Abstract and Applied Analysis
Volume 2010 (2010), Article ID 652521, 12 pages
doi:10.1155/2010/652521
Research Article

On Semi-Uniform Kadec-Klee Banach Spaces

Satit Saejung1,2 and Ji Gao3

1Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
2Centre of Excellence in Mathematics, CHE, Sriayudthaya Rd., Bangkok 10400, Thailand
3Department of Mathematics, Community College of Philadelphia, Philadelphia, PA, 19130-3991, USA

Received 30 June 2010; Accepted 31 August 2010

Academic Editor: S. Reich

Copyright © 2010 Satit Saejung and Ji Gao. 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

Inspired by the concept of U-spaces introduced by Lau, (1978), we introduced the class of semi-uniform Kadec-Klee spaces, which is a uniform version of semi-Kadec-Klee spaces studied by Vlasov, (1972). This class of spaces is a wider subclass of spaces with weak normal structure and hence generalizes many known results in the literature. We give a characterization for a certain direct sum of Banach spaces to be semi-uniform Kadec-Klee and use this result to construct a semi-uniform Kadec-Klee space which is not uniform Kadec-Klee. At the end of the paper, we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al., (1997).