The following exercise is from Deuflhard's "Newton Methods for Nonlinear Problems" :
Given a nonlinear $C^1$-mapping $F:X\to Y$ over some domain $D\subset X$ for Banach spaces $X$, $Y$, each endowed with some norm $||\cdot||$. Assume a Lipschitz condition of the form \begin{align*} ||F'(x)-F'(y)||\leq \gamma||x-y||\end{align*} for $x,y\in D$. Let the derivative at some point $x^0$ have a bounded inverse with \begin{align*} ||F'(x^0)^{-1}||\leq \beta_0.\end{align*} Show that then, for all arguments $x\in D$ in some open ball $S(x^0,\rho)$ with $\rho=\frac{1}{\beta_0\gamma}$, there exists a bounded derivative inverse with \begin{align*} ||F'(x)^{-1}||\leq \dfrac{\beta_0}{1-\beta_0\gamma||x-x_0||}.\end{align*}
I am fairly new to operator theory. I understand the hypotheses, but I don't really understand the relationship between the norms of operators and their inverses. Any hints at how to proceed with this exercise will be much appreciated.
As a consequence of the Banach Perturbation lemma, one has
If $A$ and $B$ are matrices such that $A$ nonsingular, $||A^{-1}||<\theta$, and $||A||^{-1}\,||B||<1$, then $A+B$ is nonsingular and
$||B^{-1}||\leq \dfrac{\theta}{1-\theta||B||}$.
In the case above, take $A=F'(x_0)$ and $B=F'(x)-F'(x^0)$. Then, we have $||A^{-1}||\,||B||\leq \beta_0\gamma||x-x^0||$. If $x\in S(x^0,\rho)$ where $\rho=1/\beta_0\gamma$, one has $||A^{-1}||\,||B||<1$, therefore $A+B$ is nonsingular with bound given above. Substituting in $F'$ for $A$ and $B$, we have
$||F'(x)^{-1}||\leq \dfrac{\beta_0}{1-\beta_0\gamma||x-x^0||}$.