In generalizing a result of Pełczyǹski the sequential weak to norm continuity of N-linear mappings between certain Banach spaces (including spaces of type and cotype) will be studied. In particular, it is shown that every N-linear continuous mapping from ℓP1 × ⋯ ×ℓPN into ℓq is compact if and only if Σni = 1(1/pi) < 1/q. © 1997 Academic Press.2062532546Alencar, R., Floret, K., Weak continuity of multilinear mappings on Tsirelson's space (1994) Sem. Bras. Anál., 40, pp. 509-517Aron, R.M., Herves, C., Valdivia, M., Weakly continuous mappings on Banach spaces (1983) J. Fund. Anal., 52, pp. 189-204Aron, R.M., Globevnik, J., Analytic functions on c0 (1989) Rev. Mat. (Madrid), 2, pp. 27-34Aron, R.M., Lacruz, M., Ryan, R.A., Tonge, A.M., The generalized rademacher functions (1992) Note di Mat., 12, pp. 15-25Banach, S., (1978) Théorie des Operations Linéaires, , Chelsea Publishing Company, New YorkBeauzamy, B., Lapresté, T.J., (1984) Modèles Étalés des Espaces de Banach, , Hermann, ParisBonet, J., Peris, A., On the injective tensor product of quasinormable spaces (1991) Results in Math., 20, pp. 431-443Castillo, J., Sanchez, F., Remarks on some basic properties of Tsirelson's space (1993) Note di Mat., 13, pp. 117-122Castillo, J., Sanchez, F., Weakly p-compact, p-Banach-Saks, and super-reflexive Banach spaces (1994) J. Math. Anal. Appl., 185, pp. 256-261Defant, A., Floret, K., Tensor norms and operator ideals Math. Studies, 176, p. 1993. , North Holland, AmsterdamDefant, A., Maestre, M., Property (BB) and holomorphic functions on Fréchet-Montel spaces (1993) Math. Proc. Cambridge Philos. Soc., 115, pp. 305-313Diestel, J., Sequences and series in Banach spaces (1984) Graduate Texts in Math, 92. , Springer-Verlag, New York/BerlinDineen, S., Complex analysis in locally convex spaces (1981) Math. Studies, 57. , North Holland, AmsterdamFarmer, J., Polynomial reflexivity in Banach spaces (1994) Israel J. Math., 87, pp. 257-273Farmer, J., Johnson, W.B., Polynomial Schur and polynomial Dunford-Pettis properties (1993) Contemp. Math., 144, pp. 95-105Floret, K., Weakly Compact Sets Lecture Notes in Math., 801, p. 1980. , Springer-Verlag, New York/BerlinGonzalo, R., Jaramillo, J., Compact polynomials between Banach spaces (1993) Extracta Math., 8, pp. 42-48Knaust, H., Orliez sequence spaces of Banach-Saks type (1992) Arch. Math., 59, pp. 562-565Mascioni, V., On weak cotype and weak type in Banach spaces (1988) Note di Mat., 8, pp. 67-110Mascioni, V., Matter, U., Weakly (q, p)-summing operators and weak cotype properties of Banach spaces (1988) Proc. Roy. Irish Acad., 88 A, pp. 169-177Pełczyǹski, A., A property of multilinear operations (1957) Studia Math., 16, pp. 173-182Pcłczyǹski, A., On weakly compact polynomial operators on B-spaces with Dunford-Pettis property (1963) Bull. Acad. Pol. Sci., 11, pp. 371-378Pełczyǹski, A., A theorem of Dunford-Pettis type for polynomial operators (1963) Bull. Acad. Pol. Sci., 11, pp. 379-386Pełczyǹski, A., Semadeni, Z., Spaces of continuous functions (III) (spaces C(Ω) for Ω without perfect subsets) (1959) Studia Math., 18, pp. 211-222Pisicr, G., Factorization of linear operators and geometry of Banach spaces CBMS Regional Conference Series, 60, p. 1986. , Amer. Math. Soc., Providence, RIPitt, H.R., A note on bilinear forms (1936) J. London Math. Soc., 11, pp. 174-180Rakov, S.A., Banach-Saks property of a Banach space (1980) Math. Notes, 26, pp. 909-916. , transl. from Mat. ZametkiRakov, S.A., Banach-Saks exponent of certain Banach spaces of sequences (1982) Math. Notes, 32, pp. 791-797. , transl. from Mat. ZametkiTomczak-Jacgcrmann, N., Banach-Mazur distances and finite-dimensional operator ideals (1989) Monographs and Surveys in Pure and Applied Math., 38. , Pitman, London