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 $\mathbb C^n$, and is the n-fold Cartesian product of the complex plane $\mathbb C$ with itself. Symbolically,

$\mathbb C^n=\{ (z_1,\dots,z_n)|z_i\in\mathbb C\}$

or

$\mathbb C^n = \underbrace{\mathbb C \times \mathbb C \times \cdots \times \mathbb C}_{n}.$

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 $\mathbb C^n$ with the real coordinate space $\mathbb R^{2n}$. With the standard Euclidean topology, $\mathbb C^n$ 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.

See also

• Coordinate space