Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to try to show the continuity by proving that whenever sequences $x_{n} \rightharpoonup x$ in $W^{1,p}(\Omega)$ and $y_{n} \rightarrow y$ in $L^{p}(\Omega)$ it follows that $\eta(x_{n},y_{n}) \rightarrow \eta(x,y)$ in $L^{p'}(\Omega)$?
I know this is possible if we are trying to prove continuity of a function defined on a domain which is first-countable. I'm not sure if Sobolev space $W^{1,p}(\Omega)$ with the weak topology is first-countable or if this proof is still possible without it being first-countable?
Thanks for any help.