Let $V$ be an euclidean vector space and $U,W$ subspaces with $U\cap W = \{0\}$, $U+W = V$ and $u \perp w$ for all $(u,w) \in U \times W$. Does this imply $U^\perp = W$ and $W^\perp = U$?
Obviously, $W \subseteq U^\perp$ and $U \subseteq W^\perp$, but the other inclusion is not clear to me. It seems plausible in $\mathbb{R}^n$, but I suspect this to fail for some $V$ with $\dim V = \infty$, but I cannot think of a good counterexample.
Let $x\in U^\perp $. We have $x=u+w $, with $u\in U $, $w\in W $. Then $$ 0=\langle u,x\rangle=\langle u,u\rangle +\langle u,w\rangle =\langle u,u\rangle . $$ So $u=0$ and $x=w\in W $.