View source for Real coordinate space

{{short description|Space formed by the ''n''-tuples of real numbers}}
{{no footnotes|date=February 2024}}
[[File:Cartesian-coordinate-system v2.svg|thumb|right|[[Cartesian coordinates]] identify points of the [[Euclidean plane]] with pairs of real numbers]]

In [[mathematics]], the '''real coordinate space''' or '''real coordinate ''n''-space''', of [[dimension]] {{mvar|n}}, denoted {{math|'''R'''<sup>{{mvar|n}}</sup>}} or {{nowrap|<math>\R^n</math>}}, is the set of all ordered [[tuple|{{mvar|n}}-tuples]] of [[real number]]s, that is the set of all sequences of {{mvar|n}} real numbers, also known as ''[[coordinate vector]]s''.
Special cases are called the ''[[real line]]'' {{math|'''R'''<sup>1</sup>}}, the ''real coordinate plane'' {{math|'''R'''<sup>2</sup>}}, and the ''real coordinate three-dimensional space'' {{math|'''R'''<sup>3</sup>}}.
With component-wise addition and scalar multiplication, it is a [[real vector space]].

The [[coordinate (vector space)|coordinates]] over any [[basis (vector space)|basis]] of the elements of a real vector space form a ''real coordinate space'' of the same dimension as that of the vector space. Similarly, the [[Cartesian coordinates]] of the points of a [[Euclidean space]] of dimension {{mvar|n}}, {{math|'''E'''<sup>n</sup>}} ([[Euclidean line]], {{math|'''E'''}}; [[Euclidean plane]], {{math|'''E'''<sup>2</sup>}}; [[Euclidean three-dimensional space]], {{math|'''E'''<sup>3</sup>}}) form a ''real coordinate space'' of dimension {{mvar|n}}.

These [[one to one correspondence]]s between vectors, points and coordinate vectors explain the names of ''coordinate space'' and ''coordinate vector''. It allows using [[geometric]] terms and methods for studying real coordinate spaces, and, conversely, to use methods of [[calculus]] in geometry. This approach of geometry was introduced by [[René Descartes]] in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them.

== 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]].

== The domain of a function of several variables ==
{{main|Multivariable calculus|Real multivariable function}}
Any function {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}} of {{mvar|n}} real variables can be considered as a function on {{math|'''R'''<sup>''n''</sup>}} (that is, with {{math|'''R'''<sup>''n''</sup>}} as its [[domain of a function|domain]]). The use of the real {{mvar|n}}-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for {{math|1=''n'' = 2}}, a [[function composition]] of the following form:
<math display="block"> F(t) = f(g_1(t),g_2(t)),</math>
where functions {{math|''g''<sub>1</sub>}} and {{math|''g''<sub>2</sub>}} are [[continuous function|continuous]]. If
*{{math|∀''x''<sub>1</sub> ∈ '''R''' : ''f''(''x''<sub>1</sub>, ·<!-- not a multiplication sign! -->)}} is continuous (by {{math|''x''<sub>2</sub>}})
*{{math|∀''x''<sub>2</sub> ∈ '''R''' : ''f''(·<!-- not a multiplication sign! -->, ''x''<sub>2</sub>)}} is continuous (by {{math|''x''<sub>1</sub>}})
then {{mvar|F}} is not necessarily continuous. Continuity is a stronger condition: the continuity of {{mvar|f}} in the natural {{math|'''R'''<sup>2</sup>}} topology ([[#Topological properties|discussed below]]), also called ''multivariable continuity'', which is sufficient for continuity of the composition {{mvar|F}}.

== Vector space ==
The coordinate space {{math|'''R'''<sup>''n''</sup>}} forms an {{mvar|n}}-dimensional [[vector space]] over the [[field (mathematics)|field]] of real numbers with the addition of the structure of [[linearity]], and is often still denoted {{math|'''R'''<sup>''n''</sup>}}.  The operations on {{math|'''R'''<sup>''n''</sup>}} as a vector space are typically defined by
<math display="block">\mathbf x + \mathbf y = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)</math>
<math display="block">\alpha \mathbf x = (\alpha x_1, \alpha x_2, \ldots, \alpha x_n).</math>
The [[additive identity|zero vector]] is given by
<math display="block">\mathbf 0 = (0, 0, \ldots, 0)</math>
and the [[additive inverse]] of the vector {{math|'''x'''}} is given by
<math display="block">-\mathbf x = (-x_1, -x_2, \ldots, -x_n).</math>

This structure is important because any {{mvar|n}}-dimensional real vector space is isomorphic to the vector space {{math|'''R'''<sup>''n''</sup>}}.

===Matrix notation===
{{main|Matrix (mathematics)}}
In standard [[matrix (mathematics)|matrix]] notation, each element of {{math|'''R'''<sup>''n''</sup>}} is typically written as a [[column vector]]
<math display="block">\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}</math>
and sometimes as a [[row vector]]:
<math display="block">\mathbf x = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}.</math>

The coordinate space {{math|'''R'''<sup>''n''</sup>}} may then be interpreted as the space of all {{math|''n'' × 1}} [[column vector]]s, or all {{math|1 × ''n''}} [[row vector]]s with the ordinary matrix operations of addition and [[scalar multiplication]].

[[Linear transformation]]s from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''<sup>''m''</sup>}} may then be written as {{math|''m'' × ''n''}} matrices which act on the elements of {{math|'''R'''<sup>''n''</sup>}} via [[left and right (algebra)|left]] multiplication (when the elements of {{math|'''R'''<sup>''n''</sup>}} are column vectors) and on elements of {{math|'''R'''<sup>''m''</sup>}} via right multiplication (when they are row vectors). The formula for left multiplication, a special case of [[matrix multiplication]], is:
<math display="block">(A{\mathbf x})_k = \sum_{l=1}^n A_{kl} x_l</math>

{{anchor|continuity of linear maps}}Any linear transformation is a [[continuous function]] (see [[#Topological properties|below]]). Also, a matrix defines an [[open map]] from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''<sup>''m''</sup>}} if and only if the [[rank (matrix theory)|rank of the matrix]] equals to {{mvar|m}}.

===Standard basis===
{{main|Standard basis}}
The coordinate space {{math|'''R'''<sup>''n''</sup>}} comes with a standard basis:
<math display="block">\begin{align}
\mathbf e_1 & = (1, 0, \ldots, 0) \\
\mathbf e_2 & = (0, 1, \ldots, 0) \\
& {}\;\;  \vdots \\
\mathbf e_n & = (0, 0, \ldots, 1)
\end{align}</math>

To see that this is a basis, note that an arbitrary vector in {{math|'''R'''<sup>''n''</sup>}} can be written uniquely in the form
<math display="block">\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i.</math>

== Geometric properties and uses ==

=== Orientation ===
The fact that [[real numbers]], unlike many other [[field (mathematics)|fields]], constitute an [[ordered field]] yields an [[orientation (vector space)|orientation structure]] on {{math|'''R'''<sup>''n''</sup>}}. Any [[rank (matrix theory)|full-rank]] linear map of {{math|'''R'''<sup>''n''</sup>}} to itself either preserves or reverses orientation of the space depending on the [[sign (mathematics)|sign]] of the [[determinant]] of its matrix. If one [[permutation|permutes]] coordinates (or, in other words, elements of the basis), the resulting orientation will depend on the [[parity of a permutation|parity of the permutation]].

[[Diffeomorphism]]s of {{math|'''R'''<sup>''n''</sup>}} or [[domain (mathematical analysis)|domains in it]], by their virtue to avoid zero [[Jacobian matrix and determinant|Jacobian]], are also classified to orientation-preserving and orientation-reversing. It has important consequences for the theory of [[differential form]]s, whose applications include [[electrodynamics]].

Another manifestation of this structure is that the [[point reflection]] in {{math|'''R'''<sup>''n''</sup>}} has different properties depending on [[even and odd numbers|evenness of {{mvar|n}}]]. For even {{mvar|n}} it preserves orientation, while for odd {{mvar|n}} it is reversed (see also [[improper rotation]]).

=== Affine space ===
{{details|Affine space}}
{{math|'''R'''<sup>''n''</sup>}} understood as an affine space is the same space, where {{math|'''R'''<sup>''n''</sup>}} as a vector space [[Group action (mathematics)|acts]] by [[translation (geometry)|translations]]. Conversely, a vector has to be understood as a "[[displacement (vector)|difference]] between two points", usually illustrated by a directed [[line segment]] connecting two points. The distinction says that there is no [[canonical form|canonical]] choice of where the [[origin (mathematics)|origin]] should go in an affine {{mvar|n}}-space, because it can be translated anywhere.

=== Convexity ===
[[File:2D-simplex.svg|thumb|The ''n''-simplex (see [[#Polytopes in Rn|below]]) is the standard convex set, that maps to every polytope, and is the intersection of the standard {{math|(''n'' + 1)}} affine hyperplane (standard affine space) and the standard {{math|(''n'' + 1)}} orthant (standard cone).]]
{{details|Convex analysis}}
In a real vector space, such as {{math|'''R'''<sup>''n''</sup>}}, one can define a convex [[cone (linear algebra)|cone]], which contains all ''non-negative'' linear combinations of its vectors. Corresponding concept in an affine space is a [[convex set]], which allows only [[convex combination]]s (non-negative linear combinations that sum to 1).

In the language of [[universal algebra]], a vector space is an algebra over the universal vector space {{math|'''R'''<sup>∞</sup>}} of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra over the universal [[orthant]] (of finite sequences of nonnegative numbers), and a convex set is an algebra over the universal [[simplex]] (of finite sequences of nonnegative numbers summing to 1). This geometrizes the axioms in terms of "sums with (possible) restrictions on the coordinates".

Another concept from convex analysis is a [[convex function]] from {{math|'''R'''<sup>''n''</sup>}} to real numbers, which is defined through an [[inequality (mathematics)|inequality]] between its value on a convex combination of [[point (geometry)|points]] and sum of values in those points with the same coefficients.

=== Euclidean space ===
{{main|Euclidean space|Cartesian coordinate system}}
The [[dot product]]
<math display="block">\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n</math>
defines the [[normed vector space|norm]] {{math|1={{abs|'''x'''}} = {{sqrt|'''x''' &sdot; '''x'''}}}} on the vector space {{math|'''R'''<sup>''n''</sup>}}. If every vector has its [[Euclidean norm]], then for any pair of points the distance
<math display="block">d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\| = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}</math>
is defined, providing a [[metric space]] structure on {{math|'''R'''<sup>''n''</sup>}} in addition to its affine structure.

As for vector space structure, the dot product and Euclidean distance usually are assumed to exist in {{math|'''R'''<sup>''n''</sup>}} without special explanations. However, the real {{mvar|n}}-space and a Euclidean {{mvar|n}}-space are distinct objects, strictly speaking. Any Euclidean {{mvar|n}}-space has a [[coordinate system]] where the dot product and Euclidean distance have the form shown above, called [[Renatus Cartesius|''Cartesian'']]. But there are ''many'' Cartesian coordinate systems on a Euclidean space.

Conversely, the above formula for the Euclidean metric defines the ''standard'' Euclidean structure on {{math|'''R'''<sup>''n''</sup>}}, but it is not the only possible one. Actually, any [[positive-definite quadratic form]] {{mvar|q}} defines its own "distance" {{math|{{sqrt|''q''('''x''' − '''y''')}}}}, but it is not very different from the Euclidean one in the sense that
<math display="block">\exist C_1 > 0,\ \exist C_2 > 0,\ \forall \mathbf{x}, \mathbf{y} \in \mathbb{R}^n:
 C_1 d(\mathbf{x}, \mathbf{y}) \le \sqrt{q(\mathbf{x} - \mathbf{y})} \le
 C_2 d(\mathbf{x}, \mathbf{y}). </math>
Such a change of the metric preserves some of its properties, for example the property of being a [[complete metric space]].
This also implies that any full-rank linear transformation of {{math|'''R'''<sup>''n''</sup>}}, or its [[affine transformation]], does not magnify distances more than by some fixed {{math|''C''<sub>2</sub>}}, and does not make distances smaller than {{math|1 / ''C''<sub>1</sub>}} times, a fixed finite number times smaller.{{clarify|date=October 2014}}

The aforementioned equivalence of metric functions remains valid if {{math|{{sqrt|''q''('''x''' − '''y''')}}}} is replaced with {{math|''M''('''x''' − '''y''')}}, where {{mvar|M}} is any convex positive [[homogeneous function]] of degree 1, i.e. a [[normed vector space|vector norm]] (see [[Minkowski distance]] for useful examples). Because of this fact that any "natural" metric on {{math|'''R'''<sup>''n''</sup>}} is not especially different from the Euclidean metric, {{math|'''R'''<sup>''n''</sup>}} is not always distinguished from a Euclidean {{math|''n''}}-space even in professional mathematical works.

=== In algebraic and differential geometry ===
Although the definition of a [[manifold]] does not require that its model space should be {{math|'''R'''<sup>''n''</sup>}}, this choice is the most common, and almost exclusive one in [[differential geometry]].

On the other hand, [[Whitney embedding theorem]]s state that any real [[differentiable manifold|differentiable {{mvar|m}}-dimensional manifold]] can be [[embedding|embedded]] into {{math|'''R'''<sup>2''m''</sup>}}.

=== Other appearances ===
Other structures considered on {{math|'''R'''<sup>''n''</sup>}} include the one of a [[pseudo-Euclidean space]], [[symplectic structure]] (even {{mvar|n}}), and [[contact structure]] (odd {{mvar|n}}). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates.

{{math|'''R'''<sup>''n''</sup>}} is also a real vector subspace of {{math|[[complex coordinate space|'''C'''<sup>''n''</sup>]]}} which is invariant to [[complex conjugation]]; see also [[complexification]].

=== Polytopes in R<sup>''n''</sup> ===
{{see also|Linear programming|Convex polytope}}
There are three families of [[polytope]]s which have simple representations in {{math|'''R'''<sup>''n''</sup>}} spaces, for any {{mvar|n}}, and can be used to visualize any affine coordinate system in a real {{mvar|n}}-space. Vertices of a [[hypercube]] have coordinates {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}} where each {{mvar|x<sub>k</sub>}} takes on one of only two values, typically 0 or 1. However, any two numbers can be chosen instead of 0 and 1, for example {{num|−1}} and 1. An {{mvar|n}}-hypercube can be thought of as the Cartesian product of {{mvar|n}} identical [[interval (mathematics)|intervals]] (such as the [[unit interval]] {{closed-closed|0,1}}) on the real line. As an {{mvar|n}}-dimensional subset it can be described with a [[system of inequalities|system of {{math|2''n''}} inequalities]]:
<math display="block">\begin{matrix}
0 \le x_1 \le 1 \\
\vdots \\
0 \le x_n \le 1
\end{matrix}</math> for {{closed-closed|0,1}}, and
<math display="block">\begin{matrix}
|x_1| \le 1 \\
\vdots \\
|x_n| \le 1
\end{matrix}</math> for {{closed-closed|−1,1}}.
{{clear|left}}
Each vertex of the [[cross-polytope]] has, for some {{mvar|k}}, the {{mvar|x<sub>k</sub>}} coordinate equal to [[±1]] and all other coordinates equal to 0 (such that it is the {{mvar|k}}th [[#Standard basis|standard basis vector]] up to [[sign (mathematics)|sign]]). This is a [[dual polytope]] of hypercube. As an {{mvar|n}}-dimensional subset it can be described with a single inequality which uses the [[absolute value]] operation:
<math display="block">\sum_{k=1}^n |x_k| \le 1\,,</math>
but this can be expressed with a system of {{math|[[power of two|2<sup>''n''</sup>]]}} linear inequalities as well.

The third polytope with simply enumerable coordinates is the [[standard simplex]], whose vertices are {{mvar|n}} standard basis vectors and [[origin (mathematics)|the origin]] {{math|(0, 0, ..., 0)}}. As an {{mvar|n}}-dimensional subset it is described with a system of {{math|''n'' + 1}} linear inequalities:
<math display="block">\begin{matrix}
0 \le x_1 \\
\vdots \\
0 \le x_n \\
\sum\limits_{k=1}^n x_k \le 1
\end{matrix}</math>
Replacement of all "≤" with "<" gives interiors of these polytopes.

== Topological properties ==
The [[topology (structure)|topological structure]] of {{math|'''R'''<sup>''n''</sup>}} (called '''standard topology''', '''Euclidean topology''', or '''usual topology''') can be obtained not only [[#Definition and uses|from Cartesian product]]. It is also identical to the [[natural topology]] induced by [[#Euclidean space|Euclidean metric discussed above]]: a set is [[open set|open]] in the Euclidean topology [[if and only if]] it contains an [[open ball]] around each of its points. Also, {{math|'''R'''<sup>''n''</sup>}} is a [[linear topological space]] (see [[#continuity of linear maps|continuity of linear maps]] above), and there is only one possible (non-trivial) topology compatible with its linear structure. As there are many open linear maps from {{math|'''R'''<sup>''n''</sup>}} to itself which are not [[isometry|isometries]], there can be many Euclidean structures on {{math|'''R'''<sup>''n''</sup>}} which correspond to the same topology. Actually, it does not depend much even on the linear structure: there are many non-linear [[diffeomorphism]]s (and other homeomorphisms) of {{math|'''R'''<sup>''n''</sup>}} onto itself, or its parts such as a Euclidean open ball or [[#Polytopes in Rn|the interior of a hypercube]]).

{{math|'''R'''<sup>''n''</sup>}} has the [[topological dimension]] {{mvar|n}}.

An important result on the topology of {{math|'''R'''<sup>''n''</sup>}}, that is far from superficial, is [[L. E. J. Brouwer|Brouwer]]'s [[invariance of domain]]. Any subset of {{math|'''R'''<sup>''n''</sup>}} (with its [[subspace topology]]) that is [[homeomorphic]] to another open subset of {{math|'''R'''<sup>''n''</sup>}} is itself open. An immediate consequence of this is that {{math|'''R'''<sup>''m''</sup>}} is not [[homeomorphism|homeomorphic]] to {{math|'''R'''<sup>''n''</sup>}} if {{math|''m'' ≠ ''n''}} – an intuitively "obvious" result which is nonetheless difficult to prove.

Despite the difference in topological dimension, and contrary to a naïve perception, it is possible to map a lesser-dimensional{{clarify|date=April 2016}} real space continuously and [[surjective function|surjectively]] onto {{math|'''R'''<sup>''n''</sup>}}. A continuous (although not smooth) [[space-filling curve]] (an image of {{math|'''R'''<sup>1</sup>}}) is possible.{{clarify|date=April 2016}}

== Examples ==
{| align=right style="margin: 2ex 0 2ex 2em"
 | align=center |[[Image:Real 0-space.svg|52px]]
 |-
 | style="font-size:80%" |[[empty matrix|Empty]] column vector,<br/>the only element of {{math|'''R'''<sup>0</sup>}}
 |}

=== ''n'' ≤ 1 ===
Cases of {{math|0 ≤ ''n'' ≤ 1}} do not offer anything new: {{math|'''R'''<sup>1</sup>}} is the [[real line]], whereas {{math|'''R'''<sup>0</sup>}} (the space containing the empty column vector) is a [[singleton (mathematics)|singleton]], understood as a [[zero vector space]]. However, it is useful to include these as [[triviality (mathematics)|trivial]] cases of theories that describe different {{mvar|n}}.

=== ''n'' = 2 ===
[[Image:Real 2-space, orthoplex.svg|thumb|right|264px|Both hypercube and cross-polytope in {{math|'''R'''<sup>2</sup>}} are [[square]]s, but coordinates of vertices are arranged differently]]
{{details|Two-dimensional space}}
{{see also|SL2(R)}}
The case of (''x,y'') where ''x'' and ''y'' are real numbers has been developed as the [[Cartesian plane]] ''P''. Further structure has been attached with [[Euclidean vector]]s representing directed line segments in ''P''. The plane has also been developed as the [[field extension]] <math>\mathbf{C}</math> by appending roots of X<sup>2</sup> + 1 = 0 to the real field <math>\mathbf{R}.</math> The root i acts on P as a [[quarter turn]] with counterclockwise orientation. This root generates the [[group (mathematics)|group]] <math>\{i, -1, -i, +1\} \equiv \mathbf{Z}/4\mathbf{Z}</math>. When (''x,y'') is written ''x'' + ''y'' i it is a [[complex number]].

Another [[group action]] by <math>\mathbf{Z}/2\mathbf{Z}</math>, where the actor has been expressed as j, uses the line ''y''=''x'' for the [[involution (mathematics)|involution]] of flipping the plane (''x,y'') ↦ (''y,x''), an exchange of coordinates. In this case points of ''P'' are written ''x'' + ''y'' j and called [[split-complex number]]s. These numbers, with the coordinate-wise addition and multiplication according to ''jj''=+1, form a [[ring (mathematics)|ring]] that is not a field.

Another ring structure on ''P'' uses a [[nilpotent]] e to write ''x'' + ''y'' e for (''x,y''). The action of e on ''P'' reduces the plane to a line:  It can be decomposed into the [[projection (mathematics)|projection]] into the x-coordinate, then quarter-turning the result to the y-axis: e (''x'' + ''y'' e) = ''x'' e since e<sup>2</sup> = 0. A number ''x'' + ''y'' e is a [[dual number]]. The dual numbers form a ring, but, since e has no multiplicative inverse, it does not generate a group so the action is not a group action.

Excluding (0,0) from ''P'' makes [''x'' : ''y''] [[projective coordinates]] which describe the real projective line, a one-dimensional space. Since the origin is excluded, at least one of the ratios ''x''/''y'' and ''y''/''x'' exists. Then [''x'' : ''y''] = [''x''/''y'' : 1] or [''x'' : ''y''] = [1 : ''y''/''x'']. The projective line '''P'''<sup>1</sup>('''R''') is a [[topological manifold]] covered by two [[topological manifold#Coordinate charts|coordinate charts]], [''z'' : 1]  → ''z'' or [1 : ''z''] → ''z'', which form an [[atlas (topology)|atlas]]. For points covered by both charts the ''transition function'' is multiplicative inversion on an open neighborhood of the point, which provides a [[homeomorphism]] as required in a manifold. One application of the real projective line is found in [[Cayley–Klein metric]] geometry.

=== ''n'' = 3 ===
[[Image:Duality Hexa-Okta SVG.svg|thumb|left|[[Cube]] (the hypercube) and [[octahedron]] (the cross-polytope) of {{math|'''R'''<sup>3</sup>}}. Coordinates are not shown]]
{{main|Three-dimensional space}}
{{clear|left}}

=== ''n'' = 4 ===
[[Image:4-cube 3D.png|thumb|right]]
{{details|Four-dimensional space}}
{{math|'''R'''<sup>4</sup>}} can be imagined using the fact that {{num|16}} points {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ''x''<sub>4</sub>)}}, where each {{mvar|x<sub>k</sub>}} is either 0 or 1, are vertices of a [[tesseract]] (pictured), the 4-hypercube (see [[#Polytopes in Rn|above]]).

The first major use of {{math|'''R'''<sup>4</sup>}} is a [[spacetime]] model: three spatial coordinates plus one [[time|temporal]]. This is usually associated with [[theory of relativity]], although four dimensions were used for such models since [[Galileo Galilei|Galilei]]. The choice of theory leads to different structure, though: in [[Galilean relativity]] the {{mvar|t}} coordinate is privileged, but in Einsteinian relativity it is not. Special relativity is set in [[Minkowski space]]. General relativity uses curved spaces, which may be thought of as {{math|'''R'''<sup>4</sup>}} with a [[metric tensor (general relativity)|curved metric]] for most practical purposes. None of these structures provide a (positive-definite) [[metric (mathematics)|metric]] on {{math|'''R'''<sup>4</sup>}}.

Euclidean {{math|'''R'''<sup>4</sup>}} also attracts the attention of mathematicians, for example due to its relation to [[quaternion]]s, a 4-dimensional [[algebra over a field|real algebra]] themselves. See [[rotations in 4-dimensional Euclidean space]] for some information.

In differential geometry, {{math|1=''n'' = 4}} is the only case where {{math|'''R'''<sup>''n''</sup>}} admits a non-standard [[differential structure]]: see [[exotic R4|exotic R<sup>4</sup>]].

== Norms on {{math|'''R'''<sup>''n''</sup>}}==
One could define many norms on the [[vector space]] {{math|'''R'''<sup>''n''</sup>}}. Some common examples are

* the [[p-norm]], defined by <math display="inline">\|\mathbf{x}\|_p := \sqrt[p]{\sum_{i=1}^n|x_i|^p}</math> for all <math>\mathbf{x} \in \mathbf{R}^n</math> where <math>p</math> is a positive integer. The case <math>p = 2</math> is very important, because it is exactly the [[Euclidean norm]].
* the <math>\infty</math>-norm or [[maximum norm]], defined by <math>\|\mathbf{x}\|_\infty:=\max \{x_1,\dots,x_n\}</math> for all <math>\mathbf{x} \in \mathbf{R}^n</math>. This is the limit of all the [[p-norm]]s: <math display="inline">\|\mathbf{x}\|_\infty = \lim_{p \to \infty} \sqrt[p]{\sum_{i=1}^n|x_i|^p}</math>.

A really surprising and helpful result is that every norm defined on {{math|'''R'''<sup>''n''</sup>}} is [[Equivalent norm|equivalent]]. This means for two arbitrary norms <math>\|\cdot\|</math> and <math>\|\cdot\|'</math> on {{math|'''R'''<sup>''n''</sup>}} you can always find positive real numbers <math>\alpha,\beta > 0</math>, such that
<math display="block">\alpha \cdot \|\mathbf{x}\| \leq \|\mathbf{x}\|' \leq \beta\cdot\|\mathbf{x}\|</math> for all <math>\mathbf{x} \in \R^n</math>.

This defines an [[equivalence relation]] on the set of all norms on {{math|'''R'''<sup>''n''</sup>}}. With this result you can check that a sequence of vectors in {{math|'''R'''<sup>''n''</sup>}} converges with <math>\|\cdot\|</math> if and only if it converges with <math>\|\cdot\|'</math>.

Here is a sketch of what a proof of this result may look like:

Because of the [[equivalence relation]] it is enough to show that every norm on {{math|'''R'''<sup>''n''</sup>}} is equivalent to the [[Euclidean norm]] <math>\|\cdot\|_2</math>. Let <math>\|\cdot\|</math> be an arbitrary norm on {{math|'''R'''<sup>''n''</sup>}}. The proof is divided in two steps:

* We show that there exists a <math>\beta > 0</math>, such that <math>\|\mathbf{x}\| \leq \beta \cdot \|\mathbf{x}\|_2</math> for all <math>\mathbf{x} \in \mathbf{R}^n</math>. In this step you use the fact that every <math>\mathbf{x} = (x_1, \dots, x_n) \in \mathbf{R}^n</math> can be represented as a linear combination of the standard [[Basis (linear algebra)|basis]]: <math display="inline">\mathbf{x} = \sum_{i=1}^n e_i \cdot x_i</math>. Then with the [[Cauchy–Schwarz inequality]] <math display="block">\|\mathbf{x}\| = \left\|\sum_{i=1}^n e_i \cdot x_i \right\|\leq \sum_{i=1}^n \|e_i\| \cdot |x_i|
\leq \sqrt{\sum_{i=1}^n \|e_i\|^2} \cdot \sqrt{\sum_{i=1}^n |x_i|^2} = \beta \cdot \|\mathbf{x}\|_2,</math> where <math display="inline">\beta := \sqrt{\sum_{i=1}^n \|e_i\|^2}</math>.
* Now we have to find an <math>\alpha > 0</math>, such that <math>\alpha\cdot\|\mathbf{x}\|_2 \leq \|\mathbf{x}\|</math> for all <math>\mathbf{x} \in \mathbf{R}^n</math>. Assume there is no such <math>\alpha</math>. Then there exists for every <math>k \in \mathbf{N}</math> a <math>\mathbf{x}_k \in \mathbf{R}^n</math>, such that <math>\|\mathbf{x}_k\|_2 > k \cdot \|\mathbf{x}_k\|</math>. Define a second sequence <math>(\tilde{\mathbf{x}}_k)_{k \in \mathbf{N}}</math> by <math display="inline">\tilde{\mathbf{x}}_k := \frac{\mathbf{x}_k}{\|\mathbf{x}_k\|_2}</math>. This sequence is bounded because <math>\|\tilde{\mathbf{x}}_k\|_2 = 1</math>. So because of the [[Bolzano–Weierstrass theorem]] there exists a convergent subsequence <math>(\tilde{\mathbf{x}}_{k_j})_{j\in\mathbf{N}}</math> with limit <math>\mathbf{a} \in</math> {{math|'''R'''<sup>''n''</sup>}}. Now we show that <math>\|\mathbf{a}\|_2 = 1</math> but <math>\mathbf{a} = \mathbf{0}</math>, which is a contradiction. It is <math display="block">\|\mathbf{a}\| \leq \left\|\mathbf{a} - \tilde{\mathbf{x}}_{k_j}\right\| + \left\|\tilde{\mathbf{x}}_{k_j}\right\| \leq \beta \cdot \left\|\mathbf{a} - \tilde{\mathbf{x}}_{k_j}\right\|_2 + \frac{\|\mathbf{x}_{k_j}\|}{\|\mathbf{x}_{k_j}\|_2} \ \overset{j \to \infty}{\longrightarrow} \ 0,</math> because <math>\|\mathbf{a}-\tilde{\mathbf{x}}_{k_j}\| \to 0</math> and <math>0 \leq \frac{\|\mathbf{x}_{k_j}\|}{\|\mathbf{x}_{k_j}\|_2} < \frac{1}{k_j}</math>, so <math>\frac{\|\mathbf{x}_{k_j}\|}{\|\mathbf{x}_{k_j}\|_2} \to 0</math>. This implies <math>\|\mathbf{a}\| = 0</math>, so <math>\mathbf{a}= \mathbf{0}</math>. On the other hand <math>\|\mathbf{a}\|_2 = 1</math>, because <math>\|\mathbf{a}\|_2 = \left\| \lim_{j \to \infty}\tilde{\mathbf{x}}_{k_j} \right\|_2 = \lim_{j \to \infty} \left\| \tilde{\mathbf{x}}_{k_j} \right\|_2 = 1</math>. This can not ever be true, so the assumption was false and there exists such a <math>\alpha > 0</math>.

== See also ==
* [[Exponential object]], for theoretical explanation of the superscript notation
* [[Geometric space]]
* [[Real projective space]]

==Sources==
*{{cite book | author=Kelley, John L. | title=General Topology | publisher=Springer-Verlag | year=1975 | isbn= 0-387-90125-6 }}
*{{cite book | author=Munkres, James | title=Topology | publisher=Prentice-Hall | year=1999 | isbn= 0-13-181629-2 }}

{{Real numbers}}

[[Category:Real numbers|N]]
[[Category:Topological vector spaces]]
[[Category:Analytic geometry]]
[[Category:Multivariable calculus]]
[[Category:Mathematical analysis]]