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

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.