Neumann Series Theorem/Corollary 2

Theorem
Let $X$ be a Banach space.

Let $\map {CL} X$ be the continous linear transformation space.

Let $\norm {\, \cdot \,}$ be the supremum operator norm.

Let $A \in \map {CL} X$ be such that $\norm A < 1$.

Let $I$ be the identity mapping.

The mapping $\paren {I - A}^{-1} : X \to X$ is continuous.