\begin{definition}[Definition:Cartesian Product/Cartesian Space/Real Cartesian Space]
Let $n \in \N_{>0}$.
Then $\R^n$ is the cartesian product defined as follows:
:$\ds \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$ times} } = \prod_{k \mathop = 1}^n \R$
Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:
:$\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$
\end{definition}