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Rotundity redux

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:
School of Mathematical and Physical Sciences Murdoch UniversityMurdoch, W. A. 6150, Australia
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Abstract

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Recently the concept of uniform rotundity was generalized for real Banach spaces by using a type of “area” devised for these spaces. This paper modifies the methods used for uniform rotundity and applies them to weak rotundity in real and complex spaces. This leads to the definition of k-smoothness, k-very smoothness and k-strong smoothness. As an application, several sufficient conditions for reflexivity are obtained.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 38 , Issue 2 , April 1985 , pp. 143 - 151
Copyright
Copyright © Australian Mathematical Society 1985

References

Civin, P. and Yood, B. (1957), ‘Quasi-reflexive space’, Proc. Amer. Math. Soc. 8, 906–911.CrossRefGoogle Scholar
Cudia, D. F. (1964), ‘The geometry of Banach spaces. Smoothness’, Trans. Amer. Math. Soc. 110, 284–314.CrossRefGoogle Scholar
Dixmier, J. (1948), ‘Sur un théorème de Banach’, Duke Math. J. 15, 1057–1071.CrossRefGoogle Scholar
Geremia, R. and Sullivan, F. (1981), ‘Multi-dimensional volumes and moduli of convexity in Banach spaces’, Ann. Mat. Pura. Appl. (4) 127, 231–251.CrossRefGoogle Scholar
James, R. C. (1951), ‘A non-reflexive space isometric with its second conjugate space’, Proc. Nat. Acad. Sci. U.S.A. 37, 174–177.CrossRefGoogle ScholarPubMed
Köthe, G. (1969), Topological vector spaces I, Springer-Verlag.Google Scholar
Perrott, J. (1979), ‘Transfinite duals of Banach spaces and ergodic super-properties equivalent to super-reflexivity’, Quart. J. Math. Oxford Ser. (2) 30, 99–111.CrossRefGoogle Scholar
Silverman, F. (1951), ‘Definitions of area for surfaces in metric space’, Riv. Mat. Univ. Parma 2, 47–76.Google Scholar
Sullivan, F. (1979), ‘A generalization of uniform rotund Banach spaces’, Canad. J. Math. 31, 628–636.CrossRefGoogle Scholar
Yorke, A. C. (1977), ‘Weak rotundity in Banach spaces’, J. Austral. Math. Soc. 24, 224–233.CrossRefGoogle Scholar
Zizler, V. (1968), ‘Banach spaces with differentiable norms’, Comm. Math. Univ. Carol. 8, 415–440.Google Scholar