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Question about the Schwarz inequality

  1. Oct 9, 2007 #1
    I am confused about a proof of the Schwarz inequality in my book...

    1. The problem statement, all variables and given/known data

    [tex]\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).[/tex]

    3. The attempt at a solution

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

    [tex]\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 [/tex]

    I don't understand how it got between those two steps.
  2. jcsd
  3. Oct 9, 2007 #2


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    Homework Helper

    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, [itex]z\bar{z} = |z|^2[/itex].
  4. Oct 10, 2007 #3
    Try treating this as a quadratic equation, except switch your B and C.

    Then take [tex]Ax^{2} + 2Bx + C[/tex] [tex]\leq 0 [/tex]

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