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 $\langle\cdot,\cdot\rangle$ and a mapping

 $+:E\times V\to E$

such that the following hold:

1. 1.

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

 $y=x+u,\quad d(x,y)^{2}=\langle u,u\rangle,$
2. 2.

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

 $d(x+u,y+u)=d(x,y).$
3. 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