Let $a[x,y]$ be a densely defined, symmetric sequilinear form with domain $D(a)$ on a complex Hilbert space. Suppose that $x, y \in D(a)$ and there is a sequence $x_n \in D(a)$ such that $x_n$ weakly converges to $x$. Is it true that $a[x_n,y]$ converges to $a[x,y]$?
This is certainly true if $a[x,y]$ is the form associated with a symmetric operator $A$ by $a[x,y] = \langle Ax,y \rangle$ when $D(a) = D(A)$, but what about for other symmetric forms?