 Newton iteration in a Banach space

For a $C^2$ map $\Phi$ on a Banach space, define $N(p)=p-[D\Phi (p){]}^{-1}\Phi (p)$ wherever the bounded derivative is invertible. It solves the local affine approximation to $\Phi =0$. Every zero is a fixed point of $N$. Near a zero with invertible derivative, bounded inverses and a locally Lipschitz derivative give the quadratic Newton error bound in a Banach space.

// Target: numerical-analysis.bigb