Banach spaces in which all multilinear forms are weakly sequentially continuous

We solve several problems in the theory of polynomials in Banach spaces. (i) There exist Banach spaces without the Dunford-Pettis property and without upper p-estimates in which all multilinear forms are weakly sequentially continuous: some Lorentz sequence spaces, their natural preduals and, most notably, the dual of Schreier's space. (ii) There exist Banach spaces X without the Dunford-Pettis property such that all multilinear forms on X and X* are weakly sequentially continuous; this gives an answer to a question of Dimant and Zalduendo [20]. (iii) The sum of two polynomially null sequences need not be polynomially null; this answers a question of Biström, Jaramillo and Lindström [8] and also of González and Gutiérrez [23]. (iv), (v) The absolutely convex closed hull of a pw-compact set need not be pw-compact; the projective tensor product of two polynomially null sequences need not be a polynomially null sequence. This answers two questions of González and Gutiérrez [23]. (vi) There exists a Banach space without property (P); this answers a question of Aron, Choi and Llavona [5].