Mathematical Proceedings of the Cambridge Philosophical Society

The Dunford–Pettis property on tensor products

a1 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain; e-mail:
a2 Departamento de Matemática Aplicada, ETS de Ingenieros Industriales Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain; e-mail:


We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford–Pettis property (DPP). As a consequence, we obtain that (c0 &[otimes B: multiply sign in circle]circ;π c0)** fails the DPP. Since (c0 &[otimes B: multiply sign in circle]circ;π c0)* does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if E and F are [script L]1-spaces, then E &[otimes B: multiply sign in circle]circ;ε has the DPP if and only if both E and F have the Schur property. Other results and examples are given.

(Received November 8 1999)
(Revised March 21 2000)


1 M.G. was supported in part by DGICYT Grant PB 97{0349 (Spain).

2 J.M.G. was supported in part by DGICYT Grant PB 96{0607 (Spain).