WEAK COMPACTNESS IN SPACES OF DIFFERENTIABLE MAPPINGS

We characterize the weakly compact subsets (and thereby the weak convergence) in several spaces of k-times continuously differentiable mappings between real Banach spaces. As an application, we give characterizations of the Dunford-Pettis (DP) property of a Banach space F in terms of the weak sequential continuity of the composition map (f, g) --> g o f, where f: E --> F is a differentiable mapping and g: F --> G is a linear operator. We also prove that F has the DP property if and only if whenever (x(n)) subset of F is weakly null and (P-n) is a weakly null sequence of polynomials from F to another space G, then (P-n(x(n))) converges to 0 in the weak topology of G. Finally, we derive a new proof of the fact that any weakly compact homomorphism between algebras of differentiable functions is induced by a constant mapping.