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arXiv · 1810.11278

Supports and extreme points in Lipschitz-free spaces

Ramón J. Aliaga·Eva Pernecká
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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 $\mathcal{F}(M)$.

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BibTeXRIS

Ramón J. Aliaga, Eva Pernecká. 2018-10-26. Supports and extreme points in Lipschitz-free spaces. https://doi.org/10.4171/rmi%2F1191

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