Yudin Cones and Inductive Limit Topologies

Abstract

A cone C in a vector space has a Yudin basis {ei}i∈I if every c ∈ C can be written uniquely in the form c = Σi∈Iλiei, where λi ≥ 0 for each i ∈ I and λi = 0 for all but finitely many i. A Yudin cone is a cone with a Yudin basis. Yudin cones arise naturally since the cone generated by an arbitrary family {ei}i∈I of linearly independent vectors C = {∑i∈Iλiei: λi ≥ 0 for each I and λi = 0 for all but finitely many i} is always a Yudin cone having the family { ei}iEJ as a Yudin basis. The Yudin cones possess several remarkable order and topological properties. Here is a list of some of these properties. 1. A Yudin cone C is a lattice cone in the vector subspace it generates M = C - C. 2. A closed generating cone in a two-dimensional vector space is always a Yudin cone. 3. If the cone of a Riesz space is a Yudin cone, then the lattice operations of the space are given pointwise relative to the Yudin basis. 4. If a Riesz space has a Yudin cone, then the inductive limit topology generated by the finite dimensional subspaces is a Hausdorff order continuous locally convex-solid topology. 5. In a Riesz space with a Yudin cone the order intervals lie in finite dimensional Riesz subspaces (and so they are all compact with respect to any Hausdorff linear topology on the space). The notion of a Yudin basis originated in studies on the optimality and efficiency of competitive securities markets in the provision of insurance for investors against risk or price uncertainty.• It is a natural extension to incomplete markets of Arrow's notion of a basis for complete markets, i.e., markets where full insurance against risk can be purchased. The obtained results have immediate applications to competitive securities markets. Especially, they are sufficient for establishing the efficiency of stock markets as a means for insuring against risk or price uncertainty.