Definition:Isometry (Hilbert Spaces)

From ProofWiki
Jump to navigation Jump to search

This page is about Isometry in the context of Hilbert Space. For other uses, see Isometry.


Let $H, K$ be Hilbert spaces, and denote by $\innerprod \cdot \cdot_H$ and $\innerprod \cdot \cdot_K$ their respective inner products.

A linear map $U: H \to K$ is called an isometry if and only if:

$\forall g,h \in H: \innerprod g h_H = \innerprod {U g} {U h}_K$

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.