Lattice-Subspaces and Positive Bases in Function Spaces

Lattice-Subspaces and Positive Bases in Function Spaces Let x 1,...,xn be linearly independent, positive elements of the space RΩ of the real valued functions defined on a set Ω and let X be the vector subspace of RΩ generated by the functions x i. We study the problem: Does a finite-dimensional minimal lattice-subspace (or equivalently a finite-dimensional minimal subspace with a positive basis) of RΩ which contains X exist? To this end we define the function β(t)= $$\frac{1}{{z\left( t \right)}}$$ (x1(t),x2(t),...,xn(t)), where z(t)=x1(t)+x2(t)+...+xn(t), which we call basic function and takes values in the simplex Δn of $$\mathbb{R}_ + ^n$$ . 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 β 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 Ω is a convex set and x 1,x2,...,xn 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Lattice-Subspaces and Positive Bases in Function Spaces

, Volume 7 (4) – Oct 17, 2004
18 pages

/lp/springer_journal/lattice-subspaces-and-positive-bases-in-function-spaces-92GiE0uarb
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1026274014666
Publisher site
See Article on Publisher Site

Abstract

Let x 1,...,xn be linearly independent, positive elements of the space RΩ of the real valued functions defined on a set Ω and let X be the vector subspace of RΩ generated by the functions x i. We study the problem: Does a finite-dimensional minimal lattice-subspace (or equivalently a finite-dimensional minimal subspace with a positive basis) of RΩ which contains X exist? To this end we define the function β(t)= $$\frac{1}{{z\left( t \right)}}$$ (x1(t),x2(t),...,xn(t)), where z(t)=x1(t)+x2(t)+...+xn(t), which we call basic function and takes values in the simplex Δn of $$\mathbb{R}_ + ^n$$ . 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 β 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 Ω is a convex set and x 1,x2,...,xn 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.

Journal

PositivitySpringer Journals

Published: Oct 17, 2004

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