Bessel bounds the squared Fourier coefficients by the norm; an orthonormal basis makes the bound an equality.Suggest a slogan
Let \(H\) be a Hilbert space and let \((e_k)_{k\geq1}\) be an orthonormal sequence in \(H\). For every \(x\in H\), Bessel’s inequality holds: \[ \sum_{k\geq1} \abs{\inner{x}{e_k}}^2 \le \norm{x}^2 . \] If \((e_k)_{k\geq1}\) is a basis of \(H\), then equality holds for every \(x\in H\): \[ \sum_{k\geq1} \abs{\inner{x}{e_k}}^2 = \norm{x}^2 . \]
State Parseval's identity, and say when Bessel's inequality becomes it.