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Fréchet AL-spaces have the Dunford-Pettis property

Published online by Cambridge University Press:  17 April 2009

J.C. Díaz ,
A. Fernández  and
F. Naranjo
J.C. Díaz
Affiliation:
Dpto. Matemáticas, E.T.S.I.A.M., Universidad de Córdoba, 14004 - Córdoba, Spain e-mail: maldialj@uco.es
A. Fernández
Affiliation:
Dpto. Matemática Aplicada II, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 - Sevilla, Spain e-mail: anfercar@matinc.us.es
F. Naranjo
Affiliation:
Dpto. Matemática Aplicada II, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 – Sevilla, Spain
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Abstract

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A Fréchet lattice E is an AL-space if its topology can be defined by a family of lattice seminorms that are additive in the positive cone of E. Grothendieck proved that AL-Banach spaces have the Dunford-Pettis property. This result was recently extended by Fernández and Naranjo to AL-Fréchet spaces with a continuous norm and weak order unit. In this note we show how to remove both hypotheses.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 3 , December 1998 , pp. 383 - 386
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Aliprantis, C.D. and Burkinshaw, O., Positive operators, Pure and Applied Mathematics 119 (Academic Press, Orlando, Fl., 1985).Google Scholar
[2]Díaz, J.C., ‘Continuous norms on Fréchet lattices’, Arch. Math. (Basel) 52 (1989), 155–158.CrossRefGoogle Scholar
[3]Dodds, P.G., de Pagter, B. and Ricker, W.J., ‘Reflexivity and order properties of scalar-type spectral operators in locally convex spaces’, Trans. Amer. Math. Soc. 293 (1986), 355–380.CrossRefGoogle Scholar
[4]Dunford, N. and Pettis, P.J., ‘Linear operations on summable functions’, Trans. Amer. Math. Soc. 47 (1940), 323–392.CrossRefGoogle Scholar
[5]Fernández, A. and Naranjo, F., ‘Operators and the space of integrable scalar functions with respect to a Fréchet-valued measure’, J. Austral. Math. Soc. Ser. A 64 (1998), 1–18.Google Scholar
[6]Grosse-Erdman, K.G., ‘Lebesgue's theorem of differentiation in Fréchet lattices’, Proc. Amer. Math. Soc. 112 (1991), 371–379.Google Scholar
[7]Grothendieck, A., ‘Sur les aplications linéaires faiblement compactes d'espaces du type C(K)’, Canad. J. Math. 5 (1953), 129–173.CrossRefGoogle Scholar
[8]Khurana, S.S., ‘Dunford-Pettis property’, J. Math. Anal. Appl. 65 (1978), 361–364.CrossRefGoogle Scholar
[9]Wong, Y.C., ‘Characterizations of the topology of uniform convergence on order-intervals’, Hokkaido Math. J. 5 (1976), 164–200.CrossRefGoogle Scholar
[10]Wong, Y.C. and Ng, F.K., ‘Nuclear and AL-spaces’, Southeast Asian Bull. Math. J. 5 (1981), 45–58.Google Scholar