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Article Dans Une Revue Journal of Mathematical Analysis and Applications Année : 2020

Supports in Lipschitz-free spaces and applications to extremal structure

Résumé

We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space M is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space F (M). We then use this concept to study the extremal structure of F (M). We prove in particular that (δ(x) − δ(y))/d(x, y) is an exposed point of the unit ball of F (M) whenever the metric segment [x, y] is trivial, and that any extreme point which can be expressed as a finitely supported perturbation of a positive element must be finitely supported itself. We also characterise the extreme points of the positive unit ball: they are precisely the normalized evaluation functionals on points of M .
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Dates et versions

hal-02974712 , version 1 (22-10-2020)

Identifiants

Citer

Ramón J. Aliaga, Eva Pernecka, Colin Petitjean, Antonin Prochazka. Supports in Lipschitz-free spaces and applications to extremal structure. Journal of Mathematical Analysis and Applications, 2020, 489 (1), pp.124128. ⟨10.1016/j.jmaa.2020.124128⟩. ⟨hal-02974712⟩
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