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Complex Inner-Product Space

  1. Jul 18, 2009 #1
    1. The problem statement, all variables and given/known data

    In an arbitrary complex inner-product space V which of the following is not always true?

    a. <αu + βv, w> = α<u, w> + β<v, w>
    b. |<u, v>|2 ≤ <u, u> <v, w>
    c. <u, αv> = α<u, v>
    d. <0, u> = 0

    2. Relevant equations
    None


    3. The attempt at a solution

    The correct answer must be a, but I don't know what to do with b.

    a. False. If the constants are complex, they will be conjugated.
    b. Any ideas?
    c. True. Constants in the second argument will not be conjugated.
    d. True. Multiplying a complex number by zero still results in a zero.
     
  2. jcsd
  3. Jul 18, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    If b) were |<u,v>|^2<=<u,u>*<v,v> it would be the Cauchy-Schwarz inequality. It's true. Is the 'w' a typo?
     
  4. Jul 18, 2009 #3
    Well, my professor has a 'w' on the assignment, but you must be right.

    Thanks again!
     
  5. Jul 18, 2009 #4
    Which argument gets the conjugate depends on whether it is a physics text or a math text. So consult the text.
     
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