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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Finite-dimensional lattice-subspaces
of $C(\Omega )$ and curves of $\mathbb R^n$


Author: Ioannis A. Polyrakis
Journal: Trans. Amer. Math. Soc. 348 (1996), 2793-2810
MSC (1991): Primary 46B42, 52A21, 15A48, 53A04
MathSciNet review: 1355300
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Abstract: Let $x_1,\dotsc ,x_n$ be linearly independent positive functions in $C(\Omega )$, let $X$ be the vector subspace generated by the $x_i$ and let $\beta $ denote the curve of $\mathbb R^n$ determined by the function $\beta (t)=\frac {1}{z(t)} (x_1(t),x_2(t),\dotsc ,x_n(t))$, where $z(t)=x_1(t)+x_2(t)+\dotsb +x_n(t)$. We establish that $X$ is a vector lattice under the induced ordering from $C(\Omega )$ if and only if there exists a convex polygon of $\mathbb R^n$ with $n$ vertices containing the curve $\beta $ and having its vertices in the closure of the range of $\beta $. We also present an algorithm which determines whether or not $X$ is a vector lattice and in case $X$ is a vector lattice it constructs a positive basis of $X$. The results are also shown to be valid for general normed vector lattices.


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Additional Information

Ioannis A. Polyrakis
Affiliation: Department of Mathematics, National Technical University, 157 80 Athens, Greece
Email: ypoly@math.ntua.gr

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01639-X
PII: S 0002-9947(96)01639-X
Received by editor(s): April 24, 1995
Additional Notes: This research was supported in part by the NATO Collaborative Research Grant #941059.
Article copyright: © Copyright 1996 American Mathematical Society