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

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

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