Collection of linear functions

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 $(Tx)_\alpha:=T_\alpha x$. –  Davide Giraudo Dec 4 '12 at 19:10
The $T_\alpha$ need to be uniformly bounded. –  JSchlather Dec 4 '12 at 19:18
So we can try something like $Y:=\{\{y_\alpha\},\sup_{\alpha}\frac{\lVert y_\alpha\rVert}{1+\lVert T_\alpha\rVert}<\infty\}$. –  Davide Giraudo Dec 4 '12 at 19:22
This is a partial answer. Denote $Y=\bigoplus_\infty\{Y_\alpha:\alpha\in\mathcal{A}\}$. Then we define $$\pi_\alpha:T\to Y_\alpha: y\mapsto y_\alpha$$ $$T:X\to Y:x\mapsto\bigoplus_\infty\{\Vert T_\alpha\Vert^{-1} T_\alpha(x):\alpha\in\mathcal{A}\}$$ In this case $T\in\mathcal{B}(X,Y)$ with $\Vert T\Vert\leq 1$. Unfortunately $T$ allows us to recover only $\Vert T_\alpha\Vert^{-1} T_\alpha$, not $T_\alpha$. Indeed $$\Vert T_\alpha\Vert^{-1} T_\alpha(x)=(\pi_\alpha\circ T)(x)$$ We need some tricky method to store values $\Vert T_\alpha\Vert$ for all $\alpha\in\mathcal{A}$ in $Y$. I think we need to enlarge it by some direct sum of normed spaces which can allow us to recover this values.