Let $H$ be Hilbert space, $x,y \in H$ and $x_n, y_n \in H$. Suppose $x_n$ weakly converge to $x$ and $y_n$ weakly converges to $y$. Is it true that $\langle x_n,y_n\rangle$ converges to $\langle x,y\rangle$?
I know that scalar product is a continous function, but does it help? Could if follow from Riesz representation theorem?
No. Let $H= l^2$ and let $(u_n)$ be the usual orthonormal basis of $H$.
Put $x_n=y_n=u_n$. Then $(u_n)$ converges weakly to $x=y=0$.
But $\langle u_n,u_n\rangle=||u_n||_2^2=1 $ and $\langle x,y\rangle=0$.