JournalsrmiVol. 36, No. 7pp. 2073–2089

Supports and extreme points in Lipschitz-free spaces

  • Ramón J. Aliaga

    Universidad Politècnica de València, Valencia, Spain
  • Eva Pernecká

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

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Ramón J. Aliaga, Eva Pernecká, Supports and extreme points in Lipschitz-free spaces. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2073–2089

DOI 10.4171/RMI/1191