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The Radon-Nikodým property in conjugate Banach spaces. II


Author: Charles Stegall
Journal: Trans. Amer. Math. Soc. 264 (1981), 507-519
MSC: Primary 46B22; Secondary 46G10
DOI: https://doi.org/10.1090/S0002-9947-1981-0603779-1
MathSciNet review: 603779
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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.


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  • [1] E. Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31-47. MR 0231199 (37:6754)
  • [2] J. Bourgain, Strongly exposed points in weakly compact convex sets in Banach sets, Proc. Amer. Math. Soc. 58 (1976), 197-200. MR 0415272 (54:3363)
  • [3] W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczynski, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327. MR 0355536 (50:8010)
  • [4] J. Diestel, Geometry of Banach spaces-selected topics, Lecture Notes in Math., vol. 485, Springer-Verlag, Berlin and New York, 1975. MR 0461094 (57:1079)
  • [5] I. Ekeland and G. Lebourg, Generic Fréchet differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), 193-216. MR 0431253 (55:4254)
  • [6] A. Grothendieck, Produits tensoriels et espaces nucleaires, Mem. Amer. Math. Soc. No. 16 (1955). MR 0075539 (17:763c)
  • [7] J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not weak* sequentially compact, Israel J. Math. 28 (1977), 325-330. MR 0482086 (58:2173)
  • [8] R. Haydon, An extreme point criterion for separability of a dual space, and a new proof of a theorem of Corson, Quart. J. Math. Oxford Ser. (2) 11 (1976), 379-385. MR 0493264 (58:12293)
  • [9] R. E. Huff and P. D. Morris, Dual spaces with the Krein-Milman property have the Radon-Nikodym property, Proc. Amer. Math. Soc. 49 (1975), 104-108. MR 0361775 (50:14220)
  • [10] W. B. Johnson and J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17 (1974), 219-230. MR 0417760 (54:5808)
  • [11] J. Lindenstrauss and A. Pełczynski, Absolutely summing operators in $ {\mathfrak{L}_p}$-spaces and their applications, Studia Math. 29 (1969), 275-326.
  • [12] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $ {l_1}$ and whose duals are nonseparable, Studia Math. 54 (1975), 81-105. MR 0390720 (52:11543)
  • [13] S. Mazur, Über konvexe Mengen in linearen normierten Räumen, Studia Math. 4 (1933), 70-84.
  • [14] A. Pełczynski and M. I. Kadec, Bases, lacunary sequences and complemented subspaces in the spaces $ {L_p}$, Studia Math. 21 (1962), 161-176.
  • [15] R. R. Phelps, Support cones in Banach spaces and their applications, Advances in Math. 13 (1974), 1-19. MR 0338741 (49:3505)
  • [16] -, Dentability and extreme points, J. Funct. Anal. 17 (1974), 78-90. MR 0352941 (50:5427)
  • [17] R. R. Phelps and I. Namioka, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), 735-750. MR 0390721 (52:11544)
  • [18] H. P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compositio Math. 28 (1974), 83-111. MR 0417762 (54:5810)
  • [19] V. Smulian, On some geometrical properties of the unit sphere in the space of type $ (B)$, Math. USSR-Sb. 6 (1939), 77-94. MR 0001458 (1:242c)
  • [20] -, Sur la structure de la sphere unitaire dans l'espace de Banach, Math. USSR-Sb. 9 (1941), 545-561. MR 0005775 (3:205d)
  • [21] -, Sur la dérivabilité de la norme dans l'espace de Banach, C. R. (Doklady) Acad. Sci. URSS (N.S.) 27 (1940), 643-648. MR 0002704 (2:102f)
  • [22] -, On some geometrical properties of the sphere in a space of the type $ (B)$, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 648-653.
  • [23] C. Stegall, Banach spaces whose duals contain $ {l_1}(\Gamma )$ with applications to the study of conjugate $ {L_1}(\mu )$-spaces, Trans. Amer. Math. Soc. 176 (1973), 463-477. MR 0315404 (47:3953)
  • [24] -, A theorem of Haydon and its applications, Séminaire Maurey-Schwartz, Paris, 1975-76.
  • [25] -, The duality between Asplund spaces and spaces with the Radon-Nikodym property, Israel J. Math. 29 (1978), 408-412. MR 0493268 (58:12297)
  • [26] -, The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213-223. MR 0374381 (51:10581)
  • [27] F. B. Leach and J. H. M. Whitfield, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 120-126. MR 0293394 (45:2471)

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DOI: https://doi.org/10.1090/S0002-9947-1981-0603779-1
Article copyright: © Copyright 1981 American Mathematical Society

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