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 $\mathbb {C} ^{n}$, and is the n-fold Cartesian product of the complex line $\mathbb {C}$ with itself. Symbolically, $$\mathbb {C} ^{n}=\left\{(z_{1},\dots ,z_{n})\mid z_{1},\dots ,z_{n}\in \mathbb {C} \right\}$$ 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. The special case $\mathbb {C} ^{2}$, 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 $\mathbb {C} ^{n}$ with the 2n-dimensional 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

• Complex affine space
• Coordinate space

References

• Gunning, Robert; Hugo Rossi, Analytic functions of several complex variables