Extensions of Asymmetric Norms to Linear Spaces

Abstract

Let M be a subset of a (real) linear space that is closed with respect to the sum of vectors and the product by nonnegative scalars. An asymmetric seminorm on M is a nonnegative and subbaditive positively homogeneous function q defined on M. Moreover, q is an asymmetric norm if in addition for every non zero element x such that -x belongs to M, q(x) or q(-x) are different from zero. Consider the linear expansion X of M. In this paper we characterize when (M,q) can be extended to an asymmetric normed linear space $(X,q^*)$, i.e. when there exists an asymmetric norm $q^*$ on X such that $q^*\midM = q$. As an application we study these extensions in the case of subsets of normed lattices.