Double Orthocomplement is Closed Linear Span
This article is a landmark page. It was the 4500th proof on $\mathsf{Pr} \infty \mathsf{fWiki}$!Theorem
Let $\HH$ be a Hilbert space.
Let $A \subseteq \HH$ be a subset of $\HH$.
Then the following identity holds:
- $\paren {A^\perp}^\perp = \vee A$
Here $A^\perp$ denotes orthocomplementation, and $\vee A$ denotes the closed linear span.
Corollary
Let $A \subseteq \HH$ be a closed linear subspace of $\HH$.
Then:
- $\paren {A^\perp}^\perp = A$
Proof
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Sources
As such, the following source works, along with any process flow, will need to be reviewed.
Specifically: as to whether the corollary is also given
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
When you have done this, move the citation of that source work (if it is appropriate that it stay on this page) above this message, into the usual chronological ordering.
When this has been done for all of the sources below, {{SourceReview}} should be removed from the code.
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next): $\text I.2.9$-$10$