Weakly Continuous Mappings on Banach Spaces with the Dunford-Pettis Property

Abstract

The Dunford-Pettis weak ( dw) topology on a Banach space is introduced as the finest topology that coincides with the weak topology on Dunford-Pettis sets. We characterize a wide class of polynomials between Banach spaces (including all the scalar valued polynomials) which are dw-continuous, and prove that a Banach space has the Dunford-Pettis (DP) property if and only if all these polynomials are weakly sequentially continuous. This result contains a characterization of the DP property obtained by Ryan, answering a question of Pelczyński: E has the DP property if and only if any weakly compact polynomial on E takes weak Cauchy sequences into convergent ones. It also extends other characterizations of the DP property by Operators to the case of polynomials. Similar results are given for holomorphic mappings. Other properties of polynomials and holomorphic mappings between Banach spaces are obtained.