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 $(T_\alpha)_{\alpha\in A}\to T$ ?
Banach-Steinhaus theorem for nets?
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functional-analysis
banach-spaces
1 Answers
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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 sequence of operators does not necessarily converge (in the norm topology) to the operator defined by the pointwise limit.
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0Yes, I have \sup_{\alpha \in A, \|x\|\leq 1}\|T_\alpha x\|<\infty. So, is $T$ a limit in this case? – 2011-11-20