One-dimensional space

{{short description|Space with one dimension}}

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{{General geometry}}

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.{{cite web|url=http://fmclass.ru/math.php?id=49a0390719b7b|language=ru|title=Пространство как математическое понятие|last=Гущин|first= Д. Д.|access-date=2015-06-06|publisher=fmclass.ru}}

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 $K$ is a one-dimensional vector space over itself. The projective line over $K,$ denoted $\backslash mathbf\; P^1(K),$ is a one-dimensional space. In particular, if the field is the complex numbers $\backslash mathbb\{C\},$ then the complex projective line $\backslash mathbf\; P^1(\backslash mathbb\{C\})$ is one-dimensional with respect to $\backslash mathbb\{C\}$ (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.Peter Lancaster & Miron Tismenetsky (1985) The Theory of Matrices, second edition, page 147, Academic Press {{ISBN|0-12-435560-9}}

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.P. M. Cohn (1961) Lie Groups, page 70, Cambridge Tracts in Mathematics and Mathematical Physics # 46

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.