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Inner Product Space

  1. Oct 27, 2011 #1
    1. The problem statement, all variables and given/known data

    Is the following an inner product space if the functions are real and their derivatives are continuous:

    [tex] <f(t)|g(t)> = \int_0^1 f'(t)g'(t) + f(0)g(0) [/tex]


    2. Relevant equations

    I was able to prove that it does satisfy the first 3 conditions of linearity and that
    [tex] <f(t)|g(t)> = <g(t)|f(t)> [/tex]
    But I was struggling with the last condition that:
    [tex] <f(t)|f(t)> \geq 0 [/tex]


    3. The attempt at a solution
    I was able to get the following:
    [tex] <f(t)|f(t)> = \int_0^1 [f'(t)]^2 + f(0)^2 [/tex]
    Since this is square integrable, I figure that it must be greater than or equal to zero if the functions and their derivatives are continuous.
    Am I right? It's the last term that's giving me some trouble.
    Thanks!
     
  2. jcsd
  3. Oct 27, 2011 #2

    LCKurtz

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    Science Advisor
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    Gold Member

    Everything you have written down is non-negative. So, yes, you are correct.
     
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