Euclidean vector space
1 Definition
The term Euclidean vector space is synonymous with finitedimensional^{}, real, positive definite^{}, inner product space^{}. The canonical example is ${\mathbb{R}}^{n}$, equipped with the usual dot product^{}. Indeed, every Euclidean vector space $V$ is isomorphic to ${\mathbb{R}}^{n}$, up to a choice of orthonormal basis^{} of $V$. As well, every Euclidean vector space $V$ carries a natural metric space structure given by
$$ 
2 Remarks.

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An analogous object with complex numbers^{} as the base field^{} is called a unitary space.

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Dropping the assumption of finitedimensionality we arrive at the class of real preHilbert spaces.

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If we drop the inner product^{} and the vector space^{} structure, but retain the metric space structure, we arrive at the notion of a Euclidean space.
Title  Euclidean vector space 
Canonical name  EuclideanVectorSpace 
Date of creation  20130322 15:38:24 
Last modified on  20130322 15:38:24 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  9 
Author  rmilson (146) 
Entry type  Definition 
Classification  msc 15A63 
Related topic  InnerProductSpace 
Related topic  UnitarySpace 
Related topic  PositiveDefinite 
Related topic  EuclideanDistance 
Related topic  Vector 
Related topic  EuclideanVectorSpace 