(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Calculating weak limits

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