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Journal of Function Spaces and Applications
Volume 2 (2004), Issue 2, Pages 125-173
http://dx.doi.org/10.1155/2004/678358

Local uniform convexity and Kadec-Klee type properties in K-interpolation spaces I: General Theory

1School of Informatics and Engineering, Flinders University, Bedford Park, SA 5042, Australia
2Department of Mathematics, Voronezh State University of Architecture and Civil Engineering, 394006 Voronezh, 20-letiya Oktyabrya 84, Russia

Received 1 July 2003

Academic Editor: Evgueni Semenov

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a systematic study of the interpolation of local uniform convexity and Kadec-Klee type properties in K-interpolation spaces. Using properties of the K-functional of J.Peetre, our approach is based on a detailed analysis of properties of a Banach couple and properties of a K-interpolation functional which guarantee that a given K-interpolation space is locally uniformly convex, or has a Kadec-Klee property. A central motivation for our study lies in the observation that classical renorming theorems of Kadec and of Davis, Ghoussoub and Lindenstrauss have an interpolation nature. As a partiular by-product of our study, we show that the theorem of Kadec itself, that each separable Banach space admits an equivalent locally uniformly convex norm, follows directly from our approach.