User:Caliburn/s/fa/2/Corollary

Corollary
Let $\struct {X, \norm \cdot_X}$ be a Banach space.

Let $\norm \cdot$ be the norm of a bounded linear transformation. Let $A : X \to X$ be an invertible bounded linear operator with bounded inverse $A^{-1} : X \to X$.

Let $B : X \to X$ be an invertible bounded linear operator with $\norm B \norm {A^{-1} } < 1$.

Then:


 * $(1) \quad$ $A + B$ is invertible with inverse $\paren {A + B}^{-1}$
 * $(2) \quad$ $\paren {A + B}^{-1}$ is bounded.
 * $(3) \quad$ $\norm {\paren {A + B}^{-1} } \le \norm {A^{-1} } \paren {1 - \norm {A^{-1} } \norm B}^{-1}$