Squared coefficients along an orthonormal sequence sum to at most the squared norm.Suggest a slogan
Let \(H\) be an inner product space with inner product \(\inner{\cdot}{\cdot}\) and norm \(\norm{x}\coloneqq\inner{x}{x}^{1/2}\), let \((e_k)_{k\geq1}\) be an orthonormal sequence in \(H\), and let \(x\in H\). Then \[ \sum_{k=1}^{\infty}\abs{\inner{x}{e_{k}}}^{2} \leq \norm{x}^{2} . \]