I am having problem with the following 1)Are $H^{1}$ nad $H^{1}_{0}$ a reflexive spaces?
2)If $u_{n} \rightarrow u$ weakly in $H^{1}_{0}$, can I say that it is same as $(\nabla u_{n} , \nabla w) \rightarrow (\nabla u, \nabla w)$ for any $w \in H^{1}_{0}$? Thanks a lot