Definition

The term Euclidean vector space is synonymous with finite-dimensional, real, positive definite, inner product space. The canonical example is $\reals^n$ equipped with the usual dot product. Indeed, every Euclidean vector space $V$ is isomorphic to $\reals^n$ up to a choice of orthonormal basis of $V$ As well, every Euclidean vector space $V$ carries a natural metric space structure given by $$ d(u,v) = \sqrt{\left< u-v, u-v \right>},\quad u,v\in V.$$

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.

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15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)

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