The Standard Inner Product Makes the Complex Coordinate Space an Inner Product Space

Let $n$ be a natural number, let $\mathbb{C}^{n}$ be the complex coordinate space, which is a complex vector space by The Complex Coordinate Space is a Complex Vector Space, and let $\langle\cdot,\cdot\rangle$ be the standard inner product on it. Then the following hold.

1. (Inner product) $\langle\cdot,\cdot\rangle$ satisfies the four conditions of Complex Inner Product Space; consequently $\mathbb{C}^{n}$ together with $\langle\cdot,\cdot\rangle$ is a complex inner product space.

2. (Induced norm) The norm induced by the inner product satisfies

$$\lVert u\rVert^{2}=\sum_{k=1}^{n}|u_{k}|^{2}\qquad(u\in\mathbb{C}^{n}),$$

with the modulus of a complex number and the finite sum taken in the field of real numbers.