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 $\mathcal{F}{M}$.

中文翻译：

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无Lipschitz的空间中的支撑点和极端点

对于完整的度量空间$ M $，我们证明了Lipschitz-free空间$ \ mathcal {F} {M} $的单位球的有限支撑极点恰好是基本分子$（\ delta（p）- \ delta（q））/ d（p，q）$由$ M $中的点对$ p，q $定义，使得三角形不等式$ d（p，q）<d（p，r）+ d（ q，r）$对于M $中与$ p $和$ q $不同的任何$ r \都是严格的。为此，我们证明，当$ M $具有有限直径时，在$ M $的闭合子集上的Lipschitz-free空间类别在任意交点处闭合，并且这允许对$ \ mathcal元素的支持进行自然定义{F} {M} $。