Let $H$ be a separable Hilbert Space. Suppose $B$ is a linear map with the property that if $u_n \to u$ and $B(u_n) \to v$, then $v=B(u)$. Show that $B$ is bounded.
So the issue here is to get some sequence that $B$ will be continuous on and then proceed by some sort of scaling, but I don't see how to find such a sequence...perhaps involving the basis?
Any tips helpful.
Use the closed graph theorem. If $(u_n, B(u_n)) $ converges towards $(u, v) $, $u_n$ converges towards $u$ and the hypothesis implies that $v=B(u) $ so the graph is closed.