Let $B$ be a Banach space of dimension $n$ over $\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$ or $\mathbb{C} $) and $C:B \rightarrow \mathbb{K}^{n}$ a linear operator. Let define $A:B/$~$ \rightarrow \mathbb{R}^{n}$ such that $$ A(x)_j = |C(x)_j|, $$ and $x$~$y$ iff $x=cy$ and $|c|=1$.
Suppose that A is bijective and $A^{-1}$ is continuous, then $A^{-1}$ is Lipschitz.
I can prove that $A^{-1}$ is uniformly continuous in $\overline{B(0,1)}$, but i don't know how to find the lipschitz constant.