Euclidean space

In mathematics, Euclidean space is a generalization of the 2 and 3dimensional spaces studied by Euclid. The generalization applies Euclid's concept of distance, and the related concepts of length and angle, to a coordinate system in any number of dimensions. It is the "standard" example of a finitedimensional, real, inner product space.
A Euclidean space is a particular metric space that enables the investigation of topological properties such as compactness. An inner product space is a generalization of a Euclidean space. Both inner product spaces and metric spaces are explored within functional analysis.
Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean and nonEuclidean geometry. One mathematical motivation for defining a distance function is the ability to define an open ball around points in the space. This fundamental concept justifies a differential calculus between a Euclidean space and other manifolds. Differential geometry brings such a differential calculus into play, together with a technique of launching a mobile, local Euclidean space, to explore the properties of a nonEuclidean manifolds.
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Real coordinate space
Let R denote the field of real numbers. For any nonnegative integer n, the space of all ntuples of real numbers forms an ndimensional vector space over R sometimes called real coordinate space and denoted R^{n}.
An element of R^{n} is written x = (x_{1}, x_{2}, …, x_{n}) where each x_{i} is a real number. The vector space operations on R^{n} are defined by
 <math>\mathbf{x} + \mathbf{y} = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)<math>
 <math>a\,\mathbf{x} = (a x_1, a x_2, \ldots, a x_n)<math>
Real coordinate space R^{n} comes with a standard basis:
 <math>\mathbf{e}_1 = (1, 0, \ldots, 0)<math>
 <math>\mathbf{e}_2 = (0, 1, \ldots, 0)<math>
 <math>\vdots<math>
 <math>\mathbf{e}_n = (0, 0, \ldots, 1)<math>
An arbitrary vector in R^{n} can then be written in the form
 <math>\mathbf{x} = \sum_{i=1}^n x_i \mathbf{e}_i<math>
Real coordinate space is the prototypical example of a real ndimensional vector space. In fact, every real ndimensional vector space V is isomorphic to R^{n}. This isomorphism is not canonical however. A choice of isomorphism is equivalent to a choice of basis for V (by looking at the image of the standard basis for R^{n} in V). The reason for working with arbitrary vector spaces instead of R^{n} is that it is often preferable to work in a coordinatefree manner (i.e. without choosing a preferred basis).
Euclidean structure
Euclidean space is more than just real coordinate space. In order to do Euclidean geometry one needs to be able to talk about the distance between points and the angles between lines or vectors. The natural way in which to do this is to introduce what is called an inner product or dot product on R^{n}. This product is defined by
 <math>\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n.<math>
The dot product of any two vectors x and y gives an real number. This product allows us to define the "length" of a vector x in the following way
 <math>\\mathbf{x}\ = \sqrt{\mathbf{x}\cdot\mathbf{x}} = \sqrt{\sum_{i=1}^{n}(x_i)^2}<math>
This length function satisfies the required properties of a norm and is called the Euclidean norm on R^{n}. The (interior) angle θ between x and y is then given by
 <math>\theta = \cos^{1}\left(\frac{\mathbf{x}\cdot\mathbf{y}}{\\mathbf{x}\\\mathbf{y}\}\right)<math>
where cos^{−1} is the arccosine function. Finally, one can use the norm to define a distance function (or metric) on R^{n} in the following manner
 <math>d(\mathbf{x}, \mathbf{y}) = \\mathbf{x}  \mathbf{y}\ = \sqrt{\sum_{i=1}^n (x_i  y_i)^2}.<math>
The form of this distance function is based on the Pythagorean theorem, and is called the Euclidean metric.
Real coordinate space together with the above Euclidean structure (dot product and the associated norm and metric) is called Euclidean space often denoted by E^{n}. (Many authors refer to R^{n} itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure on E^{n} gives it the structure of an inner product space (in fact a Hilbert space), a normed vector space, and a metric space.
Euclidean topology
Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric. The metric topology on E^{n} is called the Euclidean topology. A set is open in the Euclidean topology if and only if it contains an open ball around each of its points. The Euclidean topology turns out to be equivalent to the product topology on R^{n} considered as a product of n copies of the real line R (with its standard topology).
An important result on the topology of R^{n}, that is far from superficial, is Brouwer's invariance of domain. Any subset of R^{n} (with its subspace topology) which is homeomorphic to another open subset of R^{n} is itself open. An immediate consequence of this is that R^{m} is not homeomorphic to R^{n} if m ≠ n — an intuitively "obvious" result which is nonetheless difficult to prove.
Euclidean nspace is the prototypical example of an nmanifold, in fact, a smooth manifold. For n ≠ 4, any differentiable nmanifold that is homeomorphic to R^{n} is also diffeomorphic to it. The surprising fact that this is not also true for n = 4 was proved by Simon Donaldson in 1982; the counterexamples are called exotic (or fake) 4spaces.
Euclidean space is also known as linear manifold. An mdimensional linear submanifold of R^{n} is a Euclidean space of m dimensions embedded in it (as an affine subspace). For example, any straight line in some higherdimensional Euclidean space is a 1dimensional linear submanifold of that space.
See also
es:Espacio euclídeo fr:Espace euclidien ja:ユークリッド空間 pl:PrzestrzeÅ„ euklidesowa zh:欧几里德空间