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 Fa(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 Fa(X,d) coincides with the non-linear asymmetric dual of (X,d), that is, the space SLip0(X,d) of semi-Lipschitz 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 Fa(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.