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.