Orthocomplement of Subset of Orthocomplement is Superset

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Theorem

Let $H$ be a Hilbert space.

Let $A, B \subseteq H$ be subsets of $H$ such that $B \subseteq A^\perp$, where $A^\perp$ is the orthocomplement of $A$.


Then $A \subseteq B^\perp$.


Proof

Let $B \subseteq A^\perp$.

Then by Orthocomplement Reverses Subset:

$A^{\perp\perp} \subseteq B^\perp$

By Double Orthocomplement is Closed Linear Span and the definition of closed linear span:

$A \subseteq A^{\perp\perp}$


Hence, by Subset Relation is Transitive:

$A \subseteq B^\perp$

$\blacksquare$