Definition:Isometry (Hilbert Spaces)

This page is about the notion of isometry between Hilbert spaces. For other uses, see Definition:Isometry.

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:

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

See also

• 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.

Sources

• John B. Conway: A Course in Functional Analysis (1990)... (previous)... (next) $I.5.2$