I am reading a paper ('Hardy's inequalities revisited' by Brezis and Marcus). In one of the proofs, they write the following:
I am not sure regarding the "last assertion" part. Is it true that if a sequence is bounded in $L^2$ and converges to $0$ in $L^2_{\text{loc}}$ then it also converges weakly to $0$? I couldn't find such a result and was wondering where I could read about it.
If you need anymore details about the functions in the picture above let me know.
Thanks in advance.
It has been answered in the comments, so I put it here as a community-wiki.
Third-party edit: Since Feng has been encouraging me to add to this answer I'll point out that it doesn't seem necessary to acknowledge the comments, because the argument in the comments is wrong, or at least majorly incomplete. The argument below is entirely different (also correct).
Let $\Omega$ be an open set. If $u_n\to u$ in $L^2_{\text{loc}}(\Omega)$, then $\langle u_n,\varphi\rangle\to \langle u,\varphi \rangle$ for all $\varphi\in C_c^\infty(\Omega)$. Since $C_c^\infty(\Omega)$ is dense in $L^2$ and $\{u_n\}$ is bounded in $L^2$, we can prove the weakly convergent of $u_n$ to $u$.
(Talking about $\phi\in L^2$ concentrated on a set with finite measure instead of $\phi\in C^\infty_c$ gives an argument that works in any measure space...)