Consider the inner product by $\langle f,g \rangle_{H^1} = \langle f, g \rangle_{L^2} + \sum_{|\alpha|=1} \langle D^\alpha f, D^\alpha g \rangle_{L^2}$ where $\alpha$ is a multi-index and $D$ denotes the weak derivative. Define $H^1(\Omega)$ as the space of functions that are finite under the norm induced from this inner product. It can be shown that $H^1(\Omega)$ is a Hilbert space.
Now, suppose there exists a sequence of functions $\{ f_n \} \subset H^1(\Omega)$ that converges weakly in $H^1$ to some limit $f \in H^1(\Omega)$. Can I then say that this sequence converges weakly to the same limit under the $L^2$ inner product?
By an application of the Banach-Alaoglu Theorem, I know that weak convergence of this sequence in $H^1$ will imply strong convergence of a subsequence in $H^1$. And then I think strong convergence of this subsequence in $H^1$ will imply strong convergence in $L^2$ as well.
However, I'm not sure if anything can be said about the entire sequence under the $L^2$ inner product and its weak/strong convergence properties.