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Inner product

  1. Aug 16, 2011 #1
    1. The problem statement, all variables and given/known data
    Denote the inner product of f,g [itex]\in[/itex] H by <f,g> [itex]\in[/itex] R where H is some(real-valued) vector space
    a) Explain linearity of the inner product with respect to f,g. Define orthogonality.
    b) Let f(x) and g(x) be 2 real-valued vector functions on [0,1]. Could the inner product be defined as (give reasons)
    i) <f,g> = integral from 0 to 1 of f(x)g(x) dx?
    ii) <f,g> = integral from 0 to 1 [itex]\lambda[/itex](x)f'(x)g'(x) dx? where prime denotes the derivative and [itex]\lambda[/itex](x) > 0 is a smooth function (assuming f',g' [itex]\in[/itex] H)?
    iii) <f,g> = f(x)g(x)?
    2. Relevant equations



    3. The attempt at a solution
    a) That is just trivial
    b) Not too sure on any of them
     
  2. jcsd
  3. Aug 16, 2011 #2
    For i) since the integral is a real number (provided that f and g are continuous, differentiable etc.) once computed suggest that this will be an inner product, similarly for ii) and iii) (except the integral part for iii))
     
  4. Aug 16, 2011 #3

    HallsofIvy

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    An inner product must be such that
    [tex]<au+ bv, w>= a<u, w>+ b<v, w>[/tex]
    [tex]<u, v>= \overline{<v, u>}[/tex]

    Can you show that those are true for each of the given operations?
     
  5. Aug 16, 2011 #4
    Do I show positive definiteness <f,f> >= 0? and <f,f> = 0 if and only if f = 0
     
  6. Aug 16, 2011 #5
    I just proved all 3 axioms to be an inner product.
    I got that all 3 operations are all defined as inner products, is this correct (I'm a little skeptical on ii))?
    i) and iii) are easy.
     
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