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    García, Ricardo Jaramillo, Jesús A. and Llavona, José G. 2011. The Aron-Berner extension, Goldstine's theorem and P-continuity. Mathematische Nachrichten, Vol. 284, Issue. 5-6, p. 694.


    Cilia, Raffaella and Gutiérrez, Joaquín M. 2009. Weakly sequentially continuous differentiable mappings. Journal of Mathematical Analysis and Applications, Vol. 360, Issue. 2, p. 609.


    Garrido, M. Isabel Jaramillo, Jesús A. and Llavona, José G. 2005. Polynomial topologies on Banach spaces. Topology and its Applications, Vol. 153, Issue. 5-6, p. 854.


    Bombal, Fernando and Villanueva, Ignacio 2003. Polynomial sequential continuity on C(K,E) spaces. Journal of Mathematical Analysis and Applications, Vol. 282, Issue. 1, p. 341.


    Pérez-Garcı́a, David and Villanueva, Ignacio 2003. Multiple summing operators on Banach spaces. Journal of Mathematical Analysis and Applications, Vol. 285, Issue. 1, p. 86.


    Bombal, Fernando and Villanueva, Ignacio 1999. Regular multilinear operators on C(K) spaces. Bulletin of the Australian Mathematical Society, Vol. 60, Issue. 01, p. 11.


    Aron, Richard M. and Galindo, Pablo 1997. Weakly compact multilinear mappings. Proceedings of the Edinburgh Mathematical Society, Vol. 40, Issue. 01, p. 181.


    Gutiérrez, Joaquín M. and Llavona, José G. 1997. Polynomially continuous operators. Israel Journal of Mathematics, Vol. 102, Issue. 1, p. 179.


    ×
  • Bulletin of the Australian Mathematical Society, Volume 52, Issue 3
  • December 1995, pp. 475-486

Estimates by polynomials

  • R.M. Aron (a1), Y.S. Choi (a2) and J.G. Llavona (a3)
  • DOI: http://dx.doi.org/10.1017/S0004972700014957
  • Published online: 01 April 2009
Abstract

Consider the following possible properties which a Banach space X may have: (P): If (xi) and (yj) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P(xj) − (yj) → 0, then Q(xjyj) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynomial Q on X.

(RP): If (xj)and (yj) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P(xjyj) → 0, then Q(xj) − Q(yj) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynimial Q on X. We study properties (P) and (RP) and their relation with the Schur proqerty, Dunford-Pettis property, Λ, and others. Several applications of these properties are given.

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[1]R. Alencar , R.M. Aron and S. Dineen , ‘A reflexive space of holomorphic functions in infinitely many variables’, Proc. Amer. Math. Soc. 90 (1984), 407411.

[4]T. Carne , B. Cole and T. Gamelin , ‘A uniform algebra of analytic functions on a Banach space’, Trans. Amer. Math. Soc. 314 (1989), 639659.

[5]J.F. Castillo and C. Sanchez , ‘Weakly-p-compact, p-Banach-Saks and super-reflexive Banach spaces’, J. Math. Anal. Appl. 185 (1994), 256261.

[6]Y.S. Choi and S.G. Kim , ‘Polynomial properties of Banach spaces’, J. Math. Anal. Appl. 190 (1995), 203210.

[7]A.M. Davie and T.W. Gamelin , ‘A theorem on polynomial-star approximation’, Proc. Amer. Math. Soc. 106 (1989), 351358.

[8]J. Diestel , ‘A survey of results related to the Dunford-Pettis property’, in Contemp. Math. 2 (Amer. Math. Soc., Providence, R.I., 1980), pp. 1560.

[12]J.D. Farmer , ‘Polynomial reflexivity in Banach spaces’, Israel J. Math. 87 (1994), 257273.

[14]J.A. Jaramillo and A. Prieto , ‘Weak-polynomial convergence on a Banach space’, Proc. Amer. Math. Soc. 118 (1993), 463468.

[16]A. Grothendieck , ‘Sur les applications linéaires faiblement compactes d'espaces du type C(K)’, Canad. J. Math. 5 (1953), 129173.

[18]Y.I. Petunin and V.I. Savkin , ‘Convergence generated by analytic functions’, Ukranian. Math. J. 40 (1988), 676679.

[19]H.P. Rosenthal , ‘Some recent discoveries in the isomorphic theory of Banach spaces’, Bull. Amer. Math. Soc. 84 (1980), 803831.

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  • ISSN: 0004-9727
  • EISSN: 1755-1633
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