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, also known as complex vectors. The space is denoted $$\Complex^n$$, and is the n-fold Cartesian product of the complex plane $$\Complex$$ with itself. Symbolically, $$\Complex^n = \left\{ (z_1,\dots,z_n) \mid z_i \in \Complex\right\}$$ or $$ \Complex^n = \underbrace{\Complex \times \Complex \times \cdots \times \Complex}_{n}.$$ The variables $$z_i$$ 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 $$ \Complex^n$$ with the 2n-dimensional real coordinate space, $$\mathbb R^{2n}$$. With the standard Euclidean topology, $$ \Complex^n$$ 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.