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Radon-Nikodým compact spaces and fragmentability

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

I. Namioka
I. Namioka
Affiliation:
Department of Mathematics, GN-50, University of Washington, Seattle, Washington 98195, U.S.A.
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Extract

While investigating Asplund spaces in [15], R. R. Phelps and the author noticed that weak* compact subsets of the duals of Asplund spaces (or equivalently, as it turned out, weak* compact subsets of dual Banach spaces with the Radon-Nikodým property) possessed many properties in common with weakly compact subsets of Banach spaces. The topological study of the spaces homeomorphic to the latter, the so-called Eberlein compact spaces, or EC spaces for short, had flourished and had already yielded a rich collection of results. Therefore it was natural to hope that a similar study of the former might also lead to interesting discoveries. In a series of letters with S. Fitzpatrick exchanged during the summer and the fall of 1981, we started to collect properties of compact spaces that are homeomorphic to weak* compact subsets of the duals of Asplund spaces, which we tentatively called “Asplund compact spaces“. However, as far as we are aware, Reynov's paper [16] is the first study in print of the topological properties of “Asplund compact spaces” or “compacta of RN type” as Reynov termed them.

MSC classification

Secondary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties
Type
Research Article
Information
Mathematika , Volume 34 , Issue 2 , December 1987 , pp. 258 - 281
Copyright
Copyright © University College London 1987

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References

1.Bennett, H. R.Lutzer, D. J. and Wouwe, J. M. van. Linearly ordered Eberlein compact spaces, Topology Appl., 12 (1981), 11–18.CrossRefGoogle Scholar
2.Benyamini, Y.Rudin, M. E. and Wage, M.. Continuous images of weakly compact subsets of Banach spaces. Pacific J. Math., 70 (1977), 309–324.CrossRefGoogle Scholar
3.Bourgin, R.. Geometric aspects of convex sets with the Radon-Nikodym property, Lecture Notes in Mathematics 993 (Springer, Berlin, 1983).CrossRefGoogle Scholar
4.Davis, W. J.Figiel, T.Johnson, W. B. and Pelczynski, A.. Factoring weakly compact operators. J. Func. Anal., 17 (1974), 311–327.CrossRefGoogle Scholar
5.Diestel, J. and Uhl, J. J. Jr. Vector measures, Math. Surveys 15 (Amer. Math. Soc, Providence, 1977).CrossRefGoogle Scholar
6.Fitzpatrick, S.. Weak* compact convex sets with the RNP. Rainwater Seminar Notes (University of Washington, 1977).Google Scholar
7.Ghoussoub, N. and Maurey, B.. H δ-embeddings in Hilbert space and optimization on Gδ-sets. Memoirs A.M.S., 349 (1986).Google Scholar
8.Jayne, J. E.Namioka, I. and Rogers, C. A.. Properties like the Radon-Nikodym property. To appear.Google Scholar
9.Jayne, J. E. and Rogers, C. A.. Borel selectors for upper semicontinuous set-valued maps. Acta Math., 155 (1985), 41–79.CrossRefGoogle Scholar
10.Kelley, J. L.. General topology (Springer, New York, 1975).Google Scholar
11.Kelley, J. L. and Namioka, I., et al. Linear topological spaces (Springer, New York, 1976).Google Scholar
12.Larman, D. G. and Phelps, R. R.. Gateaux differentiability of convex functions on Banach spaces. J. London Math. Soc. (2), 20 (1979), 115–127.CrossRefGoogle Scholar
13.Michael, E. and Namioka, I.. Barely continuous functions. Bull. Acad. Polon. Sci. Sir. Sci. Math. Astronom. Phys., 24 (1976), 889–892.Google Scholar
14.Namioka, I.. Separate continuity and joint continuity. Pacific J. Math., 51 (1974), 515–531.CrossRefGoogle Scholar
15.Namioka, I. and Phelps, R. R.. Banach spaces which are Asplund spaces. Duke Math. J., 42 (1975), 735–750.CrossRefGoogle Scholar
16.Reynov, O. I.. On a class of Hausdorff compacts and GSG Banach spaces. Studia Math., 71 (1981), 113–126.CrossRefGoogle Scholar
17.Rudin, W.. Continuous functions on compact spaces without perfect subsets. Proc. Amer. Math. Soc., 8 (1957), 39–42.CrossRefGoogle Scholar
18.Ruess, W. M. and Stegall, C. P.. Remarks concerning the paper “on a class of Hausdorfi compacts and GSG Banach spaces”. Studia Math., 82 (1985), 97–98.CrossRefGoogle Scholar
19.Steen, L. A. and Seebach, J. A. Jr. Counter examples in topology, Second edition (Springer, New York, 1978).CrossRefGoogle Scholar
20.Stegall, C.. The duality between Asplund spaces and spaces with the Radon-Nikodym property. IsraelJ. Math., 29 (1978), 408–412.CrossRefGoogle Scholar
21.Stegall, C.. The Radon-Nikodym property in conjugate Banach spaces II. Trans. A.M.S., 264 (1981), 507–519.Google Scholar
22.Stegall, C.. A class of topological spaces and differentiation of functions on Banach spaces. Proc. Conf. on Vector Measures and Integral Representations of Operators, Vorlesungen aus dem Fachbereich Math. Heft 10, Ed. Ruess, W. Univ. Essen (1983).Google Scholar
23.Stegall, C.. More gateaux differentiability spaces. Proc. Conf. Banach Spaces, Univ. Missouri 1984, Ed. Kalton, N. and Saab, E., Lecture Notes in Math. 1166 (Springer, Berlin, 1985), 154–168.Google Scholar
24.Talagrand, M.. Deux examples de fonctions convexes. C.R. Acad. Sci. Paris Ser. A, 288 (1979), 461–464.Google Scholar