Let (e_k) be an orthonormal Hilbert-space basis, f_k=e_k+e_(k+1), and $g_j=\sum_{n=1}^j(-1)^{n+1}e_n$. Find $\sum_{k=1}^\infty|\langle g_j,f_k\rangle|^2$.
For k<j the two alternating coefficients cancel. For k=j only the coefficient of e_j remains, equal to ±1; for k>j both coefficients vanish. Exactly one squared inner product is 1, giving **1, D**.
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