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Uniform Kadec–Klee–Huff properties of vector-valued Hardy spaces

Published online by Cambridge University Press:  24 October 2008

P. N. Dowling  and
C. J. Lennard
P. N. Dowling
Affiliation:
Miami University, Oxford, Ohio 4506, USA
C. J. Lennard
Affiliation:
University of Pittsburgh, Pittsburgh, PA 15260, USA
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Abstract

In [8] Partington showed that a Banach space X is uniformly convex if and only if Lp([0, 1], X) has the uniform Kadec–Klee–Huff property with respect to the weak topology (UKKH (weak)), where 1 < p < ∞. In this note we will characterize the Banach spaces X such that HP(D, X) has UKKH (weak), where 1 ≤ p < ∞. Similar results for UKKH (weak*) are also obtained. These results (and proofs) are quite different from Partington's result (and proof).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 25 - 30
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Besbes, M., Dilworth, S., Dowling, P. and Lennard, C.. New convexity and fixed point properties in Hardy and Lebesgue–Bochner spaces. J. Fund. Anal., to appear.Google Scholar
[2]Dowling, P. N.. Duality in some vector-valued function spaces. Rocky Mountain J. Math. 22 (1992), 511–518.CrossRefGoogle Scholar
[3]Dowling, P. N. and Lennard, C. J.. On uniformly H-convex complex quasi-Banach spaces. Bull. Sci. Math., to appear.Google Scholar
[4]Van Dulst, D. and Sims, B.. Fixed points of non-expansive mappings and Chebyshev centers in Banach spaces with norms of type (KK). Banach Space Theory and its Applications, Proc. Bucharest 1981, Lecture Notes in Math. 991 (Springer-Verlag, 1983), 34–43.Google Scholar
[5]Huff, R. E.. Banach spaces which are nearly uniformly convex. Rocky Mountain J. of Math. 10 (1980), 743–749.CrossRefGoogle Scholar
[6]Kirk, W. A.. A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 72 (1965), 1004–1006.CrossRefGoogle Scholar
[7]Lennard, C. J.. On the Schur and Kadec–Klee properties in Banach spaces, preprint.Google Scholar
[8]Partington, J. R.. On nearly uniformly convex Banach spaces. Math. Proc. Camb. Phil. Soc. 93 (1983), 127–129.CrossRefGoogle Scholar
[9]Xu, Q.. Inègalités pour les martingales de Hardy et renormage des espaces quasi-normés. C.R. Acad. Sci. Paris 306, série I (1988), 601–604.Google Scholar
[10]Xu, Q.. Convexités uniformes et inégalités de martingales. Math. Ann. 287 (1990), 193–211.Google Scholar