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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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