Discussion
Let \(\{e_{k} : k \in \mathbb{N}\}\) be an orthonormal basis for a Hilbert space H.
Define \(f_{k} = e_{k} + e_{k+1}\), \(k \in \mathbb{N}\) and \(g_{j} = \sum_{n=1}^{j} (-1)^{n+1} e_{n}\), \(j \in \mathbb{N}\).
Then \(\sum_{k=1}^{\infty} |\langle g_{j}, f_{k} \rangle|^{2} =\)
Answer: D
The sum equals 1.
Write \(g_{j} = \sum c_{n}e_{n}\) with \(c_{n} = (-1)^{n+1}\) for n ≤ j and \(c_{n} = 0\) for n \(\gt\) j. By orthonormality,
\[ \langle g_{j}, f_{k} \rangle = c_{k} + c_{k+1} \]
For k ≤ j - 1 both coefficients are present with opposite signs, so the inner product is 0. For k = j only \(c_{j} = \pm 1\) survives, giving modulus 1. For k \(\gt\) j both are 0. So the series has a single non-zero term, \(|\pm 1|^{2} = 1\).
The options \(j^{2}\) and \(4j^{2}\) come from adding the coefficients without the cancellation between neighbouring terms, and 0 misses the last term k = j.
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