nspace

n-space, or n-dimensional space, usually refers to Euclidean n-space, the set R^n of all n-tuples of real numbers equipped with the standard Euclidean metric. It generalizes the line and the plane: 1-space is a line, 2-space a plane, and 3-space ordinary physical space. In general, n-space is the Cartesian product of n copies of the real line and carries the standard topology and a metric such as the Euclidean distance.

As a mathematical object, n-space is a vector space over the real numbers of dimension n, with the standard basis e_1, ..., e_n. Subspaces and affine subspaces are defined by linear equations or by fixing coordinates. More generally, one may consider F^n over any field F, yielding a finite-dimensional n-dimensional vector space with similar structure.

Topology and geometry: Open and closed sets follow from the metric, and notions such as convergence, continuity, and compactness are defined accordingly. Other norms, such as L^p norms, induce alternative but equivalent geometries on the same underlying set. n-space provides the natural setting for linear algebra, multivariable calculus, and analytic geometry, enabling coordinate representation, linear transformations, and inner products.

Generalizations and related concepts: Infinite-dimensional analogues include Hilbert spaces and Banach spaces, which extend the ideas of Euclidean space to infinite dimensions. Complex Euclidean space C^n uses complex coordinates with the Hermitian inner product. n-space also appears in physics as a configuration or phase space and in computer graphics and data analysis as a coordinate model for scenes and datasets.

See also: Euclidean space, R^n, Cartesian product, vector space, metric space.