Continuous Linear Transformation Space as Banach Algebra
Jump to navigation
Jump to search
Theorem
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 $*_X : X \times X \to X$ and $* : \map {CL} X \times \map {CL} X \to \map {CL} X$ be bilinear mappings.
Suppose $\struct {\struct {X, \norm {\, \cdot \,}_X}, *_X}$ is a Banach algebra.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Then $\struct {\struct {\map {CL} X, \norm {\, \cdot \,}}, *}$ is a Banach algebra.
This article, or a section of it, needs explaining. In particular: What is $*$? Maybe $\circ$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. 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 {{Explain}} from the code. |
Proof
This theorem requires a proof. 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