Double Orthocomplement of Closed Linear Subspace

Theorem
Let $H$ be a Hilbert space. Let $A \subseteq H$ be a closed linear subspace of $H$.

Then:


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

Proof
Let
 * $I : H \to H$

be the identity operator (viz., $Ih=h$).

Also, let
 * $P : H \to A$

be the orthogonal projection.

Then
 * $I-P : H \to A^\perp$

is the orthogonal projection onto the orthocomplement.

Now
 * $\ker(I-P)=\paren{A^\perp}^\perp$

by Kernel of Orthogonal Projection.

Note that
 * $h \in P(H) \implies h=Ph$

since orthogonal projection is idempotent.

Also
 * $0=(I-P)h \iff h=Ph$

Therefore,
 * $\ker(I-P)=P(H)$

Finally, the range of the orthogonal projection is
 * $P(H)=A$

Conclude that $\paren{A^\perp}^\perp = \ker(I-P) = P(H) = A$.