Question about the Schwarz inequality

[silimay]

I am confused about a proof of the Schwarz inequality in my book...

Homework Statement

$$\left(\sum_{j=1}^n |a_j \overline{b}_j |\right)^2\leq \left(\sum_{j=1}^n |a_j|^2\right) \left(\sum_{j=1}^n |b_j|^2\right).$$

The Attempt at a Solution

In the proof in my book (Rudin) it sets $$A = \sum a_j^2$$ and $$B = \sum b_j^2$$ and $$C = \sum a_j \overline{b}_j$$. It assumes B > 0 and then says

$$\sum {|Ba_j - Cb_j|}^2 = B^2 \sum {|a_j|}^2 - B \overline{C} \sum a_j \overline{b}_j - BC \sum \overline{a}_j b_j + |C|^2 \sum |b_j|^2 = B^2 A - B |C| ^2$$

I don't understand how it got between those two steps.

 

[morphism]

There are a lot of inconsistencies in what you've written down, but I'm going to assume they're just transcription errors. Anyway, I think what you're missing is that for a complex number z, $z\bar{z} = |z|^2$.

 

[brh2113]

Try treating this as a quadratic equation, except switch your B and C.

Then take $$Ax^{2} + 2Bx + C$$ $$\leq 0$$

Then complete the square and see what you get. (This proof is found in Apostol's Calculus)