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?