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 quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasimetric 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 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 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.

Index of symmetry and topological classification of asymmetric normed spaces

“…Quasi-metric spaces and asymmetric norms have recently attracted a lot of interest in modern mathematics, they arise naturally when considering non-reversible Finsler manifolds [10,5,13]. For an introduction and study of asymmetric free spaces (or semi-Lipschitz free spaces), we refer to the recent paper [6].…”