Double Orthocomplement is Closed Linear Span/Corollary

Corollary to Double Orthocomplement is Closed Linear Span
Let $\HH$ be a Hilbert space. Let $A \subseteq \HH$ be a closed linear subspace of $\HH$.

Then:


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

Proof
Since $A$ is a subspace of $\HH$, it is closed under linear combination, so:


 * $\map \span A = A$

We therefore have:

while from Double Orthocomplement is Closed Linear Span, we have:


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

So, we obtain:


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