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Abstract and Applied Analysis
Volume 2008, Article ID 135873, 5 pages
http://dx.doi.org/10.1155/2008/135873
Research Article

Modulus of Convexity, the Coeffcient , and Normal Structure in Banach Spaces

1Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China
2Department of Mathematics, Luoyang Normal University, Luoyang 471022, China

Received 7 April 2008; Accepted 27 May 2008

Academic Editor: William A.  Kirk

Copyright © 2008 Hongwei Jiao et al. 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

Let and be the modulus of convexity and the Domínguez-Benavides coefficient, respectively. According to these two geometric parameters, we obtain a sufficient condition for normal structure, that is, a Banach space has normal structure if for some which generalizes the known result by Gao and Prus.

1. Introduction

Let be a Banach space. Throughout the paper, denote by the unit sphere and unit ball of respectively. Recall that a Banach space is said to have normal structure (resp., weak, normal, structure) if for every closed bounded (resp., weakly compact) convex subset of with diam there exists such that where diam For a reflexive Banach space, the normal structure and weak normal structure are the same. Recently a good deal of investigations have focused on finding the sufficient conditions with various geometrical constants for a Banach space to have normal structure (see, e.g., [15]). The geometric condition sufficient for normal structure in terms of the modulus of convexity is given by Goebel [6], who proved that has normal structure provided that Here the function defined by Clarkson [7] asis called the modulus of convexity of Later Gao and Prus generalized the above results as the following (see [2, 8]).

Theorem 1.1. A Banach space has normal structure provided that for some .

In this paper, we obtain a class of Banach spaces with normal structure, which involves the coefficient . This coefficient is defined by Domínguez Benavides [9] aswhere the supremum is taken over all with and all weakly null sequence in such thatObviously,

2. Main Results

Let us begin this section with a sufficient condition for a Banach space having weak normal structure and the idea in the following proof is due to [5, Lemma 5].

Lemma 2.1. Let be a Banach space for which is -sequentially compact. If does not have weak normal structure, then for any , there exist and such that
(1)(2) for and (3).

Proof. Assume that does not have weak normal structure. It is well known that (see, e.g., [10]) there exists a sequence in satisfying
(1) is weakly convergent to ;(2)diam for all . Since is -sequentially compact, we can find in satisfying
(3) for all ;(4) for some .
Let sufficiently small and Then, by the properties of the sequence , we can choose such thatNote that the sequence is weakly null and verifies It follows from the definition of thatThe rest of the proof is similar to that of [5, Lemma 5].

Theorem 2.2. A Banach space has normal structure provided that for some , where the function is defined as

Proof. Observe that is uniformly nonsquare [11] and then reflexive. Therefore normal structure and weak normal structure coincide.
Assume first that fails to have weak normal structure. Fix sufficiently small and . It follows that there exist and satisfying the condition in Lemma 2.1. Next, denote by and consider two cases for .
Case 1 (). Now let us putand so and also thatBy the definition of modulus of convexity,or equivalently,Letting we havewhich contradicts our hypothesis.
Case 2 (). In this case otherwise . Letwhere . It follows from Case 1 that This implies thatwhich is equivalent toThis is a contradiction.

Remark 2.3. (1) It is readily seen that for any and Theorem 2.2 is therefore a generalization of Theorem 1.1. Moreover this generalization is strict whenever is the space with
(2) Consider the space with the norm It is known that [8] and then has normal structure from Theorem 2.2, but lies out of the scope of Theorem 1.1.

Corollary 2.4. Let be a Banach space with and then has normal structure.

Corollary 2.5. If is a Banach space withthen has normal structure.

Remark 2.6. Corollary 2.5 is equivalent to [4, Corollary 24].

Acknowledgments

The authors would like to express their sincere thanks to the referees for their valuable suggestions on this paper. This work is supported by NSF of Education Department of Henan Province (2008A110012).

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