Let $H$ and $K$ be Hilbert spaces over the same scalar field $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$, and let $V\in\mathcal{L}(H,K)$. Let $I_H:H\to H$ denote the identity operator on $H$. Then $V$ is a [Hilbert space](/page/Hilbert%20Space) isometry, meaning