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 $L:U\rightarrow V$ we have $\Vert Lx\Vert_V=\Vert x\Vert_U$	$U$ and $V$ are Linearly isomorphic
Metric isometry	Metric spaces	For a homeomorphism $f:(X,d)\rightarrow(Y,d')$ we have $d(x,y)=d'(f(x),f(y))$	$X$ and $Y$ are isomorphic

Examples

Linear isometry

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