Definition:Cartesian Product/Cartesian Space/Real Cartesian Space

Definition
Let $n \in \N_{>0}$.

Then $\R^n$ is the cartesian product defined as follows:


 * $\displaystyle \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$ times} } = \prod_{k \mathop = 1}^n \R$

Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:


 * $\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$

Also known as
The real cartesian space of order $n$ is sometimes seen as the (real) cartesian $n$-space.

Some sources call this euclidean $n$-space -- however, on this term is reserved for the associated metric space.

Also see

 * Definition:Real Number Plane
 * Definition:Real Vector Space

It can be shown that:
 * $\R^2$ is isomorphic to any infinite flat plane in space
 * $\R^3$ is isomorphic to the whole of space itself.