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Defining an inner product on C[-1,1]

  1. Jul 24, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider the vector space of all continuous functions on the interval C[-1,1]. That is V = C[-1,1]
    show that <f(x),g(x)> = ∫(-1,1) x2f(x)g(x)dx defines an inner product on C[-1,1].
    I have shown <f,g> = <g,f>, <kf,g> = k<f,g>, <f+g,h> = <f,h> + <g,h> and I am trying to show that given any continuous function over [-1,1] that <f,f> ≥ 0 and <f,f> = 0 iff f(x) = 0



    3. The attempt at a solution
    <f,f> = ∫(-1,1)x2[f(x)]2dx = ∫(-1,1)[xf(x)]2dx. I am stuck here, this f(x) could be anything that is continuous on [-1,1] over a symmetric interval and x2 is an even function while f(x)2 could be even or odd. Maybe I can say something about f(x)2 being even or odd, if x2f2(x) is odd then integrating it over a symmetric interval will be zero but if x2f2(x) is even then integrating it over a symmetric interval will just be twice the value of ∫[0,1](xf(x))2. I'm not sure how to proceed because I feel I need to force the integrand to be an even function and even then how would I know that the value of the integral is positive? Any help to push me in the right direction would be much appreciated, thank you.
     
  2. jcsd
  3. Jul 24, 2016 #2

    Ray Vickson

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    Can the function ##(x f(x))^2## ever be ##< 0## anywhwere in ##(-1,1)##? Can you think of a real number whose square is negative?
     
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