Invertibility of Identity Transformation Plus Product of Two Continuous Linear Transformations

Theorem
Let $\struct {X, \norm{, \cdot ,} }$ be the normed vector space.

Let $I : X \to X$ be the identity mapping.

Let $\map {CL} X := \map {CL} {X, X}$ be the continuous linear transformation space on $X$.

Suppose $I + A \circ B$ is invertible, where $\circ$ denotes the composition of mappings.

Then $I + B \circ A$ is invertible, with the inverse given by:


 * $\paren {I + B \circ A}^{-1} = I - B \circ \paren {I + A \circ B}^{-1} \circ A$