Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Ondrej F. K. Kalenda
Journal: Proc. Amer. Math. Soc. 130 (2002), 2139-2143
MSC (2000): Primary 54C60, 26E25, 54C10
DOI: https://doi.org/10.1090/S0002-9939-02-06625-X
Published electronically: February 27, 2002
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] M.Fabian, Gâteaux differentiability of convex functions and topology: weak Asplund spaces, Wiley-Interscience, New York, 1997. MR 98h:46009
  • [2] M.Fosgeram, PhD thesis, Univ. College London, 1992, pp. 52-62.
  • [3] R.Frankiewicz and K.Kunen, Solution of Kuratowski's problem on function having the Baire property I., Fund. Math. 128 (1987), 171-180. MR 89a:03090
  • [4] T.Jech, M.Magidor, W.Mitchell and K.Prikry, Precipitous ideals, J. Symb. Log. 45 (1980), 1-8. MR 81h:03097
  • [5] Y.Kakuda, On a condition for Cohen extensions which preserve precipitous ideals, J. Symb. Log. 46 (2) (1981), 296-300. MR 82i:03062
  • [6] O.Kalenda, Stegall compact spaces which are not fragmentable, Topol. Appl. 96 (2) (1999), 121-132. MR 2001i:45027
  • [7] 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.
  • [8] P.Kenderov and J.Orihuela, A generic factorization theorem, Mathematika 42 (1) (1995), 56-66. MR 96h:54014
  • [9] I.Kortezov, Fragmentability of the dual of a Banach space with smooth bump, Serdica Math. J. 24 (2) (1998), 187-198. MR 2000b:46028
  • [10] D.A.Martin and R.M.Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143-178. MR 42:5787
  • [11] I.Namioka and R.Pol, Mappings of Baire spaces into function spaces and Kadec renorming, Israel Jour. Math. 78 (1992), 1-20. MR 94f:46020
  • [12] N.K.Ribarska, The dual of a Gâteaux smooth Banach space is weak star fragmentable, Proc. Amer. Math. Soc. 114 (4) (1992), 1003-1008. MR 92g:46020
  • [13] R.M.Solovay, On the cardinality of $\Sigma ^{1}_{2}$ sets of reals, Foundation of Mathematics, Symposium papers commemorating the sixtieth birthday of Kurt Gödel, Springer-Verlag, 1969, pp. 58-73. MR 43:3115
  • [14] R.M.Solovay and S.Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. of Math. 94 (1971), 201-245. MR 45:3212
  • [15] C. Stegall, A class of topological spaces and differentiability, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen 10 (1983), 63-77. MR 85j:46026
  • [16] S.Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54C60, 26E25, 54C10

Retrieve articles in all journals with MSC (2000): 54C60, 26E25, 54C10


Additional Information

Ondrej F. K. Kalenda
Affiliation: Department of Mathematical Analysis, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: kalenda@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/S0002-9939-02-06625-X
Keywords: Weak Asplund space, fragmentable space, Stegall's class of spaces
Received by editor(s): April 5, 2000
Published electronically: February 27, 2002
Additional Notes: Partially supported by research grants GAUK 277/2001, GAČR 201/00/1466 and MSM 113200007.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society