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 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) 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.

Asymmetric normed Baire space

“…Remark 1. The above theorem extends easily to asymmetric normed cone with the same proof, replacing "biBanach" by "bicomplete" (we refer to [7,5] for more informations about asymmetric normed cones and their duals).…”

“…We are going to prove in Theorem 3 bellow, that the condition c(X) = 0 in quasi-hemi-metric space, is however always equivalent to the fact that the semi-Lipschitz free space F a (X) over (X, d) (introduced recently in [7]) is not a Baire space. We recall below the notion of semi-Lipschitz free space and we refer to [7] for more details.…”

“…The semi-Lipschitz free space. The semi-Lipschitz free space over a quasihemi-metric space X d := (X, d) was recently introduced and studied in [7]. Its construction is analogous to the classical Lipschitz free space over a metric space introduced by Godefroy-Kalton in [12].…”

“…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 [4,8,15]. For an introduction and recent study of asymmetric free spaces (or semi-Lipschitz free spaces), we refer to the recent paper [7]. Definition 1.…”

“…However, we prove using our main result, that the semi-Lipschitz free space over any quasi-metric space is a Baire space if and only if the quasimetric of the space is equivalent to its symmetrized metric. The concept of semi-Lipschitz free space over a quasi-metric space was introduced recently by A. Daniilidis, J. M. Sepulcre and F. Venegas in [7], in the same way as the classical Lipschitz free space of Godefroy-Kalton in [12] (see also Section 3). It follows easily from the results of this note that if the semi-Lipschitz-free over a bicomplete quasi-metric space is a Baire space then it will be the same for the quasi-metric space.…”