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Estimates by polynomials

Published online by Cambridge University Press:  17 April 2009

R.M. Aron ,
Y.S. Choi  and
J.G. Llavona
R.M. Aron
Affiliation:
Department of MathematicsKent State UniversityKent Oh 44242United States of America
Y.S. Choi
Affiliation:
Department of MathematicsPohang University of Science and TechnologyPohangKorea 790
J.G. Llavona
Affiliation:
Departamento de Análisis MatemáticoUniversidad Complutense de Madrid28040 MadridSpain
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Abstract

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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(xj − yj) → 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(xj − yj) → 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 3 , December 1995 , pp. 475 - 486
Copyright
Copyright © Australian Mathematical Society 1995

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