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https://texercises.com/exercise/orthogonal-complement-characteristics/
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Exercise:
Let V be an inner product space and Usubseteq V a finite dimensional subspace. Then U^perp^perpU.

Solution:
Proof. By a previous Lemma Usubseteq U^perp^perp. We'll show that we also have U^perp^perpsubseteq U. Indeed let vin U^perp^perp. Recall that VUoplus U^perp. Write vu+w with uin U win U^perp. Sinces uin Usubseteq U^perp^perp and also vin U^perp^perp we have that wv-uin U^perp^perp. But win U^perp hence win U^perpcap U^perp^perp. This shows that w Rightarrow vuin U. We've proven that U^perp^perpU.
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Exercise:
Let V be an inner product space and Usubseteq V a finite dimensional subspace. Then U^perp^perpU.

Solution:
Proof. By a previous Lemma Usubseteq U^perp^perp. We'll show that we also have U^perp^perpsubseteq U. Indeed let vin U^perp^perp. Recall that VUoplus U^perp. Write vu+w with uin U win U^perp. Sinces uin Usubseteq U^perp^perp and also vin U^perp^perp we have that wv-uin U^perp^perp. But win U^perp hence win U^perpcap U^perp^perp. This shows that w Rightarrow vuin U. We've proven that U^perp^perpU.
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