The Complex Coordinate Space

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

The 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

$$u+v=(u_{1}+v_{1},\dots,u_{n}+v_{n}),\qquad \lambda u=(\lambda u_{1},\dots,\lambda u_{n})$$

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 $k$-th component of $u$.