I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$.
I want to show that there is a sequence $v_k\rightharpoonup v$, such that
$$\langle u_k,v_k \rangle \nrightarrow \langle u, v \rangle.$$
My idea is to make a sequence $v_k\rightharpoonup 0$, such that $\langle u_k,v_k \rangle=1$
but I can't figure out how to do it.