Double Orthocomplement is Closed Linear Span

Theorem
Let $H$ be a Hilbert space, and let $A \subseteq H$ be a subset.

Then the following identity holds:


 * $(A^\perp)^\perp = \vee A$

Here $A^\perp$ denotes orthocomplementation, and $\vee A$ denotes the closed linear span.

Corollary
If $A$ is taken to be a closed linear subspace of $H$, the theorem yields:


 * $(A^\perp)^\perp = A$