Inner products on ordered cones (Q2778269)

Taking a cone \({\mathcal P}\) (i.e., a set endowed with addition and scalar multiplication for non-negative real numbers), inner products \(\langle .,.\rangle: {\mathcal P}^2\to \overline{\mathbb R}\) \((=:{\mathbb R}\cup\{\infty \}\)) on cones \({\mathcal P}\) are considered in the paper under review. After proving some basic properties of those inner products (e.g, a version of the Cauchy-Schwarz inequality for inner products on cones), several examples are provided. Introducing then a (partial) order (i.e., a reflexive, transitive relation \(\leq\) that is compatible with the algebraic operations) on \({\mathcal P}\), the locally convex topological structure induced by an inner product on an ordered cone is investigated, and an analogue of the Riesz representation theorem in Hilbert space is shown. Finally, cones \({\mathcal P}\) where the scalar multiplication is extended to all numbers in \({\mathbb R}\) or \({\mathbb C}\) are considered (in the case at hand, the extended scalar multiplication does not follow all rules for a vector space). For such cones \({\mathcal P}\) with an extended scalar multiplication it is shown that the values of the inner product \(\langle.,.\rangle\) may be identified with convex subsets of \({\mathbb R}\) or \({\mathbb C}\), and furthermore, that linear functionals on \({\mathcal P}\) may be represented by Riemann-Stieltjes type integrals over the inner product.