Editing Real coordinate space (section)

== Definition and structures ==
For any [[natural number]] {{mvar|n}}, the [[set (mathematics)|set]] {{math|'''R'''<sup>''n''</sup>}} consists of all {{mvar|n}}-[[tuple]]s of [[real number]]s ({{math|'''R'''}}). It is called the "{{mvar|n}}-dimensional real space" or the "real {{mvar|n}}-space".

An element of {{math|'''R'''<sup>''n''</sup>}} is thus a {{mvar|n}}-tuple, and is written
<math display="block">(x_1, x_2, \ldots, x_n)</math>
where each {{math|''x''<sub>''i''</sub>}} is a real number. So, in [[multivariable calculus]], the [[domain of a function|domain]] of a [[function of several real variables]] and the codomain of a real [[vector valued function]] are [[subset]]s of {{math|'''R'''<sup>''n''</sup>}} for some {{mvar|n}}.

The real {{mvar|n}}-space has several further properties, notably:
* With [[componentwise operation|componentwise]] addition and [[scalar multiplication]], it is a [[real vector space]]. Every {{mvar|n}}-dimensional real vector space is [[isomorphic]] to it.
* With the [[dot product]] (sum of the term by term product of the components), it is an [[inner product space]]. Every {{mvar|n}}-dimensional real inner product space is isomorphic to it.
* As every inner product space, it is a [[topological space]], and a [[topological vector space]].
* It is a [[Euclidean space]] and a real [[affine space]], and every Euclidean or affine space is isomorphic to it.
* It is an [[analytic manifold]], and can be considered as the prototype of all [[manifold]]s, as, by definition, a manifold is, near each point, isomorphic to an [[open subset]] of {{math|'''R'''<sup>''n''</sup>}}.
* It is an [[algebraic variety]], and every [[real algebraic variety]] is a subset of {{math|'''R'''<sup>''n''</sup>}}.

These properties and structures of {{math|'''R'''<sup>''n''</sup>}} make it fundamental in almost all areas of mathematics and their application domains, such as [[statistics]], [[probability theory]], and many parts of [[physics]].