I am trying to show that if $V$ is a finite-dimensional inner product space, then for any $X \subseteq V$ we have $(X ^{\perp})^{\perp} = \langle X \rangle$, the subspace of $V$ generated by $X$.
It is easy to show that $X \subseteq (X^{\perp})^{\perp}$ and therefore since orthogonal complements are subspaces of $V$, it easily follows that $\langle X \rangle \subseteq (X^{\perp})^{\perp}$. But I am struggling to show the reverse inclusion.
Any pointers??