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,

or

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.

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