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Math Help - Inverse function theorem

  1. #1
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    Inverse function theorem

    I'm having some problem with applying the inverse function theorem to prove this. Let T(x): R^n \rightarrow R^n be a linear isomorphism, f(x)=T(x)+h(x) such that \mid{h(x)}\mid \leq M\mid x \mid^2, and f \in C^1(R^n,R^n). I want to show f is invertible in neighborhood of x=0. Can someone help me with this?
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  2. #2
    Super Member girdav's Avatar
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    You have to show that the differential Df(x) is invertible at x=0. Since T is linear, we have DT(x).u = T(u) for all x and u and since h=f-T, h\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n).
    From the hypothesis, we have h(0)=0 and using the definition of the differential we get that Dh(0) = 0. We conclude that Df(0)v =T(v) for all v\in\mathbb{R}^n.
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