I have the following question:
Let $\Omega \subset \mathbb{C}$ be a bounded domain and let $\{u_{j}\}$ be a sequence of holomorphic functions in $\Omega$ such that $u_{j}\in L^{2}(\Omega)$, for each $j$. Assume that $u_{j}$ converges weakly to $0$ in $L^{2}(\Omega)$. Can we conclude that there exists a subsequece $u_{j_{k}}$ such that $u_{j_{k}}\rightarrow 0$ in $L^{2}_{loc}(\Omega)$?
My attempt: since $\{u_{j}\}$ is bounded and $|u_{j}|^{2}$ is subharmonic, by Hartogs lemma, there exists a subsequence ${u_{j_{k}}}$ and $u\in L^{2}_{loc}$ such that $|u_{j_{k}}|\rightarrow u$ in $L^{2}_{loc}(\Omega)$. But I cannot show that $u=0$.