Inverse function theorem: Difference between revisions

Revision as of 10:29, 18 July 2024

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=== Over a real closed field ===

The inverse function theorem also holds over a [[real closed field]] ''k'' (or more generally 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>. This can be shown directly as follows. Given a point <math>y</math> in <math>B(0, r/2)</math>, consider the function <math>P(x) = |f(x) - y|^2</math> defined on <math>B(0, r)</math>. If <math>P'(x) = 0</math>, then <math>0 = P'(x) = 2f'[f_1(x)(f - y_1 \dots f_n(x) - yy_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 be lie in <math>B(0, r)</math>. Since <math>P'(x_0) = 0</math>, <math>f(x_0) = y</math>, which proves the claimed inclusion. <math>\square</math>

==See also==