# Definition:Isometry (Hilbert Spaces)

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

## Definition

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:

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

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## Also see

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- 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

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next) $I.5.2$