Standard Inner Product on the Complex Coordinate Space

Let $n$ be a natural number, let $\mathbb{C}^{n}$ be the complex coordinate space, and for a complex number $z$ let $\overline{z}$ denote its complex conjugate.

The standard inner product on $\mathbb{C}^{n}$ assigns to each pair $u,v\in\mathbb{C}^{n}$ the complex number

$$\langle u,v\rangle=\sum_{k=1}^{n}\overline{u_{k}}\,v_{k},$$

where $u_{k}$ and $v_{k}$ are the components of $u$ and $v$ and the finite sum is taken in the field of complex numbers.