Let $H$ be a Hilbert space and $S\subseteq H$ be a finite subset. How can I able to show that $(S ^{\perp})^{\perp}$
is a finite dimensional vector space.
[Math] How to able to show that $(S ^{\perp})^{\perp}$ is a finite dimensional vector space.
functional-analysishilbert-spacesvector-spaces
Best Answer
What you want to prove is that, for any $S\subset H$, $$ S^{\perp\perp}=\overline{\mbox{span}\,S} $$ One inclusion is easy if you notice that $S^{\perp\perp}$ is a closed subspace that contains $S$. The other inclusion follows from $$ H=\overline{\mbox{span}\,S}\oplus S^\perp $$ and the uniqueness of the orthogonal complement.