Real coordinate space
In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted Rn or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors. Special cases are called the real line R1, the real coordinate plane R2, and the real coordinate three-dimensional space R3. With component-wise addition and scalar multiplication, it is a real vector space.
The coordinates over any 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 n, En (Euclidean line, E; Euclidean plane, E2; Euclidean three-dimensional space, E3) form a real coordinate space of dimension n.
These one to one correspondences 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.
For any natural number n, the set Rn consists of all n-tuples of real numbers (R). It is called the "n-dimensional real space" or the "real n-space".
An element of Rn is thus a n-tuple, and is written
The real n-space has several further properties, notably:
These properties and structures of Rn make it fundamental in almost all areas of mathematics and their application domains, such as statistics, probability theory, and many parts of physics.
Any function f(x1, x2, ..., xn) of n real variables can be considered as a function on Rn (that is, with Rn as its domain). The use of the real n-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for n = 2, a function composition of the following form:
then F is not necessarily continuous. Continuity is a stronger condition: the continuity of f in the natural R2 topology (discussed below), also called multivariable continuity, which is sufficient for continuity of the composition F.
Geometric properties and uses[edit]
Orientation[edit]
The fact that real numbers, unlike many other fields, constitute an ordered field yields an orientation structure on Rn. Any full-rank linear map of Rn 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 Rn 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 Rn 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).
Topological properties[edit]
The topological structure of Rn (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, Rn 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 Rn to itself which are not isometries, there can be many Euclidean structures on Rn 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 Rn onto itself, or its parts such as a Euclidean open ball or the interior of a hypercube).
Rn has the topological dimension n.
An important result on the topology of Rn, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rn (with its subspace topology) that is homeomorphic to another open subset of Rn is itself open. An immediate consequence of this is that Rm is not homeomorphic to Rn 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 Rn. A continuous (although not smooth) space-filling curve (an image of R1) is possible.
the , defined by for all where is a positive integer. The case is very important, because it is exactly the Euclidean norm.
![{\textstyle \|\mathbf {x} \|_{p}:={\sqrt[{p}]{\sum _{i=1}^{n}|x_{i}|^{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf35f8f8bd1290edf59ab62204c0dffefa08e3f)
p-norm
![{\textstyle \|\mathbf {x} \|_{\infty }=\lim _{p\to \infty }{\sqrt[{p}]{\sum _{i=1}^{n}|x_{i}|^{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/372b0f141ab63d3195ba2eb91c83f792729b74ff)
One could define many norms on the vector space Rn. Some common examples are
A really surprising and helpful result is that every norm defined on Rn is equivalent. This means for two arbitrary norms and on Rn you can always find positive real numbers , such that
This defines an equivalence relation on the set of all norms on Rn. With this result you can check that a sequence of vectors in Rn converges with if and only if it converges with .
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 Rn is equivalent to the Euclidean norm . Let be an arbitrary norm on Rn. The proof is divided in two steps:
