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Rudin 1.37 (d) is trivial to Rudin but hell for me

  1. Jun 13, 2012 #1
    Hi everyone,

    I am working on my own through Rudin's Principles of Mathematical Analysis and, after the demonstration of Cauchy - Schwarz Inequality, in Theorem 1.37, part (d), Rudin states:

    $$|x \cdot y| \leqslant |x||y|$$

    When he explains how to prove this, he simply states that this is an immediate consequence of Schwarz Inequality, which he defines as follows:

    $$|\sum_{j=1}^{n}a_j \overline{b_j}|^2 \leqslant \sum_{j=1}^{n}|a_j|^2 \sum_{j=1}^{n}|b_j|^2$$

    If someone can explain me how this two things are identical I would appreciate it a lot. My toughts so far:

    $$|x \cdot y| \leqslant |x||y|$$ Take the square of this, which is:

    $$ (x \cdot y)(x \cdot y) \leqslant \sum x_i^2 \sum y_i^2$$ and hence,

    $$ (\sum x_iy_i)^2 \leqslant \sum x_i^2 \sum y_i^2$$ which is NOT the Schwarz Inequality!
     
  2. jcsd
  3. Jun 13, 2012 #2
    Sure it is? He just conjugates the [itex]b_j[/itex] to make the formula compatible with complex spaces. It's the same reason he slaps absolute value bars everywhere.
     
  4. Jun 13, 2012 #3
    I understand the part of the conjugate, actually the whole part on the left is pretty clear to me. The problem is the right side of the inequality. I don't understand how

    $$\sum x_i^2 \sum y_i^2 = \sum_{j=1}^{n} |x|^2 \sum_{j=1}^{n} |y|^2$$
     
  5. Jun 13, 2012 #4
    You mean [itex]|x_i|[/itex] on the right, not [itex]|x|[/itex], right? Again, it's just slapping absolute value bars around to make the inequality compatible with complex spaces. For real numbers, there is no difference.
     
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