Vector spaces/K/Inner product/Linear isometry/Definition

Isometry

Let ${}V,W$ be vector spaces over ${}{\mathbb {K} }$, endowed with inner products, and let

$\varphi \colon V\longrightarrow W$

be s linear mapping. Then ${}\varphi$ is called an isometry if

${}\left\langle \varphi (v),\varphi (w)\right\rangle =\left\langle v,w\right\rangle \,$

holds for all ${}v,w\in V$.