# 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

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topCastillo, 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 -

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