Weak $L^2$ convergence of Sobolev functions still Sobolev

Question

Suppose $f^j$ and $\frac{\partial f^j}{\partial x}$ are sequences of functions in $H^1(\mathbb{R})$ weakly converging to the functions $f$ and $g$, respectively, in $L^2(\mathbb{R}).$ Is $f$ necessarily a Sobolev function, and is g necessarily its weak derivative? If so, how do you show this?

Answer 1

Hint: for arbitrary $\phi \in C^\infty_c(\mathbb{R})$, show that $$\begin{align*}\int f \phi' &= \lim_{j \to \infty} \int f^j \phi' \\ &= \dots \text{ you fill in} \dots \\ &= - \int g \phi\end{align*}$$ So $g$ is the weak derivative of $f$, and since $g \in L^2$ we have $f \in H^1$.