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On nearly uniformly convex and k-uniformly convex spaces

Published online by Cambridge University Press:  24 October 2008

V. I. Istrăt‚escu  and
J. R. Partington
V. I. Istrăt‚escu
Affiliation:
Fitzwilliam College, Cambridge
J. R. Partington
Affiliation:
Fitzwilliam College, Cambridge
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Abstract

In this note we prove that every nearly uniformly convex space has normal structure and that K-uniformly convex spaces are super-reflexive.

We recall [1] that a Banach space is said to be Kadec–Klee if whenever xn → x weakly and ∥n∥ = ∥x∥ = 1 for all n then ∥xn −x∥ → 0. The stronger notions of nearly uniformly convex spaces and uniformly Kadec–Klee spaces were introduced by R. Huff in [1]. For the reader's convenience we recall them here.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 2 , March 1984 , pp. 325 - 327
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Huff, R.Banach spaces which are nearly uniformly convex. Rocky Mountain J. Math. 10 (1980), 743–749.CrossRefGoogle Scholar
[2]Istrăt‚escu, V. I.Fixed Point Theory: An Introduction. (D. Reidel, 1981.)Google Scholar
[3]Istrăt‚escu, V. I.Strict Convexity and Complex Strict Convexity, Theory and Applications. Lecture Notes in Math. (M. Dekker, 1983).Google Scholar
[4]James, R. C. and Schäffer, J. J.Super-reflexivity and the girth of spheres. Israel J. Math. 11 (1972), 398–404.Google Scholar
[5]Partington, J. R.On nearly uniformly convex Banach spaces. Math. Proc. Cambridge Philos. Soc. 93 (1983), 127–129.Google Scholar