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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.

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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 $m(0,0) = 0$.

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