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In mathematics, real coordinate space of n dimensions, written R (R with superscript n, also written ℝ with blackboard bold R) is a coordinate space that allows several (n) real variables to be treated as a single variable. With various numbers of dimensions (sometimes unspecified), R is used in many areas of pure and applied mathematics, as well as in physics. It is the prototypical real vector space and a frequently used representation of Euclidean n-space. Due to the latter fact, geometric metaphors are widely used for R, namely a plane for R and three-dimensional space for R.

Definition and uses

For any natural number n, the set R consists of all n-tuples of real numbers (R). It is called (the) "n-dimensional real space". As it can be constructed as Cartesian product of n instances of the set R, it inherits some of its structure, notably:

• R has addition and scalar multiplication: see below
• R is a topological space: see below

An element of R is written

${\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})}$

where each x_i is a real number.

For each n there exists only one R, the real n-space.

Purely mathematical uses of R can be roughly classified as follows, although these uses overlap. First, linear algebra studies its own properties under vector addition and linear transformations and use in as a model of any n-dimensional real vector space. Second, it is used in mathematical analysis to represent the domain of a function of n real variables in an uniform way, as well as a space to which the graph of a real-valued function of n − 1 real variables is a subset. The third use parametrizes geometric points with elements of R; it is common in analytic, differential and algebraic geometries.

R, together with supplemental structures on it, is also extensively used in mathematical physics, dynamical systems theory, mathematical statistics and probability theory.

In applied mathematics, numerical analysis, and so on, arrays, sequences, and other collections of numbers in applications can be seen as the use of R too.

Vector space

R forms an n-dimensional vector space over the field of real numbers. The operations on R are defined by

${\displaystyle \mathbf {x} +\mathbf {y} =(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n})}$

${\displaystyle \alpha \mathbf {x} =(\alpha x_{1},\alpha x_{2},\ldots ,\alpha x_{n}).}$

The zero vector is given by

${\displaystyle \mathbf {0} =(0,0,\ldots ,0)}$

and the additive inverse of the vector x is given by

${\displaystyle -\mathbf {x} =(-x_{1},-x_{2},\ldots ,-x_{n}).}$

Matrix notation

In standard matrix notation, each element of R is typically written as a column vector

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}}$

and sometimes as a row vector:

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\dots &x_{n}\end{bmatrix}}.}$

The coordinate space R may then be interpreted as the space of all n × 1 column vectors, or all 1 × n row vectors with the ordinary matrix operations of addition and scalar multiplication.

Linear transformations from R to R may then be written as m × n matrices which act on the elements of R via left multiplication (when the elements of R are column vectors) and on elements of R via right multiplication (when they are row vectors). The formula for left multiplication, a special case of matrix multiplication, is:

${\displaystyle (A{\mathbf {x} })_{k}=\sum \limits _{l=1}^{n}A_{kl}x_{l}}$

Any linear transformation is a continuous function (see below). Also, a matrix define an open map from R to R if and only if the rank of the matrix equals to m.

Standard basis

The coordinate space R comes with a standard basis:

${\displaystyle {\begin{aligned}\mathbf {e} _{1}&=(1,0,\ldots ,0)\\\mathbf {e} _{2}&=(0,1,\ldots ,0)\\&{}\ \vdots \\\mathbf {e} _{n}&=(0,0,\ldots ,1)\end{aligned}}}$

To see that this is a basis, note that an arbitrary vector in R can be written uniquely in the form

${\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {e} _{i}.}$

Geometric properties and uses

Orientation

The fact that real numbers, unlike many other fields, constitute an ordered field yields an orientation structure on R. Any full-rank linear map of R to itself either preserves or reverses orientation of the space depending on the sign of the determinant of its matrix. If one permutes coordinates (or, in other words, elements of the basis), the resulting orientation will depend on the parity of the permutation.

Diffeomorphisms of R or domains in it, by their virtue to avoid zero Jacobian, are also classified to orientation-preserving and orientation-reversing. It has important consequences for the theory of differential forms, whose applications include electrodynamics.

Another manifestation of this structure is that the point reflection in R has different properties depending on evenness of n. For even n it preserves orientation, while for odd n it is reversed (see also improper rotation).

Affine space

R understood as an affine space is the same space, where R as a vector space acts by translations. Conversely, a vector has to be understood as a "difference between two points", usually illustrated by a directed line segment connecting two points. The distinction says that there is no canonical choice of where the origin should go in an affine n-space, because it can be translated anywhere.

Euclidean space

The dot product

${\displaystyle \mathbf {x} \cdot \mathbf {y} =\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}}$

defines the norm | x | =√x⋅x on the vector space R. If every vector has its Euclidean norm, then for any pair of points the distance

${\displaystyle d(\mathbf {x} ,\mathbf {y} )=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}}$

is defined, providing a metric space structure on R in addition to its affine structure.

Like for vector space structure, the dot product and Euclidean distance usually are assumed existing on R without special explanations. Though, the real n-space and an Euclidean n-space are distinct objects, strictly speaking. Any Euclidean n-space has a coordinate system where dot product and Euclidean distance have the form shown above, called Cartesian. But there are many Cartesian coordinate systems on an Euclidean space. Conversely, the formula for Euclidean metric above defines the standard Euclidean structure on R, but it is not the only possible one. In spite of this clauses, concepts of R and of an Euclidean n-space are not always distinguished 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 R, this choice is the most common, and almost exclusive one in differential geometry.

On the other hand, Whitney embedding theorems state that any real differentiable m-dimensional manifold can be embedded into R.

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Other appearances

Other structures considered on R include the one of a pseudo-Euclidean space and symplectic structure.

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Polytopes in R

There are three families of polytopes which have simple representations in R spaces, for any n, and can be used to visualize any affine coordinate system in a real n-space. Vertices of hypercube have coordinates (x₁, x₂, … , x_n) where each x_k is either 0 or 1. Actually, any two numbers can be chosen instead of 0 and 1, for example −1 and 1. An n-hypercube can be thought of as the Cartesian product of n identical intervals (such as the unit interval [0,1]) on the real line. As an n-dimensional subset it can be described with a system of 2n inequalities:

${\displaystyle \displaystyle {\begin{matrix}0\leq x_{1}\leq 1\\\vdots \\0\leq x_{n}\leq 1\end{matrix}}}$ (for [0,1])

${\displaystyle \displaystyle {\begin{matrix}|x_{1}|\leq 1\\\vdots \\|x_{n}|\leq 1\end{matrix}}}$ (for [−1,1])

Each vertex of the cross-polytope has, for some k, the x_k coordinate equal to ±1 and all other coordinates equal to 0 (such that it is the kth standard basis vector up to sign). This is a dual polytope of hypercube. As an n-dimensional subset it can be described with a single inequality which uses the absolute value operation:

${\displaystyle \sum \limits _{k=1}^{n}|x_{k}|\leq 1\,,}$

but this can be expressed with a system of 2 linear inequalities as well.

The third polytope with simply enumerable coordinates is the standard simplex, whose vertices are n standard basis vectors and the origin (0, 0, … , 0). As an n-dimensional subset it is described with a system of n + 1 linear inequalities:

${\displaystyle {\begin{matrix}0\leq x_{1}\\\vdots \\0\leq x_{n}\\\sum \limits _{k=1}^{n}x_{k}\leq 1\end{matrix}}}$

Replacement of all "≤" with "<" gives interiors of these polytopes.

Topological properties

The topological structure of R (called standard topology, Euclidean topology, or usual topology) can be obtained not only from Cartesian product. It is also identical to the natural topology induced by Euclidean metric discussed above: a set is open in the Euclidean topology if and only if it contains an open ball around each of its points. Also, R is a linear topological space (see #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 R to itself which are not isometries, there can be many Euclidean structures on R which correspond to the same topology. Actually, it does not depend much even on the linear structure: there are many non-linear diffeomorphisms (and other homeomorphisms) of R onto itself, or its parts such as an Euclidean open ball or the interior of a hypercube).

R has the topological dimension n. An important result on the topology of R, that is far from superficial, is Brouwer's invariance of domain. Any subset of R (with its subspace topology) that is homeomorphic to another open subset of R is itself open. An immediate consequence of this is that R is not homeomorphic to R if 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 real space continuously and surjectively onto R. A continuous (although not smooth) space-filling curve (an image of R) is possible.

Examples

n ≤ 1

Empty column vector, the only element of R

Cases of 0 ≤ n ≤ 1 do not offer anything new: R is the real line, whereas R (the space of empty vectors) is a singleton, understood as zero vector space. Though, it is useful to include these "trivial" cases to theories which are appropriate for different n.

n = 2

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n = 3

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n = 4

R can be imagined using the fact that 16 points (x₁, x₂, x₃, x₄), where each x_k is either 0 or 1, are vertices of a tesseract (pictured), the 4-hypercube (see above).

The first major use of R is a spacetime model: three spacial coordinates plus one temporal. This is usually associated with theory of relativity, although four dimensions were continuously used for such models since Galilei. The choice of theory leads to different structure, though: in Galilean relativity the 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 R with a curved metric for most practical purposes, though. Any of these structures does not provide a (positive) metric on R.

Euclidean R also attracts the attention of mathematicians, for example due to its relation to quaternions. See SO(4) for some information.

In differential geometry, n = 4 is the only case where R admits a non-standard differential structure: see exotic R.

Generalizations

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See also

• Exponential object, for theoretical explanation of the superscript notation
• Real projective space

Footnotes

1. Unlike many situations in mathematics where certain object is unique up to isomorphism, R in unique in the strong sense: any its element is described explicitly with its n real coordinates.

References

• Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
• Munkres, James (1999). Topology. Prentice-Hall. ISBN 0-13-181629-2.

• Real numbers
• Linear algebra
• Analytic geometry
• Topological spaces
• Mathematical analysis