In the vector space $\ell^2(\mathbb{Z})$ over $\mathbb{C}$ (i.e. the set of all two-sided infinite sequence of complex numbers such that $\sum_{n \in \mathbb{Z}} |a_n^2| < \infty$) why is it guaranteed that the inner product $(A, B) = \sum_{n \in \mathbb{Z}} a_n \overline{b_n}$ is absolutely convergent?
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1Do you know about the Cauchy-Schwarz inequality? That may be helpful for you. – Minus One-Twelfth Mar 23 '19 at 04:50
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Ah got it, thanks! – slothropp Mar 23 '19 at 05:23
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You're welcome! $\ddot{\smile}$ – Minus One-Twelfth Mar 23 '19 at 05:28
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Also, $2|a_n||b_n| \le |a_n|^2+|b_n|^2$, which follows from $(|a_n|-|b_n|)^2 \ge 0$. – Disintegrating By Parts Mar 24 '19 at 02:18