Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} with \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$, 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 an \reftext{def:orthonormal-family-2026a}{orthonormal family}. Sums of vectors are \reftext{def:finite-sum-vector-space-2026a}{finite sums in $V$} and sums of scalars are \reftext{def:finite-sum-field-2026b}{finite sums in a field}; $|z|$ is the \reftext{def:complex-modulus-2026a}{modulus} of a \reftext{def:complex-numbers-2026a}{complex number} $z$ and $\overline{z}$ its \reftext{def:complex-conjugate-2026a}{conjugate}. Then the following hold.

\textbf{1. (Expansion criterion)} The family $e$ is an \reftext{def:orthonormal-basis-2026a}{orthonormal basis} of $V$ if and only if

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

and in particular