Definition:Annihilator of Subspace of Banach Space
Jump to navigation
Jump to search
Definition
This page has been identified as a candidate for refactoring of medium complexity. In particular: Perhaps merge instances of annihilator into a parent plus subpages structure. Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.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 {{Refactor}} from the code. |
Let $X$ be a Banach space.
Let $M$ be a vector subspace of $X$.
Let $X^\ast$ be the normed dual space of $X$.
We define the annihilator $M^\bot$ by:
- $M^\bot = \set {g \in X^\ast : \map g x = 0 \text { for all } x \in M}$
Also see
- Annihilator of Subspace of Banach Space is Subspace of Normed Dual
- Annihilator of Subspace of Banach Space is Weak-* Closed
- Definition:Annihilator of Subspace of Normed Dual Space
- Results about annihilators of subspaces of Banach spaces can be found here.
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $4.6$: Annihilators