WEAK CONTINUITY OF MULTILINEAR MAPPINGS ON TSIRELSON'S SPACE

Abstract

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 space T, its dual T ¹ and the James space (T ¹) _J modelled over T ¹ are obtained. Our approach simplifies various known results and generalizes some of them, for example: the completed projective tensor product , all spaces P ^N (T′; F) of N-homogeneous polynomials, and the space ((H(T′; F),τω) of holomorphic functions are reflexive if F is a reflexive Banach space of some Rademacher type or cotype. Moreover, we give some more information about the properties P _α, and rank α which were studied by Pelczynski [P] and in [AF].