Inverse function theorem: Difference between revisions

Revision as of 08:50, 18 July 2024

Revision as of 09:04, 18 July 2024

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To prove existence, it can be assumed after an affine transformation that <math>f(0)=0</math> and <math>f^\prime(0)=I</math>, so that <math> a=b=0</math>.

By the [[Mean value theorem#Mean value theorem for vector-valued functions|mean value theorem for vector-valued functions]], for a (continuous and differentiable) function <math>u:[0,1]\to\mathbb R^m</math>, <math display="inline">\|u(1)-u(0)\|\le \sup_{0\le t\le 1} \|u^\prime(t)\|</math>. Setting <math>u(t)=f(x+t(x^\prime -x)) - x-t(x^\prime-x)</math>, it follows that

:<math>\|f(x) - f(x^\prime) - x + x^\prime\| \le \|x -x^\prime\|\,\sup_{0\le t \le 1} \|f^\prime(x+t(x^\prime -x))-I\|.</math>