Publication: Estimates by polynomials
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Publication Date
1995-12
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Cambridge University Press
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Abstract
Consider the following possible properties which a Banach space X may have: (P): If (x(j)) and (y(j)) are are bounded sequences in X such that for all n greater than or equal to 1 and for every continuous n-homogeneous polynomial P on X, P(x(j)) - P(y(j)) --> 0, then, Q(x(j) - y(j)) --> 0 for all m greater than or equal to 1 and for every continuous us m-homogeneous polynomial Q on X.
(RP): If (x(j)) and (y(j)) are bounded sequences in X such that for all n greater than or equal to 1 and for every continuous n-homogeneous polynomial P on X, P(x(j) - y(j)) --> 0, then Q(x(j)) - Q(y(j)) --> 0 for all m greater than or equal to 1 and for every continuous m-homogeneous polynomial Q on X. We study properties (P) and (RP) and their relation with the Schur property, Dunford-Pettis property, Lambda, and others. Several. applications of these properties are given.
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