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 {\displaystyle \mathbb {C} ^{n}} , and is the n-fold Cartesian product of the complex line C {\displaystyle \mathbb {C} } with itself. Symbolically, C n = { ( z 1 , … , z n ) ∣ z i ∈ C } {\displaystyle \mathbb {C} ^{n}=\left\{(z_{1},\dots ,z_{n})\mid z_{i}\in \mathbb {C} \right\}} or C n = C × C × ⋯ × C ⏟ n . {\displaystyle \mathbb {C} ^{n}=\underbrace {\mathbb {C} \times \mathbb {C} \times \cdots \times \mathbb {C} } _{n}.} The variables z i {\displaystyle z_{i}} are the (complex) coordinates on the complex n-space. The special case C 2 {\displaystyle \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 C n {\displaystyle \mathbb {C} ^{n}} with the 2n-dimensional real coordinate space, R 2 n {\displaystyle \mathbb {R} ^{2n}} . With the standard Euclidean topology, C n {\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.