Euclidean space

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.

Related Topics:
Mathematics - Euclid - Distance - Length - Angle - Coordinate system - Dimension - Finite-dimensional - Real - Inner product space

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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.

Related Topics:
Metric space - Topological - Compactness - Inner product space - Functional analysis

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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.

Related Topics:
Manifold - Euclidean - Non-Euclidean geometry - Open ball - Differential calculus - Differential geometry

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