Orthonormal Family in a Complex Inner Product Space

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $e\in V^{n}$ be an $n$-tuple in $V$, with components $e_{k}$.

The tuple $e$ is orthonormal if every $e_{k}$ with $k\in[n]$ is a unit vector, and $e_{j}$ and $e_{k}$ are orthogonal for all $j,k\in[n]$ with $j\ne k$.