Editing Euclidean space (section)

==Prototypical examples==
For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space.

A typical case of Euclidean vector space is <math>\R^n</math> viewed as a vector space equipped with the [[dot product]] as an [[inner product]]. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is [[isomorphism|isomorphic]] to it. More precisely, given a Euclidean space {{mvar|E}} of dimension {{mvar|n}}, the choice of a point, called an ''origin'' and an [[orthonormal basis]] of <math>\overrightarrow E</math> defines an isomorphism of Euclidean spaces from {{mvar|E}} to <math>\R^n.</math>

As every Euclidean space of dimension {{mvar|n}} is isomorphic to it, the Euclidean space <math>\R^n</math> is sometimes called the ''standard Euclidean space'' of dimension {{mvar|n}}.{{sfn|Berger|1987|loc=Section 9.1}}