Let $(\mathcal H, \langle\cdot,\cdot\rangle)$ be a Hilbertspace, $U,V \subset \mathcal H$ are closed subspaces. I want to show $$U \subset V \Leftrightarrow V^\bot \subset U^\bot$$ $\Rightarrow$ is easy to show, no problems with that. But I am stuck at $\Leftarrow $. Since $\mathcal H$ can be any Hilbertspace, it doesn't have to be of finite dimension, so usually $(U^\bot)^\bot \neq U$. I am pretty sure that I have to use the fact that $U$ and $V$ are closed subspaces, but I am not sure how.
I tried $x \in U \Rightarrow \langle x,u \rangle = 0 \forall u \in U^\bot \Rightarrow \langle x,v \rangle = 0 \forall v \in V^\bot$. But as $(V^\bot)^\bot \neq V$, I can't conclude $x \in V$.
I would appreciate hints more than answers, as I want to solve this myself.