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In mathematics, the n-dimensional complex coordinate space is the set of all ordered n-tuples of complex numbers. It is denoted C n {\displaystyle \mathbb {C} ^{n}} , and is the n-fold Cartesian product of the complex plane C {\displaystyle \mathbb {C} } with itself. Symbolically,
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 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.