Orthonormal Basis of 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 $V$, that is, a map from $[n]$ to $V$, with components $e_{k}$.

The tuple $e$ is an orthonormal basis of $V$ if it is orthonormal and a basis of $V$, the latter with respect to the field of complex numbers.