Banach spaces in which all multilinear forms are weakly sequentially continuous

• Autorzy: Jesús M.F. Castillo , Ricardo García , Raquel Gonzalo
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

multilinear forms; polynomials; weak continuity

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 46B20: Geometry and structure of normed linear spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 136
• Numer: 2
• Strony: 121-145

Daty

wydano

1999

otrzymano

1997-11-03

poprawiono

1999-03-26

Twórcy

autor

Jesús M.F. Castillo

• Departamento de Matemáticas, Universidad de Extremadura, Avda. de Elvas s/n 06071 Badajoz, Spain

autor

Ricardo García

• Departamento de Matemáticas, Universidad de Extremadura, Avda. de Elvas s/n 06071 Badajoz, Spain

autor

Raquel Gonzalo

• Facultad de Informática, Universidad Politécnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Sain

Bibliografia

• [1] R. Alencar, R. Aron and S. Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), 407-411.
• [2] R. Alencar, R. Aron and G. Fricke, Tensor products of Tsirelson's space, Illinois J. Math. 31 (1987), 17-23.
• [3] R. Alencar and K. Floret, Weak-strong continuity of multilinear mappings and the Pełczyński-Pitt theorem, J. Math Anal. Appl. 206 (1997), 532-546.
• [4] Z. Altshuler, P. G. Casazza and B. L. Lin, On symmetric basic sequences in Lorentz sequence spaces, Israel J. Math. 15 (1993), 140-155.
• [5] R. M. Aron, Y. S. Choi and J. G. Llavona, Estimates by polynomials, Bull. Austral. Math. Soc. 52 (1995), 475-486.
• [6] R. M. Aron and S. Dineen, Q-reflexive Banach spaces, Rocky Mountain J. Math. 27 (1997), 1009-1025.
• [7] B. Beauzamy et J. T. Lapresté, Modèles étalés des espaces de Banach, Hermann, Paris, 1984.
• [8] P. Biström, J. A. Jaramillo and M. Lindström, Polynomial compactness in Banach spaces, Rocky Mountain J. Math., to appear.
• [9] T. Carne, B. Cole and T. Gamelin, A uniform algebra of analytic functions on a Banach space, Trans. Amer. Math. Soc. 314 (1989), 639-659.
• [10] P. G. Casazza and B. L. Lin, On symmetric basis sequences in Lorentz sequence spaces II, Israel J. Math. 17 (1974), 191-218.
• [11] J. M. F. Castillo, On Banach spaces X such that $L(L_p, X) = K(L_p ,X)$, Extracta Math. 10 (1995), 27-36.
• [12] J. M. F. Castillo, R. García and R. Gonzalo, Stability properties of the class of Banach spaces in which all multilinear forms are weakly sequentially continuous, preprint, 1999.
• [13] J. M. F. Castillo and M. González, The Dunford-Pettis property is not a three-space property, Israel J. Math. 81 (1993), 297-299.
• [14] J. M. F. Castillo and M. González, New results on the Dunford-Pettis property, Bull. London Math. Soc. 27 (1995), 599-605.
• [15] J. M. F. Castillo, M. González and F. Sánchez, M-ideals of Schreier type and the Dunford-Pettis property, in: Non-Associative Algebra and its Applications, S. González (ed.), Kluwer, 1994, 81-85.
• [16] J. M. F. Castillo and F. Sánchez, Remarks on some basic properties of Tsirelson's space, Note Mat. 13 (1993), 117-122.
• [17] J. M. F. Castillo and F. Sánchez, Weakly p-compact, p-Banach-Saks, and super-reflexive Banach spaces, J. Math. Anal. Appl. 185 (1994), 256-261.
• [18] Y. S. Choi and S. G. Kim, Polynomial properties of Banach spaces, J. Math. Anal. Appl. 190 (1995), 203-210.
• [19] J. C. Díaz, Non-containment of $ℓ_1$ in projective tensor products of Banach spaces, Rev. Mat. Univ. Complut. Madrid 3 (1990), 121-124.
• [20] V. Dimant and I. Zalduendo, Bases in spaces of multilinear forms over Banach spaces, J. Math. Anal. Appl. 200 (1996), 548-566.
• [21] G. Emmanuele, A dual characterization of spaces not containing $ℓ_1$, Bull. Polish Acad. Sci. 34 (1986), 155-160.
• [22] J. D. Farmer, Polynomial reflexivity in Banach spaces, Israel J. Math. 87 (1994), 257-273.
• [23] M. González and J. M. Gutiérrez, Gantmacher type theorems for holomorphic mappings, Math. Nachr. 186 (1997), 131-145.
• [24] M. González and J. M. Gutiérrez, Polynomials on Schreier's space, preprint, 1997.
• [25] R. Gonzalo, Multilinear forms, subsymmetric polynomials and spreading models, J. Math. Anal. Appl. 202 (1996), 379-397.
• [26] R. Gonzalo and J. A. Jaramillo, Smoothness and estimates of sequences in Banach spaces, Israel J. Math. 89 (1995), 321-341.
• [27] R. Gonzalo and J. A. Jaramillo, Compact polynomials between Banach spaces, Proc. Roy. Irish Acad. Sect. A 95 (1995), 213-226.
• [28] J. M. Gutiérrez, J. A. Jaramillo and J. G. Llavona, Polynomials and geometry of Banach spaces, Extracta Math. 10 (1995), 79-114.
• [29] J. A. Jaramillo and A. Prieto, Weak-polynomial convergence on a Banach space, Proc. Amer. Math. Soc. 118 (1993), 463-468.
• [30] M. Jiménez Sevilla and R. Payá, Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces, Studia Math. 137 (1998), 99-112.
• [31] H. Knaust and E. Odell, Weakly null sequences with upper $ℓ_p$-estimates, in: Lecture Notes in Math. 1470, Springer, 1990, 85-107.
• [32] D. Leung, On $c_0$-saturated Banach spaces, Illinois J. Math. 39 (1995), 15-29.
• [33] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin, 1977.
• [34] J. Mujica, Complex Analysis in Banach Spaces, North-Holland Math. Stud. 120, North-Holland, Amsterdam, 1996.
• [35] A. Pełczyński, A property of multilinear operations, Studia Math. 16 (1957), 173-182.
• [36] R. A. Ryan, Dunford-Pettis properties, Bull. Acad. Polon. Sci. 27 (1979), 373-379.
• [37] C. Stegall, Duals of certain spaces with the Dunford-Pettis property, Notices Amer. Math. Soc. 19 (1972), A-799.
• [38] E. Straeuli, On Hahn-Banach extensions for certain operator ideals, Arch. Math. (Basel) 47 (1986), 49-54.
• [39] M. Valdivia, Complemented subspaces and interpolation properties in spaces of polynomials, J. Math. Anal. Appl. 208 (1997), 1-30.