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.

Proof
From Orthocomplement is Closed Linear Subspace:
 * $\paren {A^\perp}^\perp$ is a closed linear subspace of $H$.

Also:
 * $\paren {A^\perp}^\perp \supseteq A$

Indeed, for each $a \in A$, we have:
 * $\forall a' \in A^\perp : \innerprod a {a'} = 0$

by definition of $A^\perp$.

This also means that:
 * $a \in \paren{A^\perp}^\perp$

by definition of $\paren{A^\perp}^\perp$.

Hence:
 * $\vee A \subseteq \paren{A^\perp}^\perp$

For the converse direction, note:
 * $A \subseteq \vee A$

Now apply Orthocomplement Reverses Subset twice:
 * $\paren {A^\perp}^\perp \subseteq \paren {\paren {\vee A}^\perp}^\perp$

By Double Orthocomplement of Closed Linear Subspace:
 * $\paren{A^\perp}^\perp \subseteq \vee A$