Vector space/K/Inner product/Orthonormal basis/Definition

Orthonormal basis

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. A basis ${}v_{i}$, ${}i\in I$, of ${}V$ is called an orthonormal basis if

$\left\langle v_{i},v_{i}\right\rangle =1{\text{ for all }}i\in I{\text{ and }}\left\langle v_{i},v_{j}\right\rangle =0{\text{ for }}i\neq j$

holds.