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.



Question about pointwise convergence of operators
No, this works for sequences but not general nets. For a counterexample, let $T: E \to \mathbb{R}$ (or $\mathbb{C}$) be an unbounded linear functional. For each finite dimensional subspace $E\_0$ of $E$, the restriction of $T$ to $E\_0$ is bounded so by the Hahn-Banach theorem it can be extended to a bounded linear functional on $E$. Ordering by inclusion of finite dimensional subspaces then yields a net of bounded linear functionals which converges pointwise to $T$.


This is why the uniform boundedness principle for sequences is so amazing!


