) Math student should know that the Cauchy-Schwarz inequality, namely:\[
\tag{CS}
\left| \sum_{n=1}^N z_n\ w_n \right|^2 \leq \left( \sum_{n=1}^N |z_n|^2\right)\ \left( \sum_{n=1}^N |w_n|^2\right)\; ,
\]
holds for any \(N\in \mathbb{N}\) and \(z_1,\ldots ,z_N,w_1,\ldots, w_N\in \mathbb{C}\); moreover, he should also know its "folkloristic proof" (i.e., the one obtained by minimizing the function \(\mathbb{C}\ni \lambda \mapsto \sum_{n=1}^N|z_n-\lambda \overline{w_n}|^2 \in \mathbb{R}\)).
Here we present a lesser known proof of (CS) which relies on a purely algebraic equality.
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Exercise:
1. Let \(N\in \mathbb{N}\). Prove that equality:
\[
\tag{L} \left| \sum_{n=1}^N z_n w_n\right|^2 = \left( \sum_{n=1}^N |z_n|^2\right)\ \left( \sum_{n=1}^N |w_n|^2\right) - \sum_{n=1}^{N-1} \sum_{m=n+1}^N |z_n\overline{w_m} -z_m\overline{w_n}|^2
\]
holds for any \(z_1,\ldots ,z_N,w_1,\ldots ,w_N\in \mathbb{C}\).
2. Show that (CS) follows from (L).
3. Is it possible to characterize the case of equality in (CS) using (L)?
