Definition:Cartesian Product/Cartesian Space/Real Cartesian Space/Countable

Definition
The countable cartesian product defined as:
 * $\displaystyle \R^\omega := \R \times \R \times \cdots = \prod_\N \R$

is called the infinite-dimensional real cartesian space.

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


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

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