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The only few references I could find on this topic are either amateur blog posts (http://n.ethz.ch/~gbasso/download/A%20Hitchhikers%20guide%20to%20Wasserstein/A%20Hitchhikers%20guide%20to%20Wasserstein.pdf and https://vincentherrmann.github.io/blog/wasserstein/) or the 1000 page tome by Cedric Villani, (http://cedricvillani.org/wp-content/uploads/2012/08/preprint-1.pdf)

Is there any expository reference/lecture notes about this which is somewhere between the two kinds of references above? Any paper by maybe Cedric Villani himself which covers all these grounds?

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You might look at Chapter 3 of my book Lipschitz Algebras (second edition). The Banach space ${\rm Lip_0}(X)$ is already the dual of the space of finitely supported measures on $X$ satisfying $\mu(X) = 0$, equipped with Wasserstein distance (though I suppose it should then be called Arens-Eells distance). Going to Radon measures enlarges this space but you remain within the completion of the finitely supported measures, so its dual space doesn't change.

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Kantorovich himself wrote a very detailed exposition which constitutes Section 8.4 of Functional Analysis by Akilov and Kantorovich

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  • $\begingroup$ But beware --- this section only exists in the second edition of that book! $\endgroup$
    – Nik Weaver
    Feb 1, 2018 at 20:20
  • $\begingroup$ @NikWeaver - the one I linked - but I should have specified that - thank you! $\endgroup$
    – R W
    Feb 2, 2018 at 0:38

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