Let \(\mch\) be a Hilbert space and let \((u_n)_{n\in A}\) be a family in \(\mch\), indexed by a set \(A\), with \(\inner{u_n}{u_m}=0\) for \(n\neq m\) and \(\norm{u_n}=1\) for all \(n\). For each \(x\in\mch\) only countably many \(\inner{x}{u_n}\) are nonzero (T-4BDE3), and sums over \(A\) below are taken over those indices. The following are equivalent:
(Completeness) If \(x\in\mch\) and \(\inner{x}{u_n}=0\) for all \(n\in A\), then \(x=0\).
(Parseval’s identity) For every \(x\in\mch\), \[ \sum_{n\in A} \abs{ \inner{x}{u_n} }^2 = \norm{x}^2 . \]
(Expansion) For every \(x\in\mch\), \[ x = \sum_{n\in A} \inner{x}{u_n}u_n , \] with the series converging in norm in any enumeration of its nonzero terms; and if \(x=\sum_{n\in A}c_nu_n\) for scalars \(c_n\), only countably many nonzero, then \(c_n=\inner{x}{u_n}\) for all \(n\).