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

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

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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 finite-dimensionality we arrive at the class of real pre-Hilbert 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.