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 ${ℝ}^n$, equipped with the usual dot product. Indeed, every Euclidean vector space $V$ is isomorphic to ${ℝ}^n$, 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.

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

• Dropping the assumption of finite-dimensionality we arrive at the class of real pre-Hilbert spaces.

• 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	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