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      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



      K


      {\displaystyle K}

      is a one-dimensional vector space over itself. The projective line over



      K
      ,


      {\displaystyle K,}

      denoted





      P


      1


      (
      K
      )
      ,


      {\displaystyle \mathbf {P} ^{1}(K),}

      is a one-dimensional space. In particular, if the field is the complex numbers




      C

      ,


      {\displaystyle \mathbb {C} ,}

      then the complex projective line





      P


      1


      (

      C

      )


      {\displaystyle \mathbf {P} ^{1}(\mathbb {C} )}

      is one-dimensional with respect to




      C



      {\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

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