# Implicit function theorem metric spaces

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.

• 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. – Suvrit Jan 5 '18 at 15:42
• @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 – almosteverywhere Jan 6 '18 at 4:24
• 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. – Suvrit Jan 6 '18 at 15:23
• Thanks for the reference but the link seems to be broken. Can you give me the name of the book? – almosteverywhere Jan 6 '18 at 15:32
• The link above has a spurious quotation mark " -- if you remove it the link shoudl work (name: Gradient flows in metric spaces ...) – Suvrit Jan 6 '18 at 17:25