Inverse function theorem: Difference between revisions

Revision as of 00:51, 31 October 2023

Revision as of 02:02, 31 October 2023

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For functions of more than one variable, the theorem states that if {{Mvar|f}} is a continuously differentiable function from an open subset <math>A</math> of <math>\mathbb{R}^n</math> into <math>\R^n</math>, and the [[total derivative|derivative]] <math>f'(a)</math> is invertible at a point {{Mvar|a}} (that is, the determinant of the [[Jacobian matrix and determinant|Jacobian matrix]] of {{Mvar|f}} at {{Mvar|a}} is non-zero), then there exist neighborhoods <math>U</math> of <math>a</math> in <math>A</math> and <math>V</math> of <math>b = f(a)</math> such that <math>f(U) \subset V</math> and <math>f : U \to V</math> is bijective.<ref name="Hörmander">Theorem 1.1.7. in {{cite book|title=The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis|series=Classics in Mathematics|first=Lars|last= Hörmander|author-link=Lars Hörmander|publisher=Springer|year= 2015|edition=2nd|

isbn= {{Format ISBN|9783642614972}}978-3-642-61497-2}}</ref> Writing <math>f=(f_1,\ldots,f_n)</math>, this means that the system of {{Mvar|n}} equations <math>y_i = f_i(x_1, \dots, x_n)</math> has a unique solution for <math>x_1, \dots, x_n</math> in terms of <math>y_1, \dots, y_n</math> when <math>x \in U, y \in V</math>. Note that the theorem ''does not'' say <math>f</math> is bijective onto the image where <math>f'</math> is invertible but that it is locally bijective where <math>f'</math> is invertible.

Moreover, the theorem says that the inverse function <math>f^{-1} : V \to U</math> is continuously differentiable, and its derivative at <math>b=f(a)</math> is the inverse map of <math>f'(a)</math>; i.e.,

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The proof above is presented for a finite-dimensional space, but applies equally well for [[Banach space]]s. If an invertible function <math>f</math> is C^k with <math>k>1</math>, then so too is its inverse. This follows by induction using the fact that the map <math>F(A)=A^{-1}</math> on operators is C^k for any <math>k</math> (in the finite-dimensional case this is an elementary fact because the inverse of a matrix is given as the [[adjugate matrix]] divided by its [[determinant]]).

<ref name="Hörmander" /><ref>{{cite book|title=Calcul Differentiel|language=fr|first=Henri|last= Cartan|author-link= Henri Cartan|publisher=[[Éditions Hermann|Hermann]]|year= 1971|isbn={{Format ISBN|9780395120330}}978-0-395-12033-0 |pages=55–61}}</ref> The method of proof here can be found in the books of [[Henri Cartan]], [[Jean Dieudonné]], [[Serge Lang]], [[Roger Godement]] and [[Lars Hörmander]].

=== A proof using the contraction mapping principle ===

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There is a version of the inverse function theorem for [[holomorphic map]]s.

{{math_theorem|name=Theorem|math_statement=<ref>{{harvnb|Griffiths|Harris|1978|loc=p. 18.}}</ref><ref>{{cite book |first1=K. |last1=Fritzsche |first2=H. |last2=Grauert |title=From Holomorphic Functions to Complex Manifolds |publisher=Springer |year=2002 |pages=33–36 |isbn={{Format ISBN|9780387953953}}978-0-387-95395-3 |url=https://books.google.com/books?id=jSeRz36zXIMC&pg=PA33 }}</ref> Let <math>U, V \subset \mathbb{C}^n</math> be open subsets such that <math>0 \in U</math> and <math>f : U \to V</math> a holomorphic map whose Jacobian matrix in variables <math>z_i, \overline{z}_i</math> is invertible (the determinant is nonzero) at <math>0</math>. Then <math>f</math> is injective in some neighborhood <math>W</math> of <math>0</math> and the inverse <math>f^{-1} : f(W) \to W</math> is holomorphic.}}

The theorem follows from the usual inverse function theorem. Indeed, let <math>J_{\mathbb{R}}(f)</math> denote the Jacobian matrix of <math>f</math> in variables <math>x_i, y_i</math> and <math>J(f)</math> for that in <math>z_j, \overline{z}_j</math>. Then we have <math>\det J_{\mathbb{R}}(f) = |\det J(f)|^2</math>, which is nonzero by assumption. Hence, by the usual inverse function theorem, <math>f</math> is injective near <math>0</math> with continuously differentiable inverse. By chain rule, with <math>w = f(z)</math>,

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* {{cite book |last1=Renardy |first1=Michael |last2=Rogers |first2=Robert C. |title = An Introduction to Partial Differential Equations | series = Texts in Applied Mathematics 13

| edition = Second |publisher = Springer-Verlag | ___location = New York |year = 2004 |pages = 337&ndash;338 |isbn = 0-387-00444-0 }}

* {{cite book |last = Rudin|first = Walter|author-link= Walter Rudin|title = Principles of mathematical analysis|url = https://archive.org/details/principlesofmath00rudi|url-access = registration|edition = Third |series = International Series in Pure and Applied Mathematics |publisher = McGraw-Hill Book | ___location = New York |year = 1976 |pages = [https://archive.org/details/principlesofmath00rudi/page/221 221]&ndash;223 | isbn={{Format ISBN|9780070856134}}978-0-07-085613-4 }}

* {{cite book |title=Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus |last1=Spivak|first1=Michael|title-link=Calculus on Manifolds (book)|publisher= Benjamin Cummings |year=1965 |isbn=0-8053-9021-9 |___location=San Francisco |author1-link=Michael Spivak }}