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 $H$ and $K$ be Hilbert spaces.

Let their inner products be $\innerprod \cdot \cdot_H$ and $\innerprod \cdot \cdot_K$ respectively.

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.

• Results about isometries in the context of inner product spaces can be found here.

Sources

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