I'm puzzled in the bounded operator concept. Now I want to ask a simple problem about that.
If $I$ is an indexing set, $\{T_\alpha|\alpha\in I\}$ is a linear bounded operator collection from $X$ to $X_1$, where $X, X_1$ are normed spaces. Based on the above fact, I list some conclusions:
For $\forall T_\alpha,$ there exists a constant $M_\alpha\ge0$, such that $\forall x\in X$, $$\|T_\alpha x\|\le M_\alpha\|x\|$$ holds.
For $\forall T_\alpha$, I have $$ \sup_{\alpha}\|T_\alpha x\|\lt \infty. $$
I want to know whether the above two conclusions are right. where is the mistake? Could someone tell?