This question comes from the proof of Theorem 1 in E.N.Dancer's paper. (page 427 line 13-14)
Suppose $u_n$ converges weakly to $u$ in $L^2(\Omega)$.
Does $|u_n|$ converges weakly to $|u|$ in $L^2(\Omega)$?
From the facts that
1.) weak convergence $\Rightarrow$ norm-boundedness $\Rightarrow$ subsequence weak convergence.
2.) If every subsequence contains a weakly convergent subsubsequence, then the whole sequence converges weakly.
We know that $|u_n|$ converges weakly to some $v \in L^2(\Omega)$. But I don't know how to show $v = |u|$.