Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number} and let $\mathbb{C}$ be the field of \reftext{def:complex-numbers-2026a}{complex numbers}.

The \textbf{complex coordinate space} $\mathbb{C}^{n}$ is the set of all ordered $n$-tuples $u=(u_{1},\dots,u_{n})$ of complex numbers, together with the operations defined componentwise by

for $u,v\in\mathbb{C}^{n}$ and $\lambda\in\mathbb{C}$, the sums and products on the right being those of $\mathbb{C}$. The complex number $u_{k}$ is called the \textbf{$k$-th component} of $u$.