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v t e

A one-dimensional space (1D space) is a mathematical space in which location can be specified with a single coordinate. An example is the number line, each point of which is described by a single real number. Any straight line or smooth curve is a one-dimensional space, regardless of the dimension of the ambient space in which the line or curve is embedded. Examples include the circle on a plane, or a parametric space curve. In physical space, a 1D subspace is called a "linear dimension" (rectilinear or curvilinear), with units of length (e.g., metre).

In algebraic geometry there are several structures that are one-dimensional spaces but are usually referred to by more specific terms. Any field ${\displaystyle K}$ is a one-dimensional vector space over itself. The projective line over ${\displaystyle K,}$ denoted ${\displaystyle \mathbf {P} ^{1}(K),}$ is a one-dimensional space. In particular, if the field is the complex numbers ${\displaystyle \mathbb {C} ,}$ then the complex projective line ${\displaystyle \mathbf {P} ^{1}(\mathbb {C} )}$ is one-dimensional with respect to ${\displaystyle \mathbb {C} }$ (but is sometimes called the Riemann sphere, as it is a model of the sphere, two-dimensional with respect to real-number coordinates).

For every eigenvector of a linear transformation T on a vector space V, there is a one-dimensional space A ⊂ V generated by the eigenvector such that T(A) = A, that is, A is an invariant set under the action of T.

In Lie theory, a one-dimensional subspace of a Lie algebra is mapped to a one-parameter group under the Lie group–Lie algebra correspondence.

More generally, a ring is a length-one module over itself. Similarly, the projective line over a ring is a one-dimensional space over the ring. In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, even if the algebra is of higher dimensionality.

Coordinate systems in one-dimensional space

One dimensional coordinate systems include the number line.

• Number line

See also

• Univariate
• Zero-dimensional space

References

1. Гущин, Д. Д. "Пространство как математическое понятие" (in Russian). fmclass.ru. Retrieved 2015-06-06.
2. Peter Lancaster & Miron Tismenetsky (1985) The Theory of Matrices, second edition, page 147, Academic Press ISBN 0-12-435560-9
3. P. M. Cohn (1961) Lie Groups, page 70, Cambridge Tracts in Mathematics and Mathematical Physics # 46

v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes and shapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Number systems	Hypercomplex numbers Cayley–Dickson construction
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category

• Dimension
• 1 (number)