Supports and extreme points in Lipschitz-free spaces

Abstract

For a complete metric space M , we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space \mathcal{F}{M} are precisely the elementary molecules (\delta(p)-\delta(q))/d(p,q) defined by pairs of points p,q in M such that the triangle inequality d(p,q) < d(p,r)+d(q,r) is strict for any r\in M different from p and q . To this end, we show that the class of Lipschitz-free spaces over closed subsets of M is closed under arbitrary intersections when M has finite diameter, and that this allows a natural definition of the support of elements of \mathcal{F}{M} .