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 $\mathcal{F}_a(X,d)$ (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 $\mathcal{F}_a(X,d)$ coincides with the nonlinear asymmetric dual of $(X,d)$, that is, the space $\mathrm{SLip}_0(X,d)$ 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 $(X,D)$ inherits naturally a canonical asymmetrization coming from its free space $\mathcal{F}(X)$. This gives rise to a quasi-metric space $(X,D_+)$ and an asymmetric free space $\mathcal{F}_a(X,D_+)$. The symmetrization of the latter is isomorphic to the original free space $\mathcal{F}(X)$. The results of this work are illustrated with explicit examples.