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In Spivak's "Calculus on Manifolds", his proof is almost coordinate free. I think his proof could be altered (as well as preceding results that he uses) basically by using a different metric to produce bounds. I'm pretty sure this is doable, and I'm going to write it up (so no spoilers, please!). But I'd like to make sure it's correct, and I'd like to see different points of view, of course.

I'd like to know: Are there any texts that prove the Inverse Function Theorem as coordinate-freely as possible?

As a side note: I am not trying to avoid coordinates per-se for my intentions. Rather, I am trying to find different points of view for various basic constructions.

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  • $\begingroup$ "Baby Rudin" is a good book. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 11, 2019 at 19:43
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    $\begingroup$ The inverse function theorem holds for functions between Banach spaces. The essential ingredient in the proof is the Banach fixed point theorem. In this situation, the proof must needs be coordinate-free. Off-hand, I am fairly sure that this can be found, for example, in Dieudonné‘s multi-volumed „Treatise on Analysis“. $\endgroup$
    – user131781
    Commented Nov 11, 2019 at 19:53
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    $\begingroup$ why so much hate for coordinates... $\endgroup$
    – Abdelmalek Abdesselam
    Commented Nov 11, 2019 at 20:12
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    $\begingroup$ Or look up the implicit function theorem in any text on functional analysis, where it will be proved for infinite dimensional Banach spaces. The proof is virtually the same. $\endgroup$
    – Deane Yang
    Commented Nov 11, 2019 at 21:55
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    $\begingroup$ This seems like a great goal to further your understanding, but very much not a research-level question, so that it does not belong here. $\endgroup$
    – LSpice
    Commented Nov 12, 2019 at 3:26

2 Answers 2

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A coordinate-free proof of the inverse function theorem in the finite-dimensional case is provided by Theorem 19.6 in "Topological Geometry" by Ian R. Porteous.

In general, the cited book is an exposition of multivariable calculus in a coordinate-free manner.

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  • $\begingroup$ Dmitri thank you for the reference. It looks like it's got other stuff I've been wanting to learn about as well. $\endgroup$
    – ZxJx
    Commented Nov 11, 2019 at 20:38
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Since you wish to do it by yourself, here are some vague hints. The best way to prove the inverse mapping theorem is to start with a metric lemma based on the contraction principle: for an open subset $\Omega$ of a Banach space $E$ and a contracting map $g:\Omega\to E$, the map $I+g$ is a bilipschitz homeomorphism onto the open set $(I+g)(\Omega)$. In the second application listed here I put a more precise statement, which should lead you to a proof. You then apply the lemma to the situation of the inverse mapping theorem assuming w.l.o.g. that the differential of your function at a point $x_0$ is not only invertible, but in fact the identity, and taking as $\Omega$ a sufficiently small ball centered at $x_0$.

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