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Let $X$ be a Banach space. Let $\{Y_\alpha\}_\alpha$ be normed spaces. Let $\{T_\alpha:X\rightarrow Y_\alpha\}_\alpha$ be an infinite collection of bounded linear functions.

Is there a way to create one linear $T:X\rightarrow Y$ for some normed space $Y$ that will contain all the information about the collection $\{T_\alpha\}$? My problem is with finding a way to define a suitable $Y$ and a norm for it.

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    We can try $Y:=\{(x_{\alpha})_\alpha, \sup_{\alpha}\lVert x_\alpha\rVert_{Y_\alpha}<\infty\}$ and $(Tx)_\alpha:=T_\alpha x$.2012-12-04
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    The $T_\alpha$ need to be uniformly bounded.2012-12-04
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    So we can try something like $Y:=\{\{y_\alpha\},\sup_{\alpha}\frac{\lVert y_\alpha\rVert}{1+\lVert T_\alpha\rVert}<\infty\}$.2012-12-04

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