# 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

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next): $\text I.2.9$-$10$