Do you know the example of next operators: is a closed operator, is a bounded linear operator such that is not a closable operator? are Hilbert spaces.
Let be an orthonormal basis for the Hilbert space . Define A, B on H by and (for all n). Then B is closable (it is densely defined and contained in its adjoint), A is bounded (use Cauchy–Schwarz to show that), but AB is not closable (if then but ).
I should have said that you need to extend both A and B by linearity to the space of all finite linear combinations of the basis vectors, and then take their closures. So if (only finitely many s nonzero) then , and . That way, A and B are both linear.