The Standard Basis of the Complex Coordinate Space is an Orthonormal Basis

Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and 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, together with the standard inner product $\langle\cdot,\cdot\rangle$, a complex inner product space by claim 1 of The Standard Inner Product Makes the Complex Coordinate Space an Inner Product Space. Let $\lVert\cdot\rVert$ be the induced norm, let $e$ be the $n$-tuple in $\mathbb{C}^{n}$ whose components $e_{k}$ are the standard basis vectors, and let $u,v\in\mathbb{C}^{n}$ have components $u_{k}$ and $v_{k}$. Sums of vectors are finite sums in $\mathbb{C}^{n}$ and sums of scalars are finite sums in a field; $|z|$ is the modulus of a complex number $z$ and $\overline{z}$ its conjugate. Then the following hold.

1. (Components as coefficients) For every $k\in[n]$,

$$\langle e_{k},u\rangle=u_{k}.$$

2. (Expansion)

$$u=\sum_{k=1}^{n}u_{k}e_{k}.$$

3. (Orthonormal basis) The tuple $e$ is an orthonormal basis of $\mathbb{C}^{n}$.

4. (Parseval)

$$\langle u,v\rangle=\sum_{k=1}^{n}\overline{u_{k}}\,v_{k},\qquad \lVert u\rVert^{2}=\sum_{k=1}^{n}|u_{k}|^{2}.$$