Positivity 6: 115–127, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
Integral Representations for Continuous Linear
Functionals in Operator-Initiated Topologies
Department of Mathematics, Universiti Brunei Darussalam, Gadong BE 1410, Brunei Darussalam
(Received 29 June 1999; accepted 27 November 2000)
Abstract. On a given cone (resp. vector space) Q we consider an initial topology and order induced
by a family of linear operators into a second cone P which carries a locally convex topology.
We prove that monotone linear functionals on Q which are continuous with respect to this initial
topology may be represented as certain integrals of continuous linear functionals on P. Based on
the Riesz representation theorem from measure theory, we derive an integral version of the Jordan
decomposition for linear functionals on ordered vector spaces.
Mathematics Subject Classiﬁcation: 46A40, 46G10
Key words: locally convex cones, positive linear functionals
Our result, though of interest for locally convex vector spaces, is best formulated
using the more general concept of locally convex cones. This theory, as developed
in , deals with ordered cones that are not necessarily embeddable in vector
spaces. A topological structure is introduced using order theoretical concepts. We
shall review some of the main concepts and globally refer to  for details and
An ordered cone is a set P endowed with an addition and a scalar multiplication
for non-negative real numbers. The addition is associative and commutative, and
there is a neutral element 0 ∈ P. For the scalar multiplication the usual associative
and distributive properties hold, that is α(βa) = (αβ)a, (α + β)a = αa + βa,
α(a + b) = αa + αb, 1a = a and 0a = 0foralla, b ∈ P and α, β 0. The
cancellation law, stating that a + c = b + c implies a = b, however, is not required
in general. It holds if and only if the cone P may be embedded into a real vector
space. Also, P carries a (partial) order, that is, a reﬂexive transitive relation that
is compatible with the algebraic operations, i.e. a b implies a + c b + c and
αa αb for all a, b,c ∈ P and α 0. As equality in P is obviously such a
relation, all results about ordered cones apply to cones without order structures as