The asymptotic-norming and the Radon-Nikodým properties are equivalent in separable Banach spaces

Ghoussoub, N.; Maurey, B.

doi:10.1090/S0002-9939-1985-0792280-X

The asymptotic-norming and the Radon-Nikodým properties are equivalent in separable Banach spaces
HTML articles powered by AMS MathViewer

by N. Ghoussoub and B. Maurey PDF

Proc. Amer. Math. Soc. 94 (1985), 665-671 Request permission

Abstract:

We show that the asymptotic-norming and the Radon-Nikodym properties are equivalent, settling a problem of James and Ho [9]. In the process, we give a positive solution to two questions of Edgar and Wheeler [6] concerning Cech-complete Banach spaces. We also show that a separable Banach space with the Radon-Nikodym property semi-embeds in a separable dual whenever it has a norming space not containing an isomorphic copy of ${l_1}$. This gives a partial answer to a problem of Bourgain and Rosenthal [3].

References

J. Bourgain, Sets with the Radon-Nikodým property in conjugate Banach space, Studia Math. 66 (1979/80), no. 3, 291–297. MR 579734, DOI 10.4064/sm-66-3-291-297
J. Bourgain and F. Delbaen, A class of special ${\cal L}_{\infty }$ spaces, Acta Math. 145 (1980), no. 3-4, 155–176. MR 590288, DOI 10.1007/BF02414188
J. Bourgain and H. P. Rosenthal, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52 (1983), no. 2, 149–188. MR 707202, DOI 10.1016/0022-1236(83)90080-0
William J. Davis, Nassif Ghoussoub, and Joram Lindenstrauss, A lattice renorming theorem and applications to vector-valued processes, Trans. Amer. Math. Soc. 263 (1981), no. 2, 531–540. MR 594424, DOI 10.1090/S0002-9947-1981-0594424-2
William J. Davis and William B. Johnson, A renorming of nonreflexive Banach spaces, Proc. Amer. Math. Soc. 37 (1973), 486–488. MR 310595, DOI 10.1090/S0002-9939-1973-0310595-8
G. A. Edgar and R. F. Wheeler, Topological properties of Banach spaces, Pacific J. Math. 115 (1984), no. 2, 317–350. MR 765190, DOI 10.2140/pjm.1984.115.317
N. Ghoussoub and B. Maurey, $G_\delta$-embeddings in Hilbert space, J. Funct. Anal. 61 (1985), no. 1, 72–97. MR 779739, DOI 10.1016/0022-1236(85)90039-4

-embeddings in Hilbert space and optimization on

-sets

Robert C. James and Aggie Ho, The asymptotic-norming and Radon-Nikodým properties for Banach spaces, Ark. Mat. 19 (1981), no. 1, 53–70. MR 625537, DOI 10.1007/BF02384469
W. B. Johnson and J. Lindenstrauss, Examples of ${\cal L}_{1}$ spaces, Ark. Mat. 18 (1980), no. 1, 101–106. MR 608329, DOI 10.1007/BF02384683
J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Mathematical Library, Vol. 10, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed. MR 0402915
E. Odell and H. P. Rosenthal, A double-dual characterization of separable Banach spaces containing $l^{1}$, Israel J. Math. 20 (1975), no. 3-4, 375–384. MR 377482, DOI 10.1007/BF02760341
Charles Stegall, The Radon-Nikodým property in conjugate Banach spaces. II, Trans. Amer. Math. Soc. 264 (1981), no. 2, 507–519. MR 603779, DOI 10.1090/S0002-9947-1981-0603779-1