Euclidean Space

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Related to N-dimensional Euclidean space: Euclidean vector space

euclidean space

[yü′klid·ē·ən ′spās]
A space consisting of all ordered sets (x1, …, xn ) of n numbers with the distance between (x1, …, xn ) and (y1, …, yn ) being given by the number n is called the dimension of the space.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Euclidean Space


in mathematics, a space whose properties are described by the axioms of Euclidean geometry. In a more general sense, a Euclidean space is an n-dimensional vector space, into which several special Cartesian coordinates can be introduced so that its metric is defined in the following manner: If point M has the coordinates (x1x2, …, xn and point M* has the coordinates (x1*, x2*, …, xn*), then the distance between these points is

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
In order to grasp the main ideas behind a geometric analogue of HRR let us consider an orthonormal basis [b.sub.1],...,[b.sub.n] in some n-dimensional Euclidean space. Now consider two vectors x = [[summation].sup.n.sub.k=1] [x.sub.k][b.sub.k] and y = [[summation].sup.n.sub.k=1] [y.sub.k][b.sub.k].
As noted above, [A.sub.n] is contained in the n-dimensional Euclidean space
As an example, consider an N-dimensional Euclidean space where N is a large number, and a vp-tree of order three is built to index the uniformly distributed data points in that space.
By [R.sup.n] we denote the n-dimensional real linear space and by [E.sup.n] the n-dimensional Euclidean space. The linear space [R.sup.n] equipped with an arbitrary norm [parallel]*[parallel] is called a (Minkowski or) normed space.