Separation of Convex Sets and Best Approximation in Spaces with Asymmetric Norm

Abstract

The aim of the present paper is to show that some results in Banach spaces have their analogs in spaces with asymmetric seminorms. A space with asymmetric seminorm is a pair (X, p),where X is a real vector space and p a positive sublinear functional onX. Due to the asymmetry of p (it is possible that p(– x) ≠ p(x) for somex), there are differences between the symmetric (seminormed) and the asymmetric case. For instance, the set Xp b of all linear p-bounded functionals on (X, p) need not be a vector space, but rather a cone in the algebraic dual of X. For a study of Xp and of other properties of spaces with asymmetric seminorm, see the paper by L. M. Garcia-Raffi, S. Romaguera and E. A. Sánchez-Pérez, Quaest. Math. 26 (2003), 83–96, and the references quoted therein. We continue this investigation by proving some extension results for bounded linear functionals, separation results for convex sets and a Krein-Milman type theorem. As applications, some duality results for best approximation by elements of convex subsets and of subsets with bounded convex complement, are obtained.