Consider two Banach spaces $E,F$ and a net $T_\alpha : E \to F$ of continuous operators.

I know that for each $x \in E$ the net $T_\alpha (x)$ is convergent in $F$ and it is easy to show that the limit $L(x)$ is a linear function.

Can one deduce that $L$ is bounded?

I thought originally that this is an obvious consequence of the Uniform Bounded Principle, but unfortunatelly it is not.

**P.S.** this problem comes from some question on convergence in a non-metrisable LCTVS, which I reduced to the simpler looking question above. Because of this I cannot work with sequences.