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Math Help - L^2 integral problem.

  1. #1
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    L^2 integral problem.

    Suppose that fg \in L^2 ([a,b]) for all f \in L^2([a,b]). How to show that g is also in L^2([a,b])
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  2. #2
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    Re: L^2 integral problem.

    Take f(x) = 1.
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  3. #3
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    Re: L^2 integral problem.

    Quote Originally Posted by emakarov View Post
    Take f(x) = 1.
    this doesn't show for every f.
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  4. #4
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    Re: L^2 integral problem.

    Quote Originally Posted by younhock View Post
    Suppose that fg \in L^2 ([a,b]) for all f \in L^2([a,b]). How to show that g is also in L^2([a,b])
    I thought you assume that fg \in L^2 ([a,b]) for all f \in L^2([a,b]). We have 1 \in L^2([a,b]).
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  5. #5
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    Re: L^2 integral problem.

    Quote Originally Posted by emakarov View Post
    I thought you assume that fg \in L^2 ([a,b]) for all f \in L^2([a,b]). We have 1 \in L^2([a,b]).
    Sorry I not really get what you mean. Would you explain in details?
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  6. #6
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    Re: L^2 integral problem.

    You are given that fg \in L^2 ([a,b]) for all f \in L^2 ([a,b]). This is known; you don't have to prove this. Therefore, you can replace f with any function, as long as it is in L^2 ([a,b]), and still get a true statement. I suggest replacing f with a function that equals 1 everywhere.
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