Polynomial topologies on Banach spaces

On every real Banach space X we introduce a locally convex topology τ P , canonically associated to the weak-polynomial topology w P . It is proved that τ P is the finest locally convex topology on X which is coarser than w P . Furthermore, the convergence of sequences is considered, and sufficient conditions on X are obtained under which the convergent sequences for w P and for τ P 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).