Orthogonal complements in Hilbert space

Question

I have an exercise where i need to work out if:

$H$ is a Hilbert space and $F$, $G$ is closed subspaces of H where $F^{\perp}=G^{\perp}$ is $F=G$ ?

My initial thought is this is not true but i can't work out a counter example.

All help is very appreciated

Answer 1

Remark that $F = \overline{F}=(F^\perp)^\perp=(G^\perp)^\perp=\overline{G}=G$ since $F$ and $G$ are closed.