Weak limit of product of two weakly converging sequences (counterexample)

Question

Let $\Omega$ be an open, bounded subset of $\mathbb{R}^N$. I'm looking for an example of two sequences $u_n$ and $v_n$ in $L^2(\Omega)$ such that
$$u_n \to u \ \mbox{weakly in}\ L^2(\Omega),$$
$$v_n \to v \ \mbox{weakly in}\ L^2(\Omega),$$
but the product $u_n v_n$ doesn't weakly converge to $uv$.

Answer 1

Take $u_n=v_n\colon x\mapsto \sin(nx)$. Then $(u_n)_n$ weakly converges in $L^2(0,2\pi)$ to $0$ but $u_nv_n$ converges weakly to $\pi$, as $\sin^2u=\frac 12-\frac 12 \cos(2u)$.