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(Ugly?) Inequalities - Squares and sums

  1. Feb 11, 2007 #1
    Here is the question from the book:
    ------
    Let [itex]n\geq1[/itex] and let [itex]a_1,...,a_n[/itex] and [itex]b_1,...,b_n[/itex] be real numbers. Verify the identity:
    [tex]\left(\sum_{i=1}^n{a_ib_i}\right)^2 + \frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\left(a_ib_j-a_jb_i\right)^2}} = \left(\sum_{i=1}^n{a_i^2}\right)\left(\sum_{j=1}^n{b_j^2}\right)[/tex]

    and conclude the Cauchy-Schwartz inequality:

    [tex]\left|\sum_{i=1}^n{a_ib_i}\right| \leq \left(\sum_{i=1}^n{a_i^2}\right)^{1/2} \left(\sum_{j=1}^n{b_j^2}\right)^{1/2}[/tex]

    Then use Cauchy-Schwartz to prove the triangle inequality:

    [tex]\left(\sum_{i=1}^n{(a_i^2+b_i^2)}\right)^{1/2} \leq \left(\sum_{i=1}^n{a_i^2}\right)^{1/2} + \left(\sum_{j=1}^n{b_j^2}\right)^{1/2}[/tex]
    ----------------

    I have been trying to mess around with the first one, and see what is going on, but it is looking extremely ugly, even with [itex]n=3[/itex].

    Should I be trying to prove these by induction? Or are there some ways to manipulate these things easily? I guess the problem is almost notation, or that I don't know how to manipulate sums and squares properly. Any ideas would be appreciated. Thanks!
     
  2. jcsd
  3. Feb 11, 2007 #2

    Dick

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    Put it in this form:

    [tex]\sum_{i=1}^n{\sum_{j=1}^n{a_i b_i a_j b_j +\frac{1}{2}\left(a_ib_j-a_jb_i\right)^2 - a_i^2 b_j^2}} = 0[/tex]

    Now you basically just expand the middle term and wiggle some indices.
     
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