In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, which is defined by the dot product.[disputed – discuss]
More generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions. It may also refer to a seven-dimensional manifold such as a 7-sphere, or a variety of other geometric constructions.
Seven-dimensional spaces have a number of special properties, many of them related to the octonions. An especially distinctive property is that a cross product can be defined only in three or seven dimensions. This is related to Hurwitz's theorem, which prohibits the existence of algebraic structures like the quaternions and octonions in dimensions other than 2, 4, and 8. The first exotic spheres ever discovered were seven-dimensional.