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Schwarz Inequality Proof

  1. Sep 14, 2008 #1
    1. The problem statement, all variables and given/known data



    2. Relevant equations
    [itex](\langle \alpha | + \lambda^\ast\langle \beta |)(|\alpha\rangle+ \lambda|\beta\rangle) = \langle \alpha |\alpha \rangle + |\lambda|^2\langle \beta | \beta \rangle + \lambda \langle \alpha | \beta \rangle + \lambda^\ast \langle \beta | \alpha \rangle \geq 0[itex]



    3. The attempt at a solution
    This equation is the setup, and it leads to an equation that I can see is quadratic in lambda. From this, I calculate the discriminant, which must be greater than or equal to zero because all the terms are real and positive. However, when I manipulate this to get to the Schwarz inequality, I get a "less than or equal to" where I should have a "greater than or equal to". Can somone please help? Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 14, 2008 #2

    Dick

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    Use tex and /tex for the tex delimiters instead of latex and latex. And you don't want to solve a quadratic. Just put in the special value lambda=-<beta|alpha>/<beta|beta>.
     
    Last edited: Sep 14, 2008
  4. Sep 14, 2008 #3
    Thanks, and sorry about the equation format; I'm still trying to figure out the MathType translator.

    Anyway, I understand how it works for the value of lambda you gave, but shouldn't it work for ANY value of lambda?
     
  5. Sep 14, 2008 #4

    Dick

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    It works for any lambda, but it doesn't tell you anything very interesting for every lambda. E.g. lambda=0 isn't interesting at all.
     
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