Neumann Series Theorem/Corollary 1

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 $I - A : X \to X$ is bijective.

Proof

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Sources

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