of R-+(n). We prove that the answer to the problem is positive if and only if the convex hull K of the closure of the range of beta is a polytope. Also we prove that X is a lattice-subspace (or equivalently X has positive basis) if and only if, K is an (n- 1)-simplex. In both cases, using the vertices of K, we determine a positive basis of the minimal lattice-subspace. In the sequel, we study the case where Omega is a convex set and x(1), x(2),..., x(n) are linear functions. This includes the case where x(i) are positive elements of a Banach lattice, or more general the case where x(i) are positive elements of an ordered space Y. Based on the linearity of the functions x(i) we prove some criteria by means of which we study if K is a polytope or not and also we determine the vertices of K. Finally note that finite dimensional lattice-subspaces and therefore also positive bases have applications in economics.	en
heal.publisher	KLUWER ACADEMIC PUBL	en
heal.journalName	Positivity	en
dc.identifier.doi	10.1023/A:1026274014666	en
dc.identifier.isi	ISI:000186047300001	en
dc.identifier.volume	7	en
dc.identifier.issue	4	en
dc.identifier.spage	267	en
dc.identifier.epage	284	en