The Arrow-Barankin-Blackwell theorem in a dual space setting

In 1953 Arrow, Barankin, and Blackwell proved that, ifR ⁿ is equipped with its natural ordering and ifF is a closed convex subset ofR ⁿ , then the set of points inF that can be supported by strictly positive linear functionals is dense in the set of all efficient (maximal) points ofF. Many generalizations of this density result to infinite-dimensional settings have been given. In this note, we consider the particular setting where the setF is contained in the topological dualY ^* of a partially ordered, nonreflexive normed spaceY, and the support functionals are restricted to be either nonnegative or strictly positive elements in the canonical embedding ofY inY ^*. Three alternative density results are obtained, two of which generalize a space-specific result due to Majumdar for the dual system (Y,Y ^*)=(L ¹,L ).This research was supported in part by funds provided by the Provident Chair of Excellence in Applied Mathematics at the University of Tennessee, Chattanooga, Tennessee.