Continuing our investigation in [AF] about the sequential weak-to-norm continuity of multilinear mappings and polynomials between Banach spaces, some results about the Tsirelson spaceT, its dualT1and the James space (T1)Jmodelled overT1are obtained. Our approach simplifies various known results and generalizes some of them, for example: the completed projective tensor product, all spacesPN(T′; F) ofN-homogeneous polynomials, and the space ((H(T′; F),τω) of holomorphic functions are reflexive ifFis a reflexive Banach space of some Rademacher type or cotype. Moreover, we give some more information about the propertiesPα, and rank α which were studied by Pelczynski [P] and in [AF].