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.
, and is the n-fold Cartesian product of the complex line
, 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
with the 2n-dimensional real coordinate space,
With the standard Euclidean topology,
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.