Euclidean Space

A metric space that is linear and finite dimensional metric space a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality.

Euclidean space. It was introduced by the ancient greek mathematician euclid of alexandria and the qualifier euclidean is used to distinguish it from other spaces that were later discovered in physics and modern m. As a simple example one vector in three dimensional space could be represented by the finite list 7 2 9. Typical notation for x 2 rn will be x x1 x2 xn. The basic vector space we shall denote by rthe fleld of real numbers.

Euclidean space is the fundamental space of classical geometry. Euclidean space a space in which euclid s axioms and definitions apply. Euclidean space in geometry a two or three dimensional space in which the axioms and postulates of euclidean geometry apply. A euclidean space is a finite dimensional vector space over the reals rpof with an inner product middot dieffenbach 2013 the basic idea of a finite dimensional vector space is that a finite list of vectors spans across the space.

Also a space in any finite number of dimensions in which points are designated by coordinates one for each dimension and the distance between two points is given by a distance formula. Originally it was the three dimensional space of euclidean geometry but in modern mathematics there are euclidean spaces of any nonnegative integer dimension including the three dimensional space and the euclidean plane dimension two. Here x is called a point or a vector and x1 x2 xn are called the coordinates of x. Euclidean n space sometimes called cartesian space or simply n space is the space of all n tuples of real numbers x 1 x 2 x n.

The formal definition of euclidean space can be written as. Rof ordered n tuples of real numbers n factors.