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- Learn more:✕This summary was generated using AI based on multiple online sources. To view the original source information, use the "Learn more" links.A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called Euclidean affine spaces to distinguish them from Euclidean vector spaces.en.wikipedia.org/wiki/Euclidean_spaceA Euclidean space is an affine space over the reals such that the associated vector space is a finite-dimensional inner product over the real numbers. https://en.m.wikipedia.org/wiki/Euclidean_space#Technical_definitionmath.stackexchange.com/questions/4797610/termi…In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines.en.wikipedia.org/wiki/Euclidean_planes_in_three-di…
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Euclidean space - Wikipedia
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n … See more
History of the definition
Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in See moreSome basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an affine space. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections. See more
An isometry between two metric spaces is a bijection preserving the distance, that is
In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm
since the norm of a vector is its distance from the zero … See moreFor any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a … See more
The vector space $${\displaystyle {\overrightarrow {E}}}$$ associated to a Euclidean space E is an inner product space See more
The Euclidean distance makes a Euclidean space a metric space, and thus a topological space. This topology is called the See more
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