Positivity 7: 267–284, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Lattice-Subspaces and Positive Bases in Function
IOANNIS A. POLYRAKIS
Department of Mathematics, National Technical University, 157 80 Athens, Greece
Received 1 July 2001; accepted 10 February 2002
Abstract. Let x
be linearly independent, positive elements of the space R
of the real
valued functions deﬁned on a set and let X be the vector subspace of R
generated by the functions
. We study the problem: Does a ﬁnite-dimensional minimal lattice-subspace (or equivalently a
ﬁnite-dimensional minimal subspace with a positive basis) of R
which contains X exist? To this end
we deﬁne the function β(t) =
(t), . . . , x
(t)),wherez(t) = x
which we call basic function and takes values in the simplex
. 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 is a convex set and x
linear functions. This includes the case where x
are positive elements of a Banach lattice, or more
general the case where x
are positive elements of an ordered space Y . Based on the linearity of the
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.
Suppose E is a vector lattice and D is a subset of E
. In the theory of ordered
spaces we are interested in a lattice subspace Y of E that contains D and that is as
‘close’ as possible to the linear subspace [D] generated by D. The sublattice S(D)
generated by D is a lattice-subspace which contains D but in general, it is a ‘big’
subspace which is ‘very far’ from [D].
Since the intersection of lattice-subspaces
is not always a lattice-subspace, we are looking for minimal lattice-subspaces con-
taining D and not for a minimum one. Also the class of lattice-subspaces is larger
than that of sublattices, therefore a minimal lattice-subspace, if it exists, is ‘closer’
to [D] than S(D).
In Theorem 2.5 of  it is proved that if τ is a Lebesgue linear topology on
E and the positive cone E
of E is τ -closed (especially if E is a Banach lattice
This article is dedicated to the memory of my Father Andreas.
In Example 3.18 of , [D] is three-dimensional subspace of C(), [D] is not a lattice-
subspace, S(D) is dense in C() and a four-dimensional lattice-subspace containing D exists.