Vector space with inner product/Orthogonal complement/Definition

Orthogonal complement

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product, and let ${}U\subseteq V$ denote a linear subspace. Then

${}U^{\perp }={\left\{v\in V\mid \left\langle v,u\right\rangle =0{\text{ for all }}u\in U\right\}}\,$

is called the orthogonal complement of ${}U$.