One-dimensional space

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 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.

See also

• Univariate
• Zero-dimensional space

References

1. Four-dimensional space
2. Three-dimensional space
3. Dimension
4. Five-dimensional space
5. Zero-dimensional space
6. Two-dimensional space
7. Dimension (vector space)
8. Euclidean space
9. Six-dimensional space
10. Seven-dimensional space
11. Eight-dimensional space
12. Real coordinate space
13. Space (mathematics)
14. Half-space (geometry)
15. Nonlinear dimensionality reduction
16. Euclidean plane
17. Ball (mathematics)
18. Function of a real variable
19. Space Harmony
20. Vector space