Definition:Inverse of Continuous Linear Operator

Definition

Let $\struct {X, \norm {\, \cdot \,} }$ be the 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.

Suppose $A \in \map {CL} X$ is invertible.

Then the unique continuous linear operator denoted $A^{-1} \in \map {CL} X$ is called the inverse of $A$ if $A A^{-1} = A^{-1} A = I$.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations