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Norm of a function ||f|| & the root mean square of a function.

  1. Aug 23, 2009 #1
    norm of a function ||f|| & the "root mean square" of a function.

    How do I explain the connection between the norm of a function ||f|| & the "root mean square" of a function. You may like to consider as an example C[0,[tex]\pi[/tex]], the inner product space of continuous functions on the interval [0,[tex]\pi[/tex]] with the inner product
    (f,g) = [tex]\int_0^\pi[/tex] f(x)g(x) dx.
    Let f(x) = sin(x). How do I find ||f||. Also find "root mean square" of f. What do you notice?
     
  2. jcsd
  3. Aug 24, 2009 #2

    morphism

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    Re: Connection

    Well, have you tried using the definitions and applying them to this particular f?
     
  4. Aug 24, 2009 #3

    HallsofIvy

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    Re: Connection

    Morphism's point is that |f| is normally defined, in an inner product space, as the square root of the inner product of f with itself. So if
    [tex](f, g)= \int_0^\pi f(x)g(x)dx[/tex]
    Then
    [tex]|f|= \sqrt{\int_0^\pi f^2(x) dx[/tex]

    Now, what is
    [tex]\sqrt{\int_0^\pi sin^2(x) dx[/tex]?
     
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