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Estimates by polynomials

Published online by Cambridge University Press:  17 April 2009

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

Research Article
Copyright © Australian Mathematical Society 1995


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