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Homework Help: Prove scalar product of square-integrable functions

  1. Aug 2, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the vector space of continuous, complex-valued functions on the interval [−∏, ∏]. Show that

    defines a scalar product on this space. Are the following functions orthogonal with respect to this scalar product?

    squareintegrablefunction.png



    2. Relevant equations



    3. The attempt at a solution

    Definition of scalar product: A scalar product is an operation that assigns a complex number to any given pair of vectors |a> and |b> belonging to a LVS.

    Case 1: f(t) and g(t) are equal => integrand is real => scalar product is real
    Case 2: f(t) and g(t) are not equal => integrand is complex => scalar product is complex

    I'm not sure if this shows that the function is a scalar product. Is there a more mathematically rigorous way of showing it?

    squareintegrablefunction2.png
     
  2. jcsd
  3. Aug 2, 2013 #2
    To be an inner product, you need to satisfy a few axioms. Which axioms? You need to check those.
     
  4. Aug 2, 2013 #3
    Are these the axioms that are required? What's the difference between a scalar product and an inner product?

    tracescalarproduct2.png
     
  5. Aug 2, 2013 #4
    Yes, you need to check (1), (2) and (3).

    A scalar product is the same as an inner product. But I prefer the latter terminology. Sorry for the confusion.
     
  6. Aug 2, 2013 #5
    If this operation is right, then yes it is an inner product.

    squareintegrablefunction3.png

    How do you prove that this is right? My idea is this:

    Suppose the above integrand is a complex number that contains some 'i' s, and since the process of integration is only with respect to (t), the 'i' s are unaltered. Therefore, there is no difference in replacing all the 'i' s with '-i' s before or after the process of integration. (I like to think of it as the 'i' s in the integrand retains its symmetry)
     
  7. Aug 2, 2013 #6
    Yes, what you say is correct.
     
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