# Homotopy, contraction mapping and the inverse function theorem on Banach spaces

We all know the "open mapping" part of the inverse function theorem on Euclidean spaces can be proved by either contraction mapping (iterations), or homotopy methods (degree theory / algebraic topology etc.). Yet the inverse function theorem on Banach spaces can only be proved by iteration methods, and homotopy methods fail, mostly due to Kuiper's theorem.

My question is: is there any proof of the inverse function theorem on Banach spaces that does not use iteration methods? It's sad to think the homotopy methods are a geometric coincidence of finite-dimensional spaces. The closest thing I know that could extend the homotopy methods would be Leray-Schauder degree theory, but that is restricted to maps of the form $$\mathrm{Id}-K$$ where $$K$$ is a continuous map that maps the unit ball to a precompact set (essentially close to being finite-dimensional). We would need something else to handle general $$C^1$$ maps.

I am interested in this problem because there are questions in PDE, related in spirit to the inverse function theorem, where finding the right iteration methods can be very difficult. Though I don't have much hope that anyone has figured out a way past the limitations of Leray-Schauder, I just want to put this question out there to let people know about the problem and maybe share some thoughts.

• This paper seems to prove the inverse function theorem not by iterative method, although not using homotopy method you depicted.
– Z. M
May 1, 2021 at 22:36