For a fixed $p \in (1,\infty)$, consider $L^p(\Omega)$ for a bounded domain $\Omega$ in the Euclidean space and a sequence $\{ f_{n} \} \subset L^p(\Omega)$ converging weakly. That is, for each $g \in L^{p'}(\Omega)$ with $p^{-1} + p'^{-1}=1$, the limit \begin{equation} \lim\limits_{n \to \infty} \int_\Omega f_n g \end{equation} exists. In this case, it is well-known that there exists a unique $f \in L^p(\Omega)$ such that \begin{equation} \lim\limits_{n \to \infty} \int_\Omega f_n g = \int_\Omega f g \end{equation}
Moreover, we know that $C_c^\infty(\Omega)$, the space of compactly supported smooth functions on $\Omega$ is dense in $L^p(\Omega)$. ,y question is:
Can we find a double sequence $\{ \phi_{n,m} \} \in C_c^\infty(\Omega)$ such that $\phi_{n,m}$ approximates $f_n$ in the weak topology uniformly in $n$?
More specifically, can we find a double sequence $\{ \phi_{n,m} \} \in C_c^\infty(\Omega)$ such that \begin{equation} \Bigl \lvert \int_\Omega (f_n-\phi_{n,m}) g \Bigr \rvert \leq C(m,g) \end{equation} where $C(m,g) \to 0$ as $m \to \infty$ for each $g \in L^p$ and $C(m,g)$ is independent of $n$.
I think this is a quite nontrivial problem. Could anyone please help me?