The Goldstine Theorem for asymmetric normed linear spaces

Abstract

It is well known that if ( X , q ) is an asymmetric normed linear space, then the function q s defined on X by q s ( x ) = max { q ( x ) , q ( − x ) } , is a norm on the linear space X. However, the lack of symmetry in the definition of the asymmetric norm q yields an algebraic asymmetry in the dual space of ( X , q ) . This fact establishes a significant difference with the standard results on duality that hold in the case of locally convex spaces. In this paper we study some aspects of a reflexivity theory in the setting of asymmetric normed linear spaces. In particular, we obtain a version of the Goldstine Theorem to these spaces which is applied to prove, among other results, a characterization of reflexive asymmetric normed linear spaces.