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Schwarz inequality is Cauchy–Schwarz inequality?

  1. Oct 30, 2012 #1
    I found many information showed Schwarz inequality and Cauchy–Schwarz inequality are same on books and internet, but my teacher's material shows that:
    Schwarz inequality:
    [itex]\left\|[x,y]\right\|\leq\left\|x\right\|+\left\|y\right\|[/itex]

    Cauchy–Schwarz inequality:
    [itex]\left\|[x,y]\right\|\leq\left\|x\right\|\left\|y\right\|[/itex]


    They seem to be different on material, and I had sent email to teacher but having no reply.
    Therefore my question is "Are Schwarz inequality and Cauchy–Schwarz inequality same?"
     
    Last edited: Oct 30, 2012
  2. jcsd
  3. Oct 30, 2012 #2

    mfb

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    Those do not look like inequalities to me. And the first one looks wrong, independent of the inequality sign.
    Maybe you mean ##||x+y|| \leq ||x||+||y||##, but that is the triangle inequality. It follows from the Cauchy–Schwarz inequality if the norm is induced by a scalar product.
     
  4. Oct 30, 2012 #3
    Sorry about used wrong symbol, and I have modified.
    I know triangle inequality.

    But the question still is "are Schwarz inequality and Cauchy–Schwarz inequality same?"


    Thanks for your reply. :smile:
     
  5. Oct 30, 2012 #4

    micromass

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    Your Schwarz inequality simply seems false. In [itex]\mathbb{R}[/itex], we have [itex][x,y]=xy[/itex]. But it is certainly not the case that

    [tex]|2\cdot 3|\leq |2|+|3|[/tex]
     
  6. Oct 30, 2012 #5
    It looks like a typo to me. The books and the internet are right I think.
     
  7. Oct 30, 2012 #6
    Well, do you mean that:
    Schwarz inequality
    = Cauchy–Schwarz inequality
    = [itex]\left\|[x,y]\right\|\leq\left\|x\right\|\left\|y\right\|[/itex]?

    I think so either, thus I want to figure it out.


    Both of your answers are helpful, thanks. :smile:
     
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