# Math Help - Inverse function theorem

1. ## 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?

2. 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 $v\in\mathbb{R}^n$.