# Weak continuity of bilinear forms on Hilbert spaces

Let $$m:H_1\times H_2\to \mathbb C$$ be a bounded bilinear form, where $$H_1,H_2$$ are Hilbert spaces. Let $$\{x_n\}$$ and $$\{y_n\}$$ converges weakly to $$x$$ and $$y$$ respectively in $$H_1$$ and $$H_2$$. Does $$\lim_n m(x_n,y_n)=m(x,y)$$ holds true? Given that the limit exists.

No. Let $$m$$ be the inner product in $$H = H_1 = H_2$$ and consider an orthonormal sequence $$\{e_n\}$$. Then, $$m(e_n, e_n) = 1$$ but $$e_n \rightharpoonup 0$$ and $$m(0,0) = 0$$.