A weak Asplund space whose dual is not in Stegall’s class

Kalenda, Ondřej

doi:10.1090/S0002-9939-02-06625-X

A weak Asplund space whose dual is not in Stegall’s class
HTML articles powered by AMS MathViewer

by Ondřej F. K. Kalenda PDF

Proc. Amer. Math. Soc. 130 (2002), 2139-2143 Request permission

Abstract:

We show that, under some additional set-theoretical assumptions which are equiconsistent with the existence of a measurable cardinal, there is a weak Asplund space whose dual, equipped with the weak* topology, is not in Stegall’s class. This completes a result by Kenderov, Moors and Sciffer.

References

Marián J. Fabian, Gâteaux differentiability of convex functions and topology, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. Weak Asplund spaces; A Wiley-Interscience Publication. MR 1461271
M.Fosgeram, PhD thesis, Univ. College London, 1992, pp. 52-62.
Ryszard Frankiewicz and Kenneth Kunen, Solution of Kuratowski’s problem on function having the Baire property. I, Fund. Math. 128 (1987), no. 3, 171–180. MR 922569, DOI 10.4064/fm-128-3-171-180
T. Jech, M. Magidor, W. Mitchell, and K. Prikry, Precipitous ideals, J. Symbolic Logic 45 (1980), no. 1, 1–8. MR 560220, DOI 10.2307/2273349
Yuzuru Kakuda, On a condition for Cohen extensions which preserve precipitous ideals, J. Symbolic Logic 46 (1981), no. 2, 296–300. MR 613283, DOI 10.2307/2273622
S. Losinsky, Sur le procédé d’interpolation de Fejér, C. R. (Doklady) Acad. Sci. URSS (N.S.) 24 (1939), 318–321 (French). MR 0002001
P.Kenderov, W.Moors and S.Sciffer, A weak Asplund space whose dual is not weak* fragmentable, Proc. Amer. Math. Soc. 129 (2001), 3741–3747.
P. S. Kenderov and J. Orihuela, A generic factorization theorem, Mathematika 42 (1995), no. 1, 56–66. MR 1346672, DOI 10.1112/S0025579300011359
I. Kortezov, Fragmentability of the dual of a Banach space with smooth bump, Serdica Math. J. 24 (1998), no. 2, 187–198. MR 1680059
D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), no. 2, 143–178. MR 270904, DOI 10.1016/0003-4843(70)90009-4
I. Namioka and R. Pol, Mappings of Baire spaces into function spaces and Kadec renorming, Israel J. Math. 78 (1992), no. 1, 1–20. MR 1194955, DOI 10.1007/BF02801567
N. K. Ribarska, The dual of a Gâteaux smooth Banach space is weak star fragmentable, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1003–1008. MR 1101992, DOI 10.1090/S0002-9939-1992-1101992-3
Robert M. Solovay, On the cardinality of $\sum _{2}^{1}$ sets of reals, Foundations of Mathematics (Symposium Commemorating Kurt Gödel, Columbus, Ohio, 1966) Springer, New York, 1969, pp. 58–73. MR 0277382
R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin’s problem, Ann. of Math. (2) 94 (1971), 201–245. MR 294139, DOI 10.2307/1970860
Charles Stegall, A class of topological spaces and differentiation of functions on Banach spaces, Proceedings of the conferences on vector measures and integral representations of operators, and on functional analysis/Banach space geometry (Essen, 1982) Vorlesungen Fachbereich Math. Univ. Essen, vol. 10, Univ. Essen, Essen, 1983, pp. 63–77. MR 730947
S.Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140–150.