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	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]	[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

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

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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.