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Thread: 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 $\displaystyle T(x): R^n \rightarrow R^n$ be a linear isomorphism, $\displaystyle f(x)=T(x)+h(x)$ such that $\displaystyle \mid{h(x)}\mid \leq M\mid x \mid^2$, and $\displaystyle f \in C^1(R^n,R^n)$. I want to show $\displaystyle f$ is invertible in neighborhood of $\displaystyle x=0$. 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 $\displaystyle Df(x)$ is invertible at $\displaystyle x=0$. Since $\displaystyle T$ is linear, we have $\displaystyle DT(x).u = T(u)$ for all $\displaystyle x$ and $\displaystyle u$ and since $\displaystyle h=f-T$, $\displaystyle h\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$.
    From the hypothesis, we have $\displaystyle h(0)=0$ and using the definition of the differential we get that $\displaystyle Dh(0) = 0$. We conclude that $\displaystyle Df(0)v =T(v)$ for all $\displaystyle v\in\mathbb{R}^n$.
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