# Banach-Steinhaus theorem for nets?

Consider an uncountable set $I$ and let $A=\mbox{Fin}(I)$ be the family of finite subsets of $I$ ordered by inclusion. Let $E$ be a normed space and $F$ be a Banach space. Suppose moreover we have a net $(T_\alpha)_{\alpha\in A}$ of bounded operators between $E$ and $F$. I want to show that $(T_\alpha)_{\alpha\in A}$ is convergent to a certain operator $T$. Is there any version of Banach-Steinhaus theorem valid in this case? That is, what I can show is the fact that $(T_\alpha x)_{\alpha\in A}$ is convergent in $Y$ to $(Tx)_{\alpha\in A}$ for each $x\in X$. Can I conclude that $(T_\alpha)_{\alpha\in A}\to T$ ?

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You cannot conclude this at all, even when you have a sequence. See the corollary and the note here.

The pointwise limit defines a bounded operator, but the limit of the sequence of operators does not necessarily converge (in the norm topology) to the operator defined by the pointwise limit.

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Banach-Steinhaus requires $\{T_\alpha x:\alpha\in A\}$ to be bounded for each $x$, and convergent nets need not be bounded in general, so there potentially is a difference. –  Jonas Meyer Nov 19 '11 at 21:29
@JonasMeyer Nice catch. I don't immediately see how to fix this potential problem, so I'll edit the answer accordingly. –  Jonathan Gleason Nov 19 '11 at 21:52
Yes, I have $\sup_{\alpha \in A, \|x\|\leq 1}\|T_\alpha x\|<\infty$. So, is $T$ a limit in this case? –  bellotojimenez Nov 20 '11 at 7:41