Question: Let \[({{e}_{n}}\left. ) \right|_{1}^{\infty }\] be a complete orthonormal sequence in a Hilbert space H, and let \({{\lambda }_{n}}\in \mathbb{C}\) for \(n\in \mathbb{N}\) . Show that there is a bounded linear operator D on H such that \(D{{e}_{n}}={{\lambda }_{n}}{{e}_{n}}\) for all \(n\in \mathbb{N}\) , if and only if \({{\lambda }_{n}}\) is a bounded sequence. What is \(||D||\) , when defined.

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