Complex coordinate space

In mathematics, the "n"-dimensional complex coordinate space (or complex "n"-space) is the set of all ordered "n"-tuples of complex numbers. It is denoted formula_1, and is the "n"-fold Cartesian product of the complex plane formula_2 with itself. Symbolically,
or
The variables formula_5 are the (complex) coordinates on the complex "n"-space. 
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 formula_6 with the real coordinate space formula_7. With the standard Euclidean topology, formula_6 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.