Banach spaces in which all multilinear forms are weakly sequentially continuous

Jesús Castillo; Ricardo García; Raquel Gonzalo

Studia Mathematica (1999)

  • Volume: 136, Issue: 2, page 121-145
  • ISSN: 0039-3223

Abstract

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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].

How to cite

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Castillo, Jesús, García, Ricardo, and Gonzalo, Raquel. "Banach spaces in which all multilinear forms are weakly sequentially continuous." Studia Mathematica 136.2 (1999): 121-145. <http://eudml.org/doc/216664>.

@article{Castillo1999,
abstract = {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].},
author = {Castillo, Jesús, García, Ricardo, Gonzalo, Raquel},
journal = {Studia Mathematica},
keywords = {multilinear forms; polynomials; weak continuity; bilinear form; Dunford-Pettis property},
language = {eng},
number = {2},
pages = {121-145},
title = {Banach spaces in which all multilinear forms are weakly sequentially continuous},
url = {http://eudml.org/doc/216664},
volume = {136},
year = {1999},
}

TY - JOUR
AU - Castillo, Jesús
AU - García, Ricardo
AU - Gonzalo, Raquel
TI - Banach spaces in which all multilinear forms are weakly sequentially continuous
JO - Studia Mathematica
PY - 1999
VL - 136
IS - 2
SP - 121
EP - 145
AB - 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].
LA - eng
KW - multilinear forms; polynomials; weak continuity; bilinear form; Dunford-Pettis property
UR - http://eudml.org/doc/216664
ER -

References

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