Definition:Invertible Continuous Linear Operator

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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.

Sources

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