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

${\displaystyle \mathbb {C} ^{n}=\{(z_{1},\dots ,z_{n})|z_{i}\in \mathbb {C} \}}$

or

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

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