Neumann Series Theorem/Corollary 2
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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.
Proof
This theorem requires a proof. In particular: Obviously, we have invertibility in $CL(X)$, but the source mentions $\norm x \le \frac 1 {1 - \norm A} \norm y$. A chance to have two proofs? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations