Does weak convergence of (periodic) $L^2(\mathbb{R}/\mathbb{Z})$ functions imply strong convergence?
i.e. let $f_n \in L^2(\mathbb{R}/\mathbb{Z})$
assume $f_n \rightharpoonup f$ in $L^2$
does $f_n \rightarrow f$ in $L^2$?
Intuitively, if weak convergence means that part of the 'mass' of the function travels to infinity, then the constraint of periodicity should prevent this and thus imply strong convergence, but I can't prove it. I also can't find a counterexample :(