Consider the uniform boundedness principle:
UBP. Let $E$ and $F$ be two Banach spaces and let $(T_i)_{i \in I}$ be a family (not necessarily countable) of continuous linear operators from $E$ into $F$. Assume that $\sup_{i \in I} \|T_ix \| < \infty$ for all $x \in E$. Then $\sup_{i \in I} \|T_i\|_{\mathcal{L}(E,F)} < \infty$.
I don't understand the statement of the UBP. The assumption tells us that, fixed an element $u$, we surely find a $\|T_ku\|< \infty$ (in particular, for that fixed $u$ each other $T$ is limited in $u$ too). The conclusion tells us that the sup over the $i$'s of the set $ \biggl\{ \sup_{\|x\|\leq 1} \|Tx\| \biggr\} $ is limited. But isn't that clear from the assumption? I mean, if each $T$ is bounded, a fortiori the conclusion must hold... please explain me where I am wrong.