Polynomial topologies on Banach spaces

On every real Banach space X we introduce a locally convex topology τP, canonically associated to the weak-polynomial topology wP. It is proved that τP is the finest locally convex topology on X which is coarser than wP. Furthermore, the convergence of sequences is considered, and sufficient conditions on X are obtained under which the convergent sequences for wP and for τP either coincide with the weakly convergent sequences (when X has the Dunford-Pettis property) or coincide with the norm-convergent sequences (when X has nontrivial type).