inverse function theorem?
f(x)=l(x)+g(x) is function from R^n to R^n. If l(x) is linear isomorphism, f is continuously differentiable, and ||g(x)|| <= M ||x||^2 for all x, show f is invertible near zero.
I think I have to show that f is injective near 0, but l is isomorphism so it is injective. If g is injective near 0 too, then I am done? How to show this? Does the restriction on ||g(x)|| mean near zero that f(x) is almost same as l(x)? I do not know. To use inverse function theorem, I have to show that determinant of Jacobian matrix of f is not zero at zero, but I do not know what f(x) is exactly. Ack!