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.

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