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 ${ℂ}^n$, and is the n-fold Cartesian product of the complex line $ℂ$ with itself. Symbolically, $${ℂ}^n=\{(z_1,\dots ,z_n)\mid z_i\in ℂ\}$$ or $${ℂ}^n=\underset{n}{\underbrace{ℂ\times ℂ\times \cdots \times ℂ}}.$$ The variables $z_i$ are the (complex) coordinates on the complex n-space. The special case ${ℂ}^2$, 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 ${ℂ}^n$ with the 2n-dimensional real coordinate space, ${ℝ}^{2n}$. With the standard Euclidean topology, ${ℂ}^n$ 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