# Orthogonal complement of $S$

Let $V$ be an inner product space. If $S$ is subset of $V$, then the orthogonal complement of $S$ is $$S^\bot =\{v\in V \vert (v,s)=0 \text{ for all } s\in S\}$$ If $S$ is subspace of V then $S\cap S^\bot=\{0\}$. My question is that is it required that $S$ be a subspace of $V$?

No, it is true even if $S$ isn't a subspace.

Namely, let $x \in S \cap S^\perp$. Then

$$\|x\|^2 = \left\langle \underbrace{x}_{\in S}, \underbrace{x}_{\in S^\perp}\right\rangle = 0$$

so $x = 0$.