Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $e:[n]\to V$ be a map with values $e_{k}$.

The family $e$ is an \textbf{orthonormal basis} of $V$ if it is an \reftext{def:orthonormal-family-2026a}{orthonormal family} and a \reftext{def:finite-basis-2026a}{basis} of $V$, the latter with respect to the field of \reftext{def:complex-numbers-2026a}{complex numbers}.