# Homework Help: Calculating weak limits

1. Apr 14, 2012

### lmedin02

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

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

3. The attempt at a solution

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