Prove the boundedness of a bilinear continuous mapping.

Although Giuseppe mentioned in his comment, I am giving the little details of the alternative proof.Note that for $x\in X_1$ (the unit ball of $X$), the collection {${B_x| x\in X_1}$} is a set of continuous linear map (By continuity of $B$ in 2nd variable) from the Banach Space $Y$ to $Z$. And the collection is also pointwise bounded.(By contunuity of $B$ in 1st variable).Hence by uniform boundedness principle we will have {${B_x| x\in X_1}$} is uniformly norm bounded. And hence the required result follows.