%0 Journal Article
%T On Semi-Uniform Kadec-Klee Banach Spaces
%A Satit Saejung
%A Ji Gao
%J Abstract and Applied Analysis
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/652521
%X 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). 1. Introduction Let be a real Banach space with the unit sphere and the closed unit ball . In this paper, the strong and weak convergences of a sequence in to an element are denoted by and , respectively. We also let Definition 1.1 (see [1]). We say that a Banach space is a Kadec-Klee space if A uniform version of the KK property is given in the following definition. Definition 1.2 (see [2]). We say that a Banach space is uniform Kadec-Klee if for every there exists a such that Two properties above are weaker than the following one. Definition 1.3 (see [3]). We say that a Banach space is uniformly convex if for every there exists a such that Let us summarize a relationship between these properties in the following implication diagram: In the literature, there are some generalizations of UC and KK. Definition 1.4 (see [4]). We say that a Banach space is a -space if for every there exists a such that Here . Definition 1.5 (see [5]). We say that a Banach space is semi-Kadec-Klee if Some interesting results concerning semi-KK property are studied by Megginson [6]. Remark 1.6. It is clear that Remark 1.7. A Banach space is semi-KK if and only if We now introduce a property lying between -space and semi-KK. Definition 1.8. We say that a Banach space is semi-uniform Kadec-Klee if for every there exists a such that In this paper, we prove that semi-UKK property is a nice generalization of -space and semi-KK property. Moreover, every semi-UKK space has weak normal structure. We also give a characterization of the direct sum of finitely many Banach spaces which is semi-KK and semi-UKK. We use such a characterization to construct a Banach space which is semi-UKK but not UKK. Finally we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al. [7]. 2. Results 2.1. Some Implications For a sequence and satisfying for all , we let It is clear that . Theorem
%U http://www.hindawi.com/journals/aaa/2010/652521/