Double Orthocomplement is Closed Linear Span

Theorem

Let $H$ be a Hilbert space, and let $A \subseteq H$ be a subset.

Then the following identity holds:

$(A^\perp)^\perp = \vee A$

Here $A^\perp$ denotes orthocomplementation, and $\vee A$ denotes the closed linear span.

Corollary

If $A$ is taken to be a closed linear subspace of $H$, the theorem yields:

$(A^\perp)^\perp = A$

Sources

• John B. Conway: A Course in Functional Analysis (1990)... (previous)... (next) $I.2.9-10$