On the Krein-Šmulian theorem for weaker topologies

Illinois Journal of Mathematics

Volumn 47, Issue 4, 2003, Pages 957-976

  Cascales, B ^a   Shvydkoy, R ^b,c  

^a UNIVERSITY OF MURCIA   (Spain)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^c UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1642375633     PISSN: 00192082     EISSN: None     Source Type: Journal    
DOI: 10.1215/ijm/1258138086     Document Type: Article

References (31)

1. q(μ, X′)), Math. Z. 182 (1983), 409-423. MR 84m:46039
2. G. Birkhoff, Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38 (1935), 357-378. MR 1501815
3. J. Bourgain and M. Talagrand, Compacité extrémale, Proc. Amer. Math. Soc. 80 (1980), 68-70. MR 81h:46011
4. B. Cascales and G. Godefroy, Angelicity and the boundary problem, Mathematika 45 (1998), 105-112. MR 99f:46019
5. B. Cascales, G. Manjabacas, and G. Vera, A Krein-Šmulian type result in Banach spaces, Quart. J. Math. Oxford Ser. (2) 48 (1997), 161-167. MR 99c:46009
6. B. Cascales, I. Namioka, and G. Vera, The Lindelöf property and fragmentability, Proc. Amer. Math. Soc. 128 (2000), 3301-3309. MR 2001b:54024
7. B. Cascales and J. Rodriguez, Birkhoff integral and the property of Bourgain, Preprint, 2003.
8. B. Cascales and G. Vera, Topologies weaker than the weak topology of a Banach space, J. Math. Anal. Appl. 182 (1994), 41-68. MR 95c:46017
9. _, Norming sets and compactness, Rocky Mountain J. Math. 25 (1995), 919-925. MR 96i:46011
10. G. Choquet, Lectures on analysis. Vol. II: Representation theory, Edited by J. Marsden, T. Lance and S. Gelbart, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 40#3253
11. J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 85i:46020
12. J. Diestel and J. J. Uhl, Jr., Vector measures, With a foreword by B. J. Pettis, Mathematical Surveys, no. 15, American Mathematical Society, Providence, R.I., 1977. MR 56#12216
13. K. Floret, Weakly compact sets, Lectures held at S.U.N.Y. Buffalo, Spring 1978, Springer-Verlag, Berlin, 1980. MR 82b:46001
14. G. Godefroy, Boundaries of a convex set and interpolation sets, Math. Ann. 277 (1987), 173-184. MR 88f:46037
15. A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168-186. MR 13,857e
16. P. Habala, P. Hájek, and V. Zizler, Introduction to Banach spaces, Matfyz Press, Prague, 1996.
17. H. Jarchow, Locally convex spaces, Mathematische Leitfäden, B. G. Teubner, Stuttgart, 1981. MR 83h:46008
18. J. E. Jayne and C. A. Rogers, Borel selectors for upper semicontinuous set-valued maps, Acta Math. 155 (1985), 41-79. MR 87a:28011
19. V. Kadets, B. Shumyatskiy, R. Shvidkoy, L. Tseytlin, and K. Zheltukhin, Some remarks on vector-valued integration, Mat. Fiz. Anal. Geom. 9 (2002), 48-65. MR 2008g:28026
20. V. M. Kadets and L. M. Tseytlin, On "integration" of non-integrable vector-valued functions, Mat. Fiz. Anal. Geom. 7 (2000), 49-65. MR 2001e:28017
21. J. L. Kelley, General topology, Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, vol. 27, Springer-Verlag, New York, 1975. MR 51#6681
22. J. L. Kelley and I. Namioka, Linear topological spaces, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR 29#3851
23. M. Krein and V. Šmulian, On regulary convex sets in the space conjugate to a Banach space, Ann. of Math. (2) 41 (1940), 556-583. MR 1,335e
24. I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515-531. MR 51#6693
25. _, Radon-Nikodým compact spaces and fragmentability, Mathematika 34 (1987), 258-281. MR 89i:46021
26. R. R. Phelps, Lectures on Choquet's theorem, second ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001. MR 2002k:46001
27. A. P. Robertson and W. Robertson, Topological vector spaces, second ed., Cambridge Tracts in Mathematics and Mathematical Physics, no. 53, Cambridge University Press, London, 1973. MR 50#2854
28. 1, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413. MR 50#10773
29. R. Shvydkoy, Operators and integrals in Banach spaces, Ph.D. thesis, University of Missouri-Columbia, 2001.
30. S. Simons, A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703-708. MR 47#755
31. _, An eigenvector of Fatou's lemma for continuous functions, Math. Intelligencer 17 (1995), 67-70. MR 96e:26003