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Kadec–Klee 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 45056, U.S.A.
C. J. Lennard
Affiliation:
University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.
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Extract

In 1930, S. Warschawski [19] showed that H1(D), where D is the open unit disc in ℂ, has the following property: Let be a sequence of functions in H1(D) converging uniformly on compact subsets of D to a function f∈H1(D) and suppose that ‖fn‖1 = |f‖1 = 1 for all n∈ℕ. Then converges to zero. From a Banach space standpoint, this result says that H1(D) has the Kadec–Klee property with respect to uniform convergence on compact subsets of D. Warschawski's result was proved independently by Newman [16] in 1963 (see also [13] for another proof) and extended to more general domains by Hoffman [12], Goldstein and Swaminathan [8] and Godefroy [7]. A uniform version of Warschawski's result and its subsequent extensions was recently obtained by Besbes, Dilworth, Dowling and Lennard [2] (see also [1]). We mention here that these results for H1 spaces also hold for the Hp-spaces for 1 < p < ∞ because these spaces are uniformly convex.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 3 , May 1992 , pp. 535 - 544
Copyright
Copyright © Cambridge Philosophical Society 1992

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