Dear BCnrd, Thank you for your interesting comments but... 1) I was only interested in the differentiable case. I am not sure what you mean by a connection in the analytic or algebraic cases. (an analytic(algebraic) splitting of the tangent bundle?) so I am really satisfied with the isomorphism of smooth sections i was talking about. By the way, it's not mine. 2)I admit the comment to Matt lacked details, probably thinking that it was clear $X$ should be thought as a tangent vector and not a vector field (as I wrote); (retrospectively that was really not the main issue of the question);

Please read the second answer below!

Willie i didn't say it was easy:) In the background lurks the isomorphism $\Gamma(P;\pi^*E)\simeq \Gamma(P;\mathbb{C})\otimes \Gamma(B;E)$ which is by not obvious(at least not to me). But Leibniz relation does make sense and it does define the connection everywhere and it also explains the trivial connection on the trivial bundle.

Emerton, you are right.