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Let $H$ a Hilbert space and $(a_n)_{n \in \mathbb{N}}$ a complete orthonormal system, i.e. a basis. Do you know a example of a orthonormal system $(b_n)_{n\in \mathbb{N}}$ satisfying $\infty > \sum_{k \in \mathbb{N}} \lVert a_k-b_k \lVert \geq 1$, which is not complete?

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