Definition:Isometry (Hilbert Spaces)

Definition
Let $H, K$ be Hilbert spaces, and denote by $\left\langle{\cdot, \cdot}\right\rangle_H$ and $\left\langle{\cdot, \cdot}\right\rangle_K$ their respective inner products.

A linear map $U: H \to K$ is called an isometry iff:


 * $\forall g,h \in H: \left\langle{g, h}\right\rangle_H = \left\langle{Ug, Uh}\right\rangle_K$

Also see

 * Above definition of isometry is shown to be equivalent to an into isometry, when considering the Hilbert spaces as metric spaces.
 * An isomorphism between Hilbert spaces is seen to be an isometry.