Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-29T08:07:47.122Z Has data issue: false hasContentIssue false
  • English
  • Français

On the problem of non-smoothness of non-reflexive second conjugate spaces

Published online by Cambridge University Press:  17 April 2009

Ivan Singer
Ivan Singer
Affiliation:
Département d'Informatique, Université de Montréal, Canada; Institute of Mathematics, Calea Grivitei 21, Bucuresti, Romania.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that if E is a Banach space which has a subspace G such that the conjugate space G* contains a proper norm closed linear subspace V of characteristic 1, then E** is not smooth and there exist in πE(E) points of non-smoothness for E**, where πE: E → E** is the canonical embedding. We show that the spaces E having such a subspace G constitute a large proper subfamily of the family of all non-reflexive Banach spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 3 , June 1975 , pp. 407 - 416
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Bishop, Errett and Phelps, R.R., “The support functionals of a convex set”, Convexity, 27–35 (Proc. Sympos. Pure Maths 7. Amer. Math. Soc., Providence, Rhode Island, 1963).CrossRefGoogle Scholar
[2]Day, Mahlon M., “Strict convexity and smoothness of normed spaces”, Trans. Amer. Math. Soc. 78 (1955), 516–528.CrossRefGoogle Scholar
[3]Day, Mahlon M., Normed linear spaces, 3rd ed. (Ergebnisse der Mathematik und ihrer Grenzgebiete, 21. Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
[4]Dixmier, J., “Sur un théorème de Banach”, Duke Math. J. 15 (1948), 1057–1071.CrossRefGoogle Scholar
[5]van Dulst, Dick and Singer, Ivan, “On Kadec-Klee norms on Banach spaces”, Studia Math. (to appear).Google Scholar
[6]Giles, J.R., “A non-reflexive Banach space has non-smooth third conjugate space”, Canad. Math. Bull. 17 (1974), 117–119.CrossRefGoogle Scholar
[7]James, Robert C., “A non-reflexive Banach space isometric with its second conjugate space”, Proc. Nat. Acad. Sci. USA 37 (1951), 174–177.CrossRefGoogle ScholarPubMed
[8]Кадець, М.И. [M.ĭ. Kadec'], “Про зв'язок між слабою та сильною збіжністю” [On the connection between weak and strong convergence], Dopovidi Akad. Nauk Ukraïn. RSR 1959, 949–952.Google ScholarPubMed
[9]Klee, V., “Mappings into normed linear spaces”, Fund. Math. 49 (1960–1961), 25–34.Google Scholar
[10]Pełczyński, A., “A note on the paper of I. Singer ‘Basic sequences and reflexivity of Banach spaces’”, Studia Math. 21 (1961–1962), 371–374.Google Scholar
[11]Pełczyński, A., “A proof of Eberlein-Šmulian theorem by an application of basic sequences”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 12 (1964), 543–548.Google Scholar
[12]Rainwater, John, “A non-reflexive Banach space has non-smooth third conjugate space”, preprint.Google Scholar
[13]Singer, Ivan, Bases in Banach spaces I (Die Grundlehren der mathematischen Wissenschaften, 154. Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
[14]Tacon, D.G., “The conjugate of a smooth Banach space”, Bull. Austral. Math. Soc. 2 (1970), 415–425.Google Scholar