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Inverse Function Theorem via Mean Value Equality

  1. Nov 13, 2013 #1
    I know the mean value theorem for [itex]\mathbb{R}^1[/itex] expressed in terms of an equality doesn't strictly generalize to [itex]\mathbb{R}^n[/itex], as the example on page 2 of http://artsci.wustl.edu/~e511jn/InvFT.pdf [Broken] shows. The mean value inequality is used to prove the inverse function theorem, and (I think) because we're using the inequality form of the mean value theorem the proof's get quite complicated, requiring contraction mappings & Banach's fixed point theorem to alleviate the uncertainty introduced by the inequality. If we had some kind of mean value equality on [itex]\mathbb{R}^n[/itex] then the proof would be immensely simpler.

    I found a really old book which apparently claims to prove some form of the mean value theorem in terms of equalities, & then goes on to give an extremely simple proof of the inverse function theorem on [itex]\mathbb{R}^n[/itex] almost exactly following the lines of the proof in the [itex]\mathbb{R}^1[/itex] case.

    I don't know if the arguments are sound, I can't really follow the proof because the notation is old, & I don't know if the proof of the inverse function theorem is sound since no heavy machinery is invoked like the other proofs, so I'd like to post pictures of the proofs & get peoples input. I think this would be of interest to a lot of people so hopefully you guys can help me write out these proofs in detail & make up examples or counter-examples where necessary:

    https://www.physicsforums.com/attachment.php?attachmentid=63894&stc=1&d=1384331357

    https://www.physicsforums.com/attachment.php?attachmentid=63895&stc=1&d=1384331473

    attachment.php?attachmentid=63898&stc=1&d=1384331661.jpg

    attachment.php?attachmentid=63897&stc=1&d=1384331576.jpg

    attachment.php?attachmentid=63899&stc=1&d=1384331702.jpg

    attachment.php?attachmentid=63900&stc=1&d=1384332173.jpg

    attachment.php?attachmentid=63901&stc=1&d=1384332230.jpg

    attachment.php?attachmentid=63903&stc=1&d=1384332313.jpg
     

    Attached Files:

    Last edited by a moderator: May 6, 2017
  2. jcsd
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