Lagrange's Identity and Cauhchy-Schwarz Inequality for complex numbers

[rmcknigh]

I guess the best way to start this is by admitting that my conceptual understanding of the Cauchy-Schwarz Inequality and the Lagrange Identity, as the title suggests, is not as deep as it could be.

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

 

[Opalg]

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.

Hi Anthony and welcome to MHB!

The Cauchy–Schwarz inequality is absolutely central to the study of inner-product spaces. To take just one example, it gives the most natural proof of the triangle inequality $\|a+b\|\leqslant \|a\| + \|b\|$. But I have to admit that in 50 years of doing complex and functional analysis I have never until now come across Lagrange's inequality, so I'm not surprised that you are having trouble finding applications of it. According to Lagrange's identity - Wikipedia, the free encyclopedia, it crops up in exterior algebra, so it seems that maybe it is more of an algebraic than an analytical tool.