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Polynomial Schur and polynomial Dunford-Pettis properties. (English) Zbl 0822.46013
Lin, Bor-Luh (ed.) et al., Banach spaces. Proceedings of an international workshop on Banach space theory, held at the Universidad de Los Andes, Merida, Venezuela, January 6-17, 1992. Providence, RI: American Mathematical Society. Contemp. Math. 144, 95-105 (1993).
Summary: A Banach space is polynomially Schur if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is Schur. Herein is defined a reasonable generalization of the Dunford-Pettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach space has the \(P_ N\) Dunford- Pettis 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.
For the entire collection see [Zbl 0782.00080].

46B20 Geometry and structure of normed linear spaces
46G20 Infinite-dimensional holomorphy
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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