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Norm of a Function vs. Length of a Vector

  1. Mar 10, 2008 #1
    Suppose f(x)= -2x+1 is a vector in the vector space C[0,1].

    Calculating the norm (f,f) results in 1/3.

    I'm a little confused.

    So on [0,1] the function is a straight line from (0,1) to (0,-1).

    So I thought I could simply takes this line segment and turn it into a directed line segment originating from the origin. So it would be equivalent to the vector v= 0i - 2j (right?)

    , so then ||v|| = sqr(0^2 + (-2)^2) = 2

    So the length of vector v is 2.

    Why is this different from the norm (f,f)? Shouldn't they be the same?

    ...or am I completely missing the point here of the norm / inner product of the function?
  2. jcsd
  3. Mar 10, 2008 #2
    There is more then one definition for the norm of a function.
  4. Mar 11, 2008 #3


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    The length of a curve (or straight line) has little to do with its "norm". Why do you think they should be the same?

    Your title, "Norm of a Vector Versus Length of a Vector" is somewhat misleading. You are actually talking about the norm of a function (thought of as a vector) and the length of its graph which is not at all a vector.
    Last edited by a moderator: Mar 13, 2008
  5. Aug 27, 2009 #4
    The norm of the function defined in this case is (f.f) = [tex]\oint f^2 dx[/tex]
    The limits are from 0 to 1.
    The above integral turns out to be 1/3 which is the correct answer.
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