Inverse function theorem: Difference between revisions

Revision as of 12:45, 18 July 2024

Revision as of 12:55, 18 July 2024

Line 104:

Here is a proof based on the [[contraction mapping theorem]]. Specifically, following T. Tao,<ref>Theorem 17.7.2 in {{cite book|mr=3310023|last1=Tao|first1=Terence|title=Analysis. II|edition=Third edition of 2006 original|series=Texts and Readings in Mathematics|volume=38|publisher=Hindustan Book Agency|___location=New Delhi|year=2014|isbn=978-93-80250-65-6|zbl=1300.26003}}</ref> it uses the following consequence of the contraction mapping theorem.

{{math_theorem|name=Lemma|math_statement=Let <math>B(0, r)</math> denote an open ball of radius ''r'' in <math>\mathbb{R}^n</math> with center 0. Ifand <math>g : B(0, r) \to \mathbb{R}^n</math> is a map such that there existswith a constant <math>0 < c < 1</math> such that

:<math>|g(y) - g(x)| \le c|y-x|</math>

for all <math>x, y</math> in <math>B(0, r)</math>,. thenThen for <math>f = I + g</math> on <math>B(0, r)</math>, we have

:<math>(1-c)|x - y| \le |f(x) - f(y)|,</math>

in particular, ''f'' is injective. If, moreover, <math>g(0) = 0</math>, then