Complex line

English

Noun

1. Template:Lb A 1-dimensional affine subspace of a vector space over the complex numbers.
   • 1990, Template:W, Introduction to Holomorphic Functions of Several Variables, Volume 1 Function Theory, Wadsworth & Brooks/Cole, page 102,

     However, it is possible to characterize the real parts of holomorphic functions of several variables at least locally as those continuous functions such that their restrictions to all complex lines, not just to complex lines parallel to the coordinate axes, are real parts of holomorphic functions. The complex line in ${ℂ}^n$ through a point $A\in {ℂ}^n$ in the direction of a vector $B\in {ℂ}^n$ is the one-dimensional complex submanifold of ${ℂ}^n$ described parametrically as $\{A+tB:t\in ℂ\}$.

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Usage notes

• The term emphasises the structure's 1-dimensional aspect. The structure is 1-dimensional strictly in the sense that it is defined (as a vector space) over a single dimension of the complex numbers: topologically, it is equivalent to the real plane.

Synonyms

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Derived terms

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Further reading

• complex line on Template:W