Double Orthocomplement is Closed Linear Span

Theorem
Let $H$ be a Hilbert space.

Let $A \subseteq H$ be a subset of $H$.

Then the following identity holds:


 * $\paren {A^\perp}^\perp = \vee A$

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