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Square integrable functions

  1. Nov 18, 2009 #1
    Hi i have a question about [itex]L^2[/itex] spaces and convergence.
    Here it goes:
    Let [itex]K\subset \mathbb{R}^2[/itex] be bounded.
    Let [itex]g,h\in L^2(K)[/itex], and a sequence [itex]f_n\in L^2(K)[/itex] such that [itex]f_n[/itex] converges strongly to [itex]f\in L^2[/itex].
    Is it true that [itex]\lim_{n\rightarrow \infty} \int_{K} f_n g h = \int_{K} f g h[/itex]? If it is how?
    Thank you.
     
    Last edited by a moderator: Dec 2, 2009
  2. jcsd
  3. Nov 19, 2009 #2
    [itex]g,h \in L^2[/itex], then [itex]gh\in L^1[/itex] by Holder inequality.

    and so I do not know the integral [itex]\int_K fgh[/itex] is well-defined?
     
    Last edited by a moderator: Dec 2, 2009
  4. Dec 2, 2009 #3
    I didn't get what you mean. Are you asking it as a question?
     
  5. Dec 2, 2009 #4

    HallsofIvy

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    Yes. [itex]L^2[/itex] consists of functions whose square is integrable. The product of any two such functions is integrable but the product of three of them may not be.
    ([itex]L^2[/itex] is not closed under multiplication.)
     
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