Definition:Euclidean Space

Definition
Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$.

Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$.

Let $d: S^n \times S^n \to \R$ be the usual (Euclidean) metric on $S^n$.

Then $\left({S^n, d}\right)$ is a Euclidean space.

Also see

 * Definition:Dot Product
 * Definition:Euclidean Norm
 * Definition:Euclidean Metric