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Math Help - L^2 convergence problems.

  1. #1
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    L^2 convergence problems.

    Suppose f_n,f,g \in L^2 (A) and f_n \rightarrow f in L^2. Is is true if i say lim_{n\rightarrow \infty} \int_A f_n g dm = \int_A f g dm?
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  2. #2
    Super Member girdav's Avatar
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    Re: L^2 convergence problems.

    Apply Cauchy-Schwarz inequality to see it's true.
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  3. #3
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    Re: L^2 convergence problems.

    Quote Originally Posted by girdav View Post
    Apply Cauchy-Schwarz inequality to see it's true.
    Sorry, can you tell me more detail, because i cant prove it. THanks very much.
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  4. #4
    Super Member girdav's Avatar
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    Re: L^2 convergence problems.

    \left|\int_Af_ng-\int_Afg\right|\leq \int_A |f-f_n||g|\leq \sqrt{\int_A |f-f_n|^2}\sqrt{\int_A g^2}, and you can conclude (the hypothesis is the strong convergence, whereas the conclusion is called weak convergence).
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