Definition:Invertible Continuous Linear Operator

Definition
Let $\struct {X, \norm {\, \cdot\,}_X}$ be a normed vector space.

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

Let $I \in \map {CL} X$ be the identity element.

Let $A \in \map {CL} X$.

Suppose:


 * $\exists B \in \map {CL} X : A \circ B = B \circ A = I$

where $\circ$ denotes the composition of mappings.

Then $A$ is said to be invertible.