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Let $(X,0)$ be a pointed metric space and let $\mathcal{F}(X)$ be the natural predual of ${\rm Lip}_0(X)$, the space of Lipschitz functions on $X$ that map $0$ to $0$; here $\mathcal{F}(X)$ is really the closed linear span of the Dirac delta functionals on ${\rm Lip}_0(X)$ and it goes under many names: Lipschitz-free space, Arens-Eells space, transportation cost space, 1-Wasserstein space, and many others.

Are there any natural criteria for weak compactness in $\mathcal{F}(X)$ that one could call `concrete'?

Equi-integrability of pre-weakly compact sets in $L_1$ is such a criterion.

My feeling is that some version of Prokhorov's theorem should be possible to apply in this context.

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  • $\begingroup$ Probably this is known to you and you're looking for something more tangible, but here's my 2 cents I can lay on the table: Kalton had given a criterion for weakly null sequences in Lemma 4.5 in eudml.org/doc/44337 that can be generalized for weakly precompact sets, although it doesn't characterize weakly precompact sets. For specific metric spaces we could perhaps say more using isomorphic Banach space properties of $F(X)$, for instance when $X$ is complete and countable, or when $X$ has the Heine-Borel property. $\endgroup$
    – Onur Oktay
    Commented Apr 14, 2022 at 20:27

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