Euclidean space
1 Definition
Euclidean^{} $n$space is a metric space $(E,d)$ with the property that the group of isometries is transitive^{} and is isomorphic^{} to an $n$dimensional Euclidean vector space. To be more precise, we are saying that there exists an $n$dimensional Euclidean vector space $V$ with inner product $\u27e8\cdot ,\cdot \u27e9$ and a mapping
$$+:E\times V\to E$$ 
such that the following hold:

1.
For all $x,y\in E$ there exists a unique $u\in V$ satisfying
$$y=x+u,d{(x,y)}^{2}=\u27e8u,u\u27e9,$$ 
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 is one of loss of structure^{}. Euclidean space is a Euclidean vector space that has “forgotten” its origin.

•
A 2dimensional Euclidean space is often called a Euclidean plane.
Title  Euclidean space 
Canonical name  EuclideanSpace 
Date of creation  20130322 14:17:19 
Last modified on  20130322 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 