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I am looking for a proof or counter example to the statement \begin{align*} \sum_{j=-\infty}^{\infty}|\psi_j| < \infty \Rightarrow \sum_{j=-\infty}^{\infty}\psi_j^2 < \infty, \end{align*}

where $\psi_j \in \mathbb{R}$. Any help will be appreciated. Thanks in advance.

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1 Answer 1

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$\sum_{j} |\psi_j| < \infty$ implies that $M:= \sup_j|\psi_j| < \infty$. Thus

$$\sum_j \psi_j^2 \leq \sum_j |\psi_j|M = M \sum_j|\psi_j| < \infty$$

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