I am trying to find an example of a sequence of bounded operators in an Hilbert space such that the limit is unbounded. For a total family $(e_n)_n$ I thought of defining $u_N(e_n)$ as $ne_n$ if $n<N$ and zero otherwise. But what I think as of the "limit", $u(e_n)=ne_n$ for all $n$, is not truly the limit, right?
Can we more generally fully characterize the sequences of bounded operators giving an unbounded limit?
if $(T_n)$ is sequence of bounded operators such that $Tx=\lim T_n x$ exists for every $x \in H$ then $T$ is automatically a bounded operator. This is a consequence of Uniform Boundedness Principle, aka Banach Steinhaus Theorem.