Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T10:10:45.112Z Has data issue: false hasContentIssue false
  • English
  • Français

Differentiability and local rotundity

Published online by Cambridge University Press:  09 April 2009

A. C. Yorke
A. C. Yorke
Affiliation:
Mathematics Department The University of New EnglandArmidale, N.S.W. 2351, 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 concept of very weak local uniform rotundity (very WLUR) is introduced. It is shown that if a Banach space E is WLUR, very WLUR, or LUR then its dual E* is smooth, very smooth, or Fréchet differentiable, respectively, on a norm dense subset. This leads to examples of nonreflexive spaces which are Fréchet differentiable at every nonzero point of a (relatively) norm dense subset of the embedding of E** in the fourth dual. When E is reflexive, necessary and sufficient conditions for E and E* to be WLUR and LUR are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 46 B 99.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 2 , September 1979 , pp. 205 - 213
Copyright
Copyright © Australian Mathematical Society 1979

References

Bollobás, B. (1970), ‘An extension to the theorem of Bishop and Phelps’, Bull. London Math. Soc. 2, 181–182.CrossRefGoogle Scholar
Cudia, D. F. (1963), ‘Rotundity’, Proc. Amer. Math. Soc. (Convexity) 7, 73–97.Google Scholar
Diestel, J. and Faires, B. (1974), ‘On vector measures’, Trans. Amer. Math. Soc. 198, 253–271.CrossRefGoogle Scholar
Dixmier, J. (1948), ‘Sur un théorème de Banach’, Duke Math. J. 15, 1057–1071.CrossRefGoogle Scholar
Giles, J. R. (1971), ‘On a characterization of differentiability of the norm of a normed linear space’, J. Austral. Math. Soc. 12, 106–114.CrossRefGoogle Scholar
Giles, J. R. (1975), ‘On smoothness of the Banach space embedding’, Bull. Austral. Math. Soc. 13, 69–74.CrossRefGoogle Scholar
Giles, J. R. (1976), ‘Uniformly weak differentiability of the norm and condition of Vlasov’, J. Austral. Math. Soc. Ser. A 21, 393–409.CrossRefGoogle Scholar
James, R. C. (1964), ‘Characterizations of reflexivity’, Studia Math. 23, 205–216.CrossRefGoogle Scholar
Lovaglia, A. (1955), ‘Locally uniformly convex Banach spaces’, Trans. Amer. Math. Soc. 78, 225–238.CrossRefGoogle Scholar
Smith, M. A. (1978), ‘Some examples concerning rotundity in Banach spaces’, Math. Ann. 233, 155–161.CrossRefGoogle Scholar
Sullivan, F. (1975), ‘Some geometric properties of higher duals in Banach spaces’, preprint.Google Scholar
Yorke, A. C. (1977), ‘Weak rotundity in Banach spaces’, J. Austral. Math. Soc. Ser. A 24, 224–233.CrossRefGoogle Scholar
Zizler, V. (1969), ‘Some notes on various rotundity and smoothness properties of separable Banach spaces’, Comm. Math. Univ. Canol. 10, 2, 195–206.Google Scholar