For an orthonormal basis $(e_k)$ of $H$, $\norm{x}^2=\sum_k\abs{\inner{x}{e_k}}^2$ for every $x\in H$.Suggest a slogan
Let \(H\) be a Hilbert space and let \((e_k)_{k\geq 1}\) be an orthonormal sequence in \(H\) that is a basis of \(H\). Then for every \(x\in H\), \[ \sum_{k\geq 1} \abs{ \inner{x}{e_k} }^2 = \norm{x}^2. \]
This is the case of equality in Bessel’s inequality T-4BDE3. For an orthonormal sequence, being a basis is equivalent to being complete; see T-5AALA.