Definition

Let $V$ be an inner product space and let $W\subset V$ be a subspace. We define the orthogonal complement to $W$ to be $$W^{\perp}:=\{ v\in V: \langle v, w \rangle =0, \forall w\in W \}$$ That is, $W^{\perp}$ is the set of vectors that are orthogonal to all vectors in $W$.