New Convexity and Fixed Point Properties in Hardy and Lebesgue-Bochner Spaces

✍ Scribed by M. Besbes; S.J. Dilworth; P.N. Dowling; C.J. Lennard

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 607 KB
• Volume: 119
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1994.1014

✦ Synopsis

We show that for the Hardy class of functions (H^{\prime}) with domain the ball or polydisc in (\mathbf{C}^{N}), a certain type of uniform convexity property (the uniform Kadec-Klee-Huff property) holds with respect to the topology of pointwise convergence on the interior, which coincides with both the topology of uniform convergence on compacta and the weak * topology on bounded subsets of (H^{1}). Also, we show that if a Banach space (X) has a uniform Kadec-Klee-Huff property, then the Lebesgue-Bochner space (L_{p}(\mu, X), 1 \leqslant p<\infty), must have a related uniform Kadec-Klee-Huff property. Consequently. by known results, the above spaces have normal structure properties and fixed point properties for non-expansive mappings. " 1994 Academic Press, Ine.