Euclidean space

1 Definition

EuclideanPlanetmathPlanetmath n-space is a metric space (E,d) with the property that the group of isometries is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and is isomorphicPlanetmathPlanetmathPlanetmath 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 , and a mapping


such that the following hold:

  1. 1.

    For all x,yE there exists a unique uV satisfying

  2. 2.

    For all x,yE and all uV we have

  3. 3.

    For all xE and all u,vV we have


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


  • The differencePlanetmathPlanetmath between Euclidean spaceMathworldPlanetmath and a Euclidean vector space is one of loss of structureMathworldPlanetmath. 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