Let $H$ and $K$ be Hilbert spaces over the same scalar field $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$, and where complex inner products are linear in the first variable. Let $V \in \mathcal{L}(H,K)$. Then $V$ is a [Hilbert space](/page/Hilbert%20Space) isometry, meaning $\|Vx\|_K = \|x\|_H$ for every $x \in H$, if and only if $(Vx,Vy)_K = (x,y)_H$ for every $x,y \in H$.