Definition:Isometry (Inner Product Spaces)

This page is about isometry in the context of inner product spaces. For other uses, see isometry.

Definition

Let $V$ and $W$ be inner product spaces.

Let their inner products be $\innerprod \cdot \cdot_V$ and $\innerprod \cdot \cdot_W$ respectively.

Let the mapping $F : V \to W$ be a vector space isomorphism that preserves inner products:

$\forall v_1, v_2 \in V : \innerprod {v_1} {v_2}_V = \innerprod {\map F {v_1}} {\map F {v_2}}_W$

Then $F$ is called a (linear) isometry.

Hilbert Spaces

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:

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

Also see

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

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions