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I'm working through Marsden's 3e "Basic Complex Analysis" and it contains a proof of the Cauchy Schwarz Inequality using what it calls a clever math trick- manipulating \(\displaystyle \sum_{k=1}^n\mid z_k-d\bar{w_k}\mid^2\) where \(\displaystyle z_k,w_k\in\mathbb{C} \ \forall \ k \ s.t. \ 1\leq k \leq n\) and \(\displaystyle d=\frac{\sum_{k=1}^n{z_kw_k}}{\sum_{k=1}^n{ \mid w_k\mid^2}}\). Then, as an exercise, we are asked to prove Lagrange's Identity and then deduce the Cauchy-Schwarz Inequality from it.

Although these aren't extremely difficult proofs, I don't understand what's so important about them that I can't find anything else in either the supplement or index of the Marsden text that uses these results at all .

I'm pretty sure that the Cauchy-Schwarz Inequality is important because it validates the definition of the angle between two complex vectors. Or perhaps restated, it solidifies the geometric interpretation of complex numbers as vectors.

But what about Lagrange's Identity? I'm having trouble finding applications of this to complex analysis besides a few different proofs of it, and its implication of the Cauchy Schwarz Inequality.

Halp.

Anthony