Polynomial Schur and Polynomial DunfordPettis Properties
Abstract
A Banach space is {\it polynomially Schur} if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the DunfordPettis property if and only if it is Schur. Herein is defined a reasonable generalization of the DunfordPettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach space will has the $P_N$ DunfordPettis property if and only if every weakly compact $N$homogeneous polynomial (in the sense of Ryan) on the space is completely continuous. A certain geometric condition, involving estimates on spreading models and implied by nontrivial type, is shown to be sufficient to imply that a space is polynomially Schur.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 1992
 arXiv:
 arXiv:math/9211210
 Bibcode:
 1992math.....11210F
 Keywords:

 Mathematics  Functional Analysis;
 46B