Journal of the Mathematical Society of Japan

The separable quotient problem and the strongly normal sequences

Wiesław ŚLIWA

Full-text: Open access

Abstract

We study the notion of a strongly normal sequence in the dual E* of a Banach space E. In particular, we prove that the following three conditions are equivalent:

(1) E* has a strongly normal sequence,

(2) (E*, σ(E*,E)) has a Schauder basic sequence,

(3) E has an infinite-dimensional separable quotient.

Article information

Source
J. Math. Soc. Japan, Volume 64, Number 2 (2012), 387-397.

Dates
First available in Project Euclid: 26 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1335444397

Digital Object Identifier
doi:10.2969/jmsj/06420387

Mathematical Reviews number (MathSciNet)
MR2916073

Zentralblatt MATH identifier
1253.46025

Subjects
Primary: 46B26: Nonseparable Banach spaces 46B10: Duality and reflexivity [See also 46A25]

Keywords
Banach space separable quotient problem normal sequence Josefson-Nissenzweig theorem

Citation

ŚLIWA, Wiesław. The separable quotient problem and the strongly normal sequences. J. Math. Soc. Japan 64 (2012), no. 2, 387--397. doi:10.2969/jmsj/06420387. https://projecteuclid.org/euclid.jmsj/1335444397


Export citation

References

  • G. Bennett and N. Kalton, Inclusion theorems for $K$-spaces, Canad. J. Math., 25 (1973), 511–524.
  • R. E. Edwards, Functional analysis, Theory and applications, Dover Publications, Inc., New York, 1995.
  • K. John and V. Zizler, Projections in dual weakly compactly generated Banach spaces, Studia Math., 49 (1973), 41–50.
  • W. B. Johnson and H. P. Rosenthal, On $\omega^*$-basic sequences and their applications to the study of Banach spaces, Studia Math., 43 (1972), 77–95.
  • B. Josefson, Weak sequential convergence in the dual of a Banach space does not imply norm convergence, Ark. Mat., 13 (1975), 78–89.
  • J. Kąkol and W. Śliwa, Remarks concerning the separable quotient problem, Note Mat., 13 (1993), 277–282.
  • G. Köthe, Topological vector spaces I, Springer-Verlag, Berlin, 1969.
  • H. E. Lacey, Separable quotients of Banach spaces, An. Acad. Brasil. Ciênc., 44 (1972), 185–189.
  • R. E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, New York, 1998.
  • J. Mujica, Separable quotients of Banach spaces, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 299–330.
  • P. P. Narayanaswami, The separable quotient problem for barrelled spaces, In: Functional analysis and related topics, Springer-Verlag, Berlin, 1993, pp.,289–308.
  • A. Nissenzweig, On $\omega^*$ sequential convergence, Israel J. Math., 22 (1975), 266–272.
  • H. P. Rosenthal, On quasicomplemented subspaces of Banach spaces with an appendix on compactness of operators from $L^{p}(\mu)$ to $L^{r}(\nu)$, J. Funct. Anal., 4 (1969), 176–214.
  • W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
  • S. A. Saxon and P. P. Narayanaswami, Metrizable (LF)-spaces, (db)-spaces and the separable quotient problem, Bull. Austral. Math. Soc., 23 (1981), 65–80.
  • S. A. Saxon and P. P. Narayanaswami, Metrizable [normable] (LF)-spaces and two classical problems in Fréchet [Banach] spaces, Studia Math., 93 (1989), 1–16.
  • S. A. Saxon and A. Wilansky, The equivalence of some Banach space problems, Colloq. Math., 37 (1977), 217–226.
  • W. Śliwa, (LF)-spaces and the separable quotient problem, Thesis, in Polish, unpublished, Poznań, 1996.
  • W. Śliwa, The separable quotient problem for symmetric function spaces, Bull. Polish Acad. Sci. Math., 48 (2000), 13–27.
  • W. Śliwa and M. Wójtowicz, Separable quotients of locally convex spaces, Bull. Polish Acad. Sci. Math., 43 (1995), 175–185.
  • M. Wójtowicz, Generalizations of the $c_0-l_1-l_{\infty}$ Theorem of Bessaga and Pełczyński, Bull. Polish Acad. Sci. Math., 50 (2002), 373–382.
  • M. Wójtowicz, Reflexivity and the Separable Quotient Problem for a Class of Banach Spaces, Bull. Polish Acad. Sci. Math., 50 (2002), 383–394.