How can I able to show that $(S ^{\perp})^{\perp}$ is a finite dimensional vector space. 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.

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. 
