Orthocomplement of Subset of Orthocomplement is Superset
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Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Let $A, B \subseteq V$ be subsets of $V$ 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:
- $A \subseteq B^\perp$
$\blacksquare$