Editing Inverse function theorem (section)

==Generalizations==

===Banach spaces===
The inverse function theorem can also be generalized to differentiable maps between [[Banach space]]s ''{{Mvar|X}}'' and ''{{Mvar|Y}}''.<ref>{{cite book |first=David G. |last=Luenberger |author-link=David Luenberger |title=Optimization by Vector Space Methods |location=New York |publisher=John Wiley & Sons |year=1969 |isbn=0-471-55359-X |pages=240–242 |url=https://books.google.com/books?id=lZU0CAH4RccC&pg=PA240 }}</ref> Let ''{{Mvar|U}}'' be an open neighbourhood of the origin in ''{{Mvar|X}}'' and <math>F: U \to Y\!</math> a continuously differentiable function, and assume that the Fréchet derivative <math>dF_0: X \to Y\!</math> of ''{{Mvar|F}}'' at 0 is a [[bounded linear map|bounded]] linear isomorphism of ''{{Mvar|X}}'' onto ''{{Mvar|Y}}''. Then there exists an open neighbourhood ''{{Mvar|V}}'' of <math>F(0)\!</math> in ''{{Mvar|Y}}'' and a continuously differentiable map <math>G: V \to X\!</math> such that <math>F(G(y)) = y</math> for all ''{{Mvar|y}}'' in ''{{Mvar|V}}''. Moreover, <math>G(y)\!</math> is the only sufficiently small solution ''{{Mvar|x}}'' of the equation <math>F(x) = y\!</math>.

There is also the inverse function theorem for [[Banach manifold]]s.<ref>{{cite book |first=Serge |last=Lang |author-link=Serge Lang |title=Differential Manifolds |location=New York |publisher=Springer |year=1985 |isbn=0-387-96113-5 |pages=13–19 }}</ref>

===Constant rank theorem===
The inverse function theorem (and the [[implicit function theorem]]) can be seen as a special case of the constant rank theorem, which states that a smooth map with constant [[rank (differential topology)|rank]] near a point can be put in a particular normal form near that point.<ref name="boothby">{{cite book |first=William M. |last=Boothby |title=An Introduction to Differentiable Manifolds and Riemannian Geometry |url=https://archive.org/details/introductiontodi0000boot |url-access=registration |edition=Second |year=1986 |publisher=Academic Press |location=Orlando |isbn=0-12-116052-1 |pages=[https://archive.org/details/introductiontodi0000boot/page/46 46–50] }}</ref> Specifically, if <math>F:M\to N</math> has constant rank near a point <math>p\in M\!</math>, then there are open neighborhoods {{Mvar|U}} of {{Mvar|p}} and {{Mvar|V}} of <math>F(p)\!</math> and there are diffeomorphisms <math>u:T_pM\to U\!</math> and <math>v:T_{F(p)}N\to V\!</math> such that <math>F(U)\subseteq V\!</math> and such that the derivative <math>dF_p:T_pM\to T_{F(p)}N\!</math> is equal to <math>v^{-1}\circ F\circ u\!</math>. That is, {{Mvar|F}} "looks like" its derivative near {{Mvar|p}}.  The set of points <math>p\in M</math> such that the rank is constant in a neighborhood of <math>p</math> is an open dense subset of {{Mvar|M}}; this is a consequence of [[semicontinuity]] of the rank function. Thus the constant rank theorem applies to a generic point of the domain.

When the derivative of {{Mvar|F}} is injective (resp. surjective) at a point {{Mvar|p}}, it is also injective (resp. surjective) in a neighborhood of {{Mvar|p}}, and hence the rank of {{Mvar|F}} is constant on that neighborhood, and the constant rank theorem applies.

===Polynomial functions===
If it is true, the [[Jacobian conjecture]] would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a [[Jacobian determinant]] that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unknown whether this is true or false, even in the case of two variables. This is a major open problem in the theory of polynomials.

===Selections===
When <math>f: \mathbb{R}^n \to \mathbb{R}^m</math> with <math>m\leq n</math>, <math>f</math> is <math>k</math> times [[continuously differentiable]], and the Jacobian <math>A=\nabla f(\overline{x})</math> at a point <math>\overline{x}</math> is of [[rank (linear algebra)|rank]] <math>m</math>, the inverse of <math>f</math> may not be unique. However, there exists a local [[Choice function#Choice function of a multivalued map|selection function]] <math>s</math> such that <math>f(s(y)) = y</math> for all <math>y</math> in a [[neighborhood (mathematics)|neighborhood]] of <math>\overline{y} = f(\overline{x})</math>, <math>s(\overline{y}) = \overline{x}</math>, <math>s</math> is <math>k</math> times continuously differentiable in this neighborhood, and <math>\nabla s(\overline{y}) = A^T(A A^T)^{-1}</math> (<math>\nabla s(\overline{y})</math> is the [[Moore–Penrose pseudoinverse]] of <math>A</math>).<ref>{{cite book |last1=Dontchev |first1=Asen L. |last2=Rockafellar |first2=R. Tyrrell |title=Implicit Functions and Solution Mappings: A View from Variational Analysis |date=2014 |publisher=Springer-Verlag |location=New York |isbn=978-1-4939-1036-6 |page=54 |edition=Second}}</ref>

=== Over a real closed field ===
The inverse function theorem also holds over a [[real closed field]] ''k'' (or an [[O-minimal structure]]).<ref>Theorem 2.11. in {{cite book |doi=10.1017/CBO9780511525919|title=Tame Topology and O-minimal Structures. London Mathematical Society lecture note series, no. 248|year=1998 |last1=Dries |first1=L. P. D. van den |authorlink = Lou van den Dries|isbn=9780521598385|publisher=Cambridge University Press|location=Cambridge, New York, and Oakleigh, Victoria }}</ref> Precisely, the theorem holds for a semialgebraic (or definable) map between open subsets of <math>k^n</math> that is continuously differentiable.

The usual proof of the IFT uses Banach's fixed point theorem, which relies on the Cauchy completeness. That part of the argument is replaced by the use of the [[extreme value theorem]], which does not need completeness. Explicitly, in {{section link||A_proof_using_the_contraction_mapping_principle}}, the Cauchy completeness is used only to establish the inclusion <math>B(0, r/2) \subset f(B(0, r))</math>. Here, we shall directly show <math>B(0, r/4) \subset f(B(0, r))</math> instead (which is enough). Given a point <math>y</math> in <math>B(0, r/4)</math>, consider the function <math>P(x) = |f(x) - y|^2</math> defined on a neighborhood of <math>\overline{B}(0, r)</math>. If <math>P'(x) = 0</math>, then <math>0 = P'(x) = 2[f_1(x) - y_1 \cdots f_n(x) - y_n]f'(x)</math> and so <math>f(x) = y</math>, since <math>f'(x)</math> is invertible. Now, by the extreme value theorem, <math>P</math> admits a minimal at some point <math>x_0</math> on the closed ball <math>\overline{B}(0, r)</math>, which can be shown to lie in <math>B(0, r)</math> using <math>2^{-1}|x| \le |f(x)|</math>. Since <math>P'(x_0) = 0</math>, <math>f(x_0) = y</math>, which proves the claimed inclusion. <math>\square</math>

Alternatively, one can deduce the theorem from the one over real numbers by [[Tarski's principle]].{{citation needed|date=December 2024}}