Double Orthocomplement is Closed Linear Span

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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:

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


Proof


Sources