Difference between revisions of "Isometry"
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Latest revision as of 13:13, 9 July 2015
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
- ↑ Unconfirmed name, "isometric" is simply used
References
- ↑ Functional Analysis - George Bachman and Lawrence Narici
