Sufficient Conditions for Determining the Orthogonal Complement

Question

Let $X$ be a Hilbert space with inner product denoted $(\cdot, \cdot)$, $V$ a closed subspace of $X$. Let $$X=V \oplus U$$ and suppose $(u,v)=0$ for any $u \in U$ and any $v \in V$. Then, clearly $U \subset V^\perp$. Is it necessarily true that $U=V^\perp$?

Answer 1

$X=V\oplus V^{\perp}\tag{1}$

$X=V\oplus U\tag{2}$

Let $x\in V^{\perp}$, then $x\overset{\text{(1)}}=v+u$

for unique $v\in V, u\in U$

$\begin{align}&\langle x, v\rangle=0\\&\langle v+u, v\rangle=0\\&\|v\|^2=0\\&v=0\end{align}$

Hence $x=0+u=u\in U$

That's conclude $V^{\perp}\subset U$