A Riesz Representation Theorem for Cone-valued Functions

The theory of locally convex cones, as developed in [3], deals with ordered cones that are not necessarily embeddable in vector spaces. A topological structure is introduced using order theoretical concepts. We will review some of the main concepts and globally refer to [3] for details and proofs. An ordered cone is a set endowed with an addition and a scalar multiplication for nonnegative real numbers. The addition is associative and commutative, and there is a neutral element 0 ∈ . 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 = 0 for all a,b ∈ 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 may be embedded into a real vector space. Also, carries a (partial) order, that is, a reflexive transitive relation ≤ such that a ≤ b implies a+c ≤ b+c and αa ≤ αb for all a,b,c ∈ and α ≥ 0. As equality in is such a relation, all results about ordered cones apply to cones without order structures as well. A linear functional on a cone is a mapping μ : → R = R∪{+∞} such that μ(a+ b) = μ(a)+μ(b) and μ(αa) = αμ(a) for all a,b ∈ and α ≥ 0. In R we consider the usual algebraic operations, in particular α +∞ = +∞ for all α ∈ R, α · (+∞) = +∞ for all α > 0 and 0 · (+∞) = 0. Note that linear functionals can assume only finite values at invertible elements of .