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Inner products

  1. Nov 25, 2006 #1

    daniel_i_l

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    1) C[-1,2] is a space of all continues functions f: [-1,2] -> C (complex)
    Is:
    [tex] <f,g> = \int_{-1}^{2}|f(t) + g(t)|dt [/tex]
    an inner product of C[-1,2]?
    I think that the answer is no because:
    [tex] <f+g, h> \neq <f,h> + <g,h>[/tex]
    for some f and g. this can happen when all the functions are positive and so:
    |f(t) + h(t) + g(t)| doesn't equal |f(t) + h(t)| + |g(t) + h(t)|

    2)
    V is a space of all real functions with defined double derivatives in the interval
    [tex] [-\pi, \pi] [/tex]
    we define:
    [tex] <f,g> = f(-\pi)g(-\pi) + \int_{-\pi}^{\pi}f''(x)g''(x)dx [/tex]
    is <f,g> an inner product of V?
    also here i think that the answer is no because <f,f> can equal zero even if all of f isn't 0, this can happen if f(-pi) is zero and the double derivative is 0 everywere (contiues slope).
    am i correct?
    Thanks.
     
    Last edited: Nov 25, 2006
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  3. Nov 25, 2006 #2

    matt grime

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    2) there is nothing in the definition of inner product that means <f,f>=0 if and only if f=0.

    Wht is the definition of an inner product? do these satisfy, or not satisfy the definitions?
     
  4. Nov 25, 2006 #3

    daniel_i_l

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    Isn't that the 3rd property of the inner product? look here for example:
    http://planetmath.org/encyclopedia/InnerProduct.html

    did i misunderstand something?

    and is (1) correct?
    Thanks.
     
  5. Nov 25, 2006 #4

    matt grime

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    Sorry, my mistake - I was thinking of a bilinear pairing, not an inner product.

    So, you have the definitions, and the counter examples: what was the question?
     
    Last edited: Nov 25, 2006
  6. Nov 25, 2006 #5

    daniel_i_l

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    I just wanted to make sure that my counter examples were correct.
    Thanks
     
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