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Homework Help: Calculating weak limits

  1. Apr 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that the sequence [itex]\{sin(kx)\}[/itex] converges weakly to [itex]0[/itex] in [itex]L^2(0,1)[/itex].

    2. Relevant equations
    A sequence of elements [itex]\{f_k\}[/itex] in a Banach space [itex]X[/itex] is to converge weakly to an element [itex]x\in X[/itex] if [itex]L(f_k)→L(f)[/itex] as [itex]k→∞[/itex] for each [itex]L[/itex] in the dual of [itex]X[/itex].

    3. The attempt at a solution

    If the sequence was orthogonal on [itex](0,1)[/itex] then I can apply Bessel's inequality to show that the sequence does converge to [itex]0[/itex]. But this sequence in not orthogonal on [itex](0,1)[/itex]. So I don't know how to approach it anymore. Showing that a sequence is weakly convergent involves calculating an integral in [itex]L^p(a,b)[/itex]. What general idea can I use to calculate these integrals?
     
  2. jcsd
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