# Definition:Isometry (Inner Product Spaces)

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

## Definition

Let $V$ and $W$ be inner product spaces with inner products $\innerprod \cdot \cdot_V$ 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.