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Are there versions of the implicit function theorem in spaces that lack a natural linear structure, e.g. metric spaces. A quick google search has found me no results.

I am specifically interested in applying such results to the space of probability measures on some Polish space equipped with the $p$-Wasserstein metric.

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  • $\begingroup$ My guess is: since the IFT may be viewed as an application of fixed-point theory, and fixed-point theory has been studied in general metric spaces, there should be some version of this possible in your settings. $\endgroup$
    – Suvrit
    Commented Jan 5, 2018 at 15:42
  • $\begingroup$ @suvrit - I agree with your comment. It's just that one would need some object in a general metric space to fulfil the role played by the Frechet derivative in a Banach space $\endgroup$
    – almosteverywhere
    Commented Jan 6, 2018 at 4:24
  • $\begingroup$ Have a look at the book "springer.com/us/book/9783764387211" for doing "gradients" in metric spaces; Wasserstein spaces etc. are a special focus too. $\endgroup$
    – Suvrit
    Commented Jan 6, 2018 at 15:23
  • $\begingroup$ Thanks for the reference but the link seems to be broken. Can you give me the name of the book? $\endgroup$
    – almosteverywhere
    Commented Jan 6, 2018 at 15:32
  • $\begingroup$ The link above has a spurious quotation mark " -- if you remove it the link shoudl work (name: Gradient flows in metric spaces ...) $\endgroup$
    – Suvrit
    Commented Jan 6, 2018 at 17:25

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