Asymmetric free spaces and canonical asymmetrizations

Abstract

A construction analogous to that of Godefroy-Kalton for metric spaces allows one to embed isometrically, in a canonical way, every quasi-metric space (X, d) in an asymmetric normed space F-a (X, d) (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasi-metric free space satisfies a universal property (linearization of semi-Lipschitz functions). The (conic) dual of F-a (X, d) coincides with the non-linear asymmetric dual of (X, d), that is, the space SLip(0)(X, d) of semiLipschitz functions on (X, d), vanishing at a base point. In particular, for the case of a metric space (X, D), the above construction yields its usual free space. On the other hand, every metric space (X, D) naturally inherits 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 F-a (X, D+) . The symmetrization of the latter is isomorphic to the original free space F(X). The results of this work are illustrated with explicit examples.