Euclidean space

1 Definition

Euclidean $n$-space is a metric space $(E,d)$with the property that the group of isometries is transitive and isisomorphic to an $n$-dimensional Euclidean vector space. To be moreprecise, we are saying that there exists an $n$-dimensional Euclideanvector space $V$ with inner product $⟨\cdot ,\cdot ⟩$ and amapping

$$+:E\times V\to E$$

such that the following hold:

1. For all $x,y\in E$ there exists a unique $u\in V$ satisfying

   $$y=x+u,d{(x,y)}^2=⟨u,u⟩,$$

2. For all $x,y\in E$ and all $u\in V$ we have

   $$d(x+u,y+u)=d(x,y).$$

3. For all $x\in E$ and all $u,v\in V$ we have

   $$(x+u)+v=x+(u+v).$$

Putting it more succinctly: $V$ acts transitively and effectively on$E$ by isometries.

Remarks.

• The difference between Euclidean space and a Euclidean vector space isone of loss of structure. Euclidean space is a Euclidean vector spacethat has “forgotten” its origin.

• A 2-dimensional Euclidean space is often called aEuclidean plane.

Title	Euclidean space
Canonical name	EuclideanSpace
Date of creation	2013-03-22 14:17:19
Last modified on	2013-03-22 14:17:19
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	16
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A03
Classification	msc 51M05
Related topic	EuclideanVectorProperties
Related topic	InnerProduct
Related topic	PositiveDefinite
Related topic	EuclideanDistance
Related topic	Vector
Defines	Euclidean plane