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 $⟨\cdot ,\cdot ⟩$ 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

   $$y=x+u,d{(x,y)}^2=⟨u,u⟩,$$

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 is one of loss of structure. Euclidean space is a Euclidean vector space that has “forgotten” its origin.

• A 2-dimensional Euclidean space is often called a Euclidean 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