Asymmetric free spaces and canonical asymmetrizations

Abstract

A construction analogous to that of Godefroy-Kalton for metric spaces allows to embed isometrically, in a canonical way, every quasi-metric space (X, d) to an asymmetric normed space Fa(X, d), its semi-Lipschitz free space. The quasi-metric free space satisfies a universal property (linearization of semi-Lipschitz functions). The (conic) dual of Fa(X, d) coincides with the nonlinear asymmetric dual of (X, d), that is, the space SLip0(X, d) of semi-Lipschitz functions on (X, d), vanishing at a base point. If (X, D) is a metric space, the above construction yieldsits usual free space. Based on this construction, every metric space (X, D) inherits naturally a canonical asymmetrization coming from its free space F(X). This gives rise to a quasi-metric space (X, D+) and an asymmetric free space Fa(X, D+). The symmetrization of the latter is isomorphic to the original free space F(X).