{{short description|Space formed by the ''n''-tuples of complex numbers}}
In mathematics, the ''n''-dimensional '''complex coordinate space''' (or '''complex ''n''-space''') is the set of all ordered ''n''-tuples of complex numbers, also known as ''complex vectors''. The space is denoted <math>\Complex^n</math>, and is the ''n''-fold Cartesian product of the complex line <math>\Complex</math> with itself. Symbolically, <math display="block">\Complex^n = \left\{ (z_1,\dots,z_n) \mid z_1, \dots, z_n \in \Complex\right\}</math> or <math display="block"> \Complex^n = \underbrace{\Complex \times \Complex \times \cdots \times \Complex}_{n}.</math> The variables <math>z_i</math> are the (complex) coordinates on the complex ''n''-space. The special case <math>\Complex^2</math>, called the ''complex coordinate plane'', is not to be confused with the complex plane, a graphical representation of the complex line.
Complex coordinate space is a vector space over the complex numbers, with componentwise addition and scalar multiplication. The real and imaginary parts of the coordinates set up a bijection of <math> \Complex^n</math> with the 2''n''-dimensional real coordinate space, <math>\mathbb R^{2n}</math>. With the standard Euclidean topology, <math> \Complex^n</math> is a topological vector space over the complex numbers.
A function on an open subset of complex ''n''-space is holomorphic if it is holomorphic in each complex coordinate separately. Several complex variables is the study of such holomorphic functions in ''n'' variables. More generally, the complex ''n''-space is the target space for holomorphic coordinate systems on complex manifolds.
==See also== * Complex affine space * Coordinate space
==References== * {{citation| author1-link = Robert Gunning (mathematician)| first = Robert | last = Gunning|author2=Hugo Rossi | title=Analytic functions of several complex variables}}
Category:Several complex variables Category:Topological vector spaces