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EUCLIDEAN SPACE

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The three dimensional space that we live in is called a three dimensional Euclidean space and the properties of objects in this space are described by solid geometry. A flat surface (such as a piece of paper) is a two dimensional Euclidean space, and the properties of objects in that space are described by plane geometry. These properties were first stated explicitly by Euclid around 300 BC from which we get the name "Euclidean space" as well as "Euclidean geometry". An essential property of a Euclidean space is its flatness. Other spaces exist which are not Euclidean. For example, the surface of a sphere is not a Euclidean space, nor is the four-dimensional spacetime described by the theory of relativity. The properties of the familiar two- and three- dimensional Euclidean space can be extended to form a Euclidean space of any number of dimensions, which is called an n-dimensional Euclidean space. In order for this process to be correct, the properties of the familiar Euclidean spaces must be very carefully expressed and then extended to an arbitrary dimension. Although the resulting mathematics is rather abstract, it nevertheless captures the essential nature of the Euclidean spaces we are all familiar with.

Mathematically an n-dimensional Euclidean space (also called Cartesian space or n-space) refers to the space of ordered n-tuples of real numbers along with the associated operations of component-wise addition and scalar multiplication which make it into a vector space, and the dot product which makes it into an inner product space and a normed space with the Euclidean norm. Euclidean space is a generalization of the 2- and 3-dimensional metric 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 finite-dimensional, real, inner product space.

A Euclidean space 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 non-Euclidean 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 non-Euclidean manifolds.

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Real coordinate space

Let R denote the field of real numbers. For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R sometimes called real coordinate space and denoted Rn.

An element of Rn is written x = (x1, x2, …, xn) where each xi is a real number. The vector space operations on Rn are defined by

\mathbf{x} + \mathbf{y} = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n)
a\,\mathbf{x} = (a x_1, a x_2, \ldots, a x_n)

Real coordinate space Rn comes with a standard basis:

\mathbf{e}_1 = (1, 0, \ldots, 0)
\mathbf{e}_2 = (0, 1, \ldots, 0)
\vdots
\mathbf{e}_n = (0, 0, \ldots, 1)

An arbitrary vector in Rn can then be written in the form

\mathbf{x} = \sum_{i=1}^n x_i \mathbf{e}_i

Real coordinate space is the prototypical example of a real n-dimensional vector space. In fact, every real n-dimensional vector space V is isomorphic to Rn. 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 Rn in V). The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free 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 Rn. This product is defined by

\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n.

The dot product of any two vectors x and y gives a real number. This product allows us to define the "length" of a vector x in the following way

\|\mathbf{x}\| = \sqrt{\mathbf{x}\cdot\mathbf{x}} = \sqrt{\sum_{i=1}^{n}(x_i)^2}

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rn. The (interior) angle θ between x and y is then given by

\theta = \cos^{-1}\left(\frac{\mathbf{x}\cdot\mathbf{y}}{\|\mathbf{x}\|\|\mathbf{y}\|}\right)

where cos−1 is the arccosine function. Finally, one can use the norm to define a distance function (or metric) on Rn in the following manner

d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\| = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}.

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 En. (Many authors refer to Rn itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure on En gives it the structure of an inner product space (in fact a Hilbert space), a normed vector space, and a metric space.

Alternative definition

In fact, Euclidean space En is a real n-dimensional affine space such as its corresponding linear or vector space ( isomorphic to the linear or vector space Rn ) has an inner product.

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 En 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 Rn considered as a product of n copies of the real line R (with its standard topology).

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) which 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.

Euclidean n-space is the prototypical example of an n-manifold, in fact, a smooth manifold. For n ≠ 4, any differentiable n-manifold that is homeomorphic to Rn 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) 4-spaces.

Euclidean space is also known as a linear manifold. An m-dimensional linear submanifold of Rn is a Euclidean space of m dimensions embedded in it (as an affine subspace). For example, any straight line in some higher-dimensional Euclidean space is a 1-dimensional linear submanifold of that space.

See also