Problems on Annihilator and Adjoint of a linear map

Aug 2009
1)Let V be vector space of Polynomial of degree 2 over R.Let D:V-->V be the differential operator on V.Find Annihilator of Image D in V* i.e,Dual of V.

2)Let V be a vector space of all polynomial function over R.Let f be the linear functional on V defined by f(p(x)) = integration of p(x)dx over 0 to 1.If Dis the differential operator on V,compute D*f, i.e; f composite D and the map D* is adjoint of D.

Thank you in advance