I thought so far that the uniform boundedness principle applies (according to the proof I know) to any net of bounded operators. Now I read that this works for sequences only. Can you shortly explain me an example without going into details? Just to make sure we're talking about the same: $$T_\lambda\in\mathcal{B}(E,F):\quad\|T_\lambda x\|<\infty\quad(x\in E)\implies \|T_\lambda\|_{\lambda\in\Lambda}<\infty$$ where $\|T_\lambda\|_{\lambda\in\Lambda}:=\sup_{\lambda\in\Lambda}\|T_\lambda\|$.
Thank you very much!
The uniform bounded principle does work for any collection of operators, see wiki or any book on functional analysis.
Note that convergent nets may not be bounded (in contrary to convergent sequences), see the post to which David Mitra is referring.