# Euclidean vector space

## 1 Definition

The term Euclidean vector space is synonymous with finite-dimensional, 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

 $d(u,v)=\sqrt{\left},\quad u,v\in V.$

## 2 Remarks.

 Title Euclidean vector space Canonical name EuclideanVectorSpace Date of creation 2013-03-22 15:38:24 Last modified on 2013-03-22 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