Isometry
From Maths
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