The Radon-Nikodým property in conjugate Banach spaces. II

Stegall, Charles

doi:10.1090/S0002-9947-1981-0603779-1

The Radon-Nikodým property in conjugate Banach spaces. II
HTML articles powered by AMS MathViewer

by Charles Stegall PDF

Trans. Amer. Math. Soc. 264 (1981), 507-519 Request permission

Abstract:

In the first part of this article the following result was proved. Theorem. The dual of a Banach space $X$ has the Radon-Nikodym property if and only if for every closed, linear separable subspace $Y$ of $X$, ${Y^ \ast }$ is separable. We find other, more detailed descriptions of Banach spaces whose duals have the Radon-Nikodym property.

References

Edgar Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31–47. MR 231199, DOI 10.1007/BF02391908
J. Bourgain, Strongly exposed points in weakly compact convex sets in Banach spaces, Proc. Amer. Math. Soc. 58 (1976), 197–200. MR 415272, DOI 10.1090/S0002-9939-1976-0415272-3
W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311–327. MR 0355536, DOI 10.1016/0022-1236(74)90044-5
Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
Ivar Ekeland and Gérard Lebourg, Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), no. 2, 193–216 (1977). MR 431253, DOI 10.1090/S0002-9947-1976-0431253-2
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not weak* sequentially compact, Israel J. Math. 28 (1977), no. 4, 325–330. MR 482086, DOI 10.1007/BF02760638
Richard Haydon, An extreme point criterion for separability of a dual Banach space, and a new proof of a theorem of Corson, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 107, 379–385. MR 493264, DOI 10.1093/qmath/27.3.379
R. E. Huff and P. D. Morris, Dual spaces with the Krein-Milman property have the Radon-Nikodým property, Proc. Amer. Math. Soc. 49 (1975), 104–108. MR 361775, DOI 10.1090/S0002-9939-1975-0361775-9
W. B. Johnson and J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17 (1974), 219–230. MR 417760, DOI 10.1007/BF02882239

Absolutely summing operators in

-spaces and their applications

29

J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $\ell _{1}$ and whose duals are non-separable, Studia Math. 54 (1975), no. 1, 81–105. MR 390720, DOI 10.4064/sm-54-1-81-105

Über konvexe Mengen in linearen normierten Räumen

4

Bases, lacunary sequences and complemented subspaces in the spaces

21

R. R. Phelps, Support cones in Banach spaces and their applications, Advances in Math. 13 (1974), 1–19. MR 338741, DOI 10.1016/0001-8708(74)90062-0
R. R. Phelps, Dentability and extreme points in Banach spaces, J. Functional Analysis 17 (1974), 78–90. MR 0352941, DOI 10.1016/0022-1236(74)90005-6
I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), no. 4, 735–750. MR 390721
Haskell P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compositio Math. 28 (1974), 83–111. MR 417762
V. Šmulian, On some geometrical properties of the unit sphere in the space of the type (B), Rec. Math. N. S. [Mat. Sbornik] 6(48) (1939), 77–94 (Russian, with English summary). MR 0001458
V. L. Šmulian, Sur la structure de la sphère unitaire dans l’espace de Banach, Rec. Math. [Mat. Sbornik] N.S. 9 (51) (1941), 545–561 (French). MR 0005775
V. Šmulian, Sur la dérivabilité de la norme dans l’espace de Banach, C. R. (Doklady) Acad. Sci. URSS (N. S.) 27 (1940), 643–648 (French). MR 0002704

On some geometrical properties of the sphere in a space of the type

24

C. Stegall, Banach spaces whose duals contain $l_{1}(\Gamma )$ with applications to the study of dual $L_{1}(\mu )$ spaces, Trans. Amer. Math. Soc. 176 (1973), 463–477. MR 315404, DOI 10.1090/S0002-9947-1973-0315404-3

A theorem of Haydon and its applications

Charles Stegall, The duality between Asplund spaces and spaces with the Radon-Nikodým property, Israel J. Math. 29 (1978), no. 4, 408–412. MR 493268, DOI 10.1007/BF02761178
Charles Stegall, The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213–223. MR 374381, DOI 10.1090/S0002-9947-1975-0374381-1
E. B. Leach and J. H. M. Whitfield, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 120–126. MR 293394, DOI 10.1090/S0002-9939-1972-0293394-4