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Let $H$ be a Hilbert space. Suppose that $\{u_n\} \subseteq B(H)$ is a sequence which converges to some operator $u$ in the weak operator topology, which means that for all $x,y\in H$ one has ‎$$‎‎‎‎\langle ‎u_n(x),y\rangle \rightarrow‎ ‎‎‎\langle u(x),y \rangle$$

Q1: Why must this sequence be norm-bounded?

Q2: Does it hold for nets?

Q2: Does it hold for the strong operator topology?

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As for Q1, the uniform bounded principle should be applied. Let $\{u_n\}$ be a net of operators in $B(H)$ which is also weakly convergent. Any vector $y\in H$, induces the sequence of bounded linear functionals $\phi_n$ on $H$ given by $x\to \langle u_nx,y\rangle$. Since $u_n$ is weakly convergent then $\phi_n$ is point-wise convergent and so it is also point-wise bounded. Therefore UBP uimplies that $\phi_n$ is uniformly bounded. Similar argument on the first variable implies the boundedness of the sequence $\{u_n\}$.

As for Q2, It is not true even when dim$H$=1. For example, let us consider the net $\{u_r\}_{r\in \mathbb{R}}$ where $u_r=r id_H$. Then $u_r\to0$ when $r\to0$ even in the norm topology but this net is unbounded!

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