# Isometry

## Definitions

There are several kinds of isometries

Type Acts on Definition We say Comment
Linear isometry Vector spaces
(normed ones)
For a lin map [ilmath]L:U\rightarrow V[/ilmath]
we have [ilmath]\Vert Lx\Vert_V=\Vert x\Vert_U[/ilmath]
[ilmath]U[/ilmath] and [ilmath]V[/ilmath] are
Linearly isomorphic
Metric isometry[Note 1] Metric spaces For a homeomorphism [ilmath]f:(X,d)\rightarrow(Y,d')[/ilmath]
we have [ilmath]d(x,y)=d'(f(x),f(y))[/ilmath][1]
[ilmath]X[/ilmath] and [ilmath]Y[/ilmath] are isomorphic

## Examples

### Linear isometry

• Consider the map [ilmath]f:\mathbb{R}^n\rightarrow\mathbb{R}^n [/ilmath] where [ilmath]f[/ilmath] is a rotation. Under the Euclidean norm this is an isometry

TODO: Consider the box norm, so forth! - afterall norms are equiv!

### Metric isometry

• Consider the map [ilmath]f:\mathbb{R}\rightarrow\mathbb{R} [/ilmath] with [ilmath]f:x\mapsto x+a[/ilmath] where [ilmath]\mathbb{R} [/ilmath] is equipped with the usual Absolute value as the distance. This is an isometry.

## Notes

1. Unconfirmed name, "isometric" is simply used

## References

1. Functional Analysis - George Bachman and Lawrence Narici