Orthocomplement Reverses Subset
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Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space space.
Let $A, B$ be subsets of $V$ with $A \subseteq B$.
Then:
- $B^\perp \subseteq A^\perp$
where $\perp$ denotes orthocomplementation.
Proof
Let:
- $h \in B^\perp$
Then, from the definition of orthocomplement, we have:
- $h \perp b$ for each $b \in B$.
Since $A \subseteq B$, we in particular have:
- $h \perp a$ for each $a \in A$.
So, from the definition of orthocomplement:
- $h \in A^\perp$
So:
- $h \in B^\perp$ implies $h \in A^\perp$.
So, from the definition of subset:
- $B^\perp \subseteq A^\perp$
$\blacksquare$