In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional 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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Euclid of Alexandria (Greek: ) (ca. ... personal space, proxemics. ... In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth... This article is about angles in geometry. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

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. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set (possibly infinite) of first-order sentences is satisfiable, i. ... In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...

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 a non-Euclidean manifolds. This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... The solid interior of a sphere or circle; in mathematics, latter terms refer specifically to the (n-1)-dimensional surface of an n-dimensional solid ball. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...

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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects (an infinite sequence is a family). ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

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

Real coordinate space Rⁿ comes with a standard basis: In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ...

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

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). In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... Canonical--an adjective derived from canon--essentially means standard or generally accepted or part of the backstory. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

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 In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... personal space, proxemics. ... This article is about angles in geometry. ... In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

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

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 In linear algebra, functional analysis and related areas of mathematics a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

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 For distance between people, see proxemics. ... See: International System of Units, colloquially called the Metric System, and also metrication. ...

The form of this distance function is based on the Pythagorean theorem, and is called the Euclidean metric. The Pythagorean theorem : a2 + b2 = c2 In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry between the three sides of a right-angled triangle. ... The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...

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. In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

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). In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, the real line is simply the set of real numbers. ...

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 R^m is not homeomorphic to Rⁿ if m ≠ n — an intuitively "obvious" result which is nonetheless difficult to prove. Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ... Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... This word should not be confused with homomorphism. ...

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. This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... This word should not be confused with homomorphism. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is a mathematician famous for his work on exotic four-dimensional spaces in differential geometry using instantons, and the discovery of new differential invariants. ... 1982 is a common year starting on Friday of the Gregorian calendar. ...

Euclidean space is also known as 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. An affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. ...

See also

• Euclidean geometry
• Euclidean distance
• Minkowski space