Definition:Isometry (Inner Product Spaces)
Jump to navigation
Jump to search
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.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions