Orthonormal Expansion and Parseval's Identity in Finite Dimensions

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with induced norm $\lVert\cdot\rVert$, let $n$ be a natural number, and let $e\in V^{n}$ be an $n$-tuple in $V$ that is orthonormal, with components $e_{k}$. Sums of vectors are finite sums in $V$ 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. (Expansion criterion) The tuple $e$ is an orthonormal basis of $V$ if and only if

$$u=\sum_{k=1}^{n}\langle e_{k},u\rangle e_{k}\qquad\text{for every }u\in V.$$

2. (Parseval's identity) If $e$ is an orthonormal basis of $V$, then for all $u,w\in V$,

$$\langle u,w\rangle=\sum_{k=1}^{n}\overline{\langle e_{k},u\rangle}\,\langle e_{k},w\rangle,$$

and in particular

$$\lVert u\rVert^{2}=\sum_{k=1}^{n}\bigl|\langle e_{k},u\rangle\bigr|^{2}.$$