Definition:Cartesian Product/Cartesian Space/Real Cartesian Space

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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:

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

Countable-Dimensional Real Cartesian Space

The countable cartesian product defined as:

$\displaystyle \R^\omega := \R \times \R \times \cdots = \prod_\N \R$

is called the countable-dimensional real cartesian space.

Thus, $\R^\omega$ can be defined as the set of all real sequences:

$\R^\omega = \left\{{\left({x_1, x_2, \ldots}\right): x_1, x_2, \ldots \in \R}\right\}$

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 $\mathsf{Pr} \infty \mathsf{fWiki}$ this term is reserved for the associated metric space.

Also see

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.

Source of Name

This entry was named for René Descartes.