1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted