Duality in Spaces of Nuclear and Integral Polynomials

We study the spaces of nuclear and integral vector-valued.polynomials and their duals. We prove that, if E is Asplund, PN nE; F.sPI nE; F. isometrically, for any Banach space F. We describe PN nE; F.9 as a subspace of P nE9; F9. in terms of weak-star continuous polynomials. We show that Pw* nE9; F. s Pc* nE9; F., when E has the approximation property.