Problem
Let $V$ be a real inner product space and $U \subset V$. Show that $(U^\perp)^\perp=U$.
Progress
Clearly for $x\in U$ we have that $\langle x,v \rangle=0$ for all $v \in U^\perp$. This immediately yields that $x \in (U^\perp)^\perp$ and so $U \subset (U^\perp)^\perp$.
Taking $x \in (U^\perp)^\perp$, we have that $\langle x,v \rangle=0$ for all $v \in U^\perp$. Not sure how to move it on from here though; any assistance would be much appreciated. Regards.