Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T17:48:58.384Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometry in quotient reflexive spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

A. C. Yorke
A. C. Yorke
Affiliation:
Department of MathematicsThe University of Western Australia Nedlands, W. A. 6009, Australia
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.

The structure and geometry of Banach spaces with the property that E(4) = Ê** + E⊥ ⊥ are investigated: such spaces are called quotient reflexive spaces here. For these spaces, if E is very smooth, Ê is also very smooth, and if E* is weakly locally uniformly rotund (WLUR), E(4) is smooth on a certain (relatively) norm dense subset of Ê**. Consequently, for quotient reflexive spaces, WLUR and very-WLUR are equivalent in E*.

MSC classification

Secondary: 46B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 2 , October 1983 , pp. 263 - 268
Copyright
Copyright © Australian Mathematical Society 1983

References

Bishop, E. and Phelps, R. R. (1961), ‘A proof that every Banach space is subreflexive’, Bull. Amer. Math. Soc. 67, 97–98.Google Scholar
Civin, P. and Yood, B. (1957), ‘Quasi-reflexive spaces’, Proc. Amer. Math. Soc. 8, 906–911.Google Scholar
Clarke, J. R. (1972), ‘Coreflexive and somewhat reflexive Banach spaces’, Proc. Amer. Math. Soc. 36, 421–427.Google Scholar
Davis, W. J., Figiel, T., Johnson, W. B. and Pelczynski, A. (1971), ‘Factoring weakly compact operators’, J. Functional Analysis 17, 311–327.Google Scholar
Dixmier, J. (1948), ‘Sur un théoréme de Banach’, Duke Math. J. 15, 1057–1071.Google Scholar
James, R. C. (1951), ‘A non-reflexive Banach space isometric with its second conjugate space’, Proc. Nat. Acad. Sci. U. S. A. 37, 174–177.CrossRefGoogle ScholarPubMed
Smith, M. A. (1976), ‘A smooth, non-reflexive second conjugate space’, Bull. Austral. Math. Soc. 15, 29–31.Google Scholar
Smith, M. A. and Sullivan, F. (1977), ‘Extremely smooth Banach spaces’, preprint.Google Scholar
Sullivan, F. (1976), ‘Some geometric properties of higher duals in Banach spaces, Illinois J. Math. 21, 315–331.Google Scholar
Yorke, A. C. (1978), ‘Differentiability and local rotundity’, J. Austral. Math. Soc. Ser. A 28, 205–213.Google Scholar