Double Orthocomplement is Closed Linear Span
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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
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Sources
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Specifically: as to whether the corollary is also given
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- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next): $\text I.2.9$-$10$
