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Math Help - how to prove this inner product

  1. #1
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    how to prove this inner product

    f:[0,2]-> C
    C[0,2]
    <f,g>=\int_{1}^{2}f(x)\overline{g(x)}dx

    i was told that its not an inner product
    because the integral interval is from 1 to 2
    so we dont know what on 0 to 1 interval
    why its interfering?
    what could happen that could make it a non inner product
    ?
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  2. #2
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    Quote Originally Posted by transgalactic View Post
    f:[0,2]-> C
    C[0,2]
    <f,g>=\int_{1}^{2}f(x)\overline{g(x)}dx

    i was told that its not an inner product
    because the integral interval is from 1 to 2
    so we dont know what on 0 to 1 interval
    why its interfering?
    what could happen that could make it a non inner product
    ?

    It may be that <f,f> = INT{1,2} ( |f(x)|^2) dx = 0 without f(x) = 0 on all [0,1]

    Tonio
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  3. #3
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    so from 0 to 1 it could be a graph which integral is not 0
    but from 1 to 2 its integral is 0
    correct?
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  4. #4
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    Quote Originally Posted by transgalactic View Post
    so from 0 to 1 it could be a graph which integral is not 0
    but from 1 to 2 its integral is 0
    correct?

    Yup. For example, let f(x) = -x + 1 in [0,1] and f(x) = 0 in [1,2]. This function is cont. on [0,2] and <f,f> = 0, but f =/= 0 .

    Tonio
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