\begin{definition}[Definition:Euclidean Space]
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 $\tuple {S^n, d}$ is a '''Euclidean space'''.
\end{definition}