Double Orthocomplement is Closed Linear Span

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:

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