Research Article Some Inequalities Concerning the Weakly Convergent Sequence Coefficient in Banach Spaces

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1 Hindawi Publishing Corporation Abstract and Applied Analysis Volume 008, Article ID 80387, 8 pages doi:0.55/008/80387 Research Article Some Inequalities Concerning the Weakly Convergent Sequence Coefficient in Banach Spaces Hongwei Jiao and Yunrui Guo Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 53003, China Correspondence should be addressed to Hongwei Jiao, jiaohongwei@6.com Received August 008; Accepted 6 September 008 Recommended by Simeon Reich We establish two inequalities concerning the weakly convergent sequence coefficient and other parameters, which enable us to obtain some sufficient conditions for normal structure. Copyright q 008 H. Jiao and Y. Guo. 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.. Introduction Throughout this paper, we denote by B X and S X the unit ball and the unit sphere of a Banach space X, respectively. Let C be a nonempty bounded subset of X. The numbers diam C sup{ x y : x, y C} and r C inf{sup{ x y : y C} : x C} are, respectively, called the diameter and the Chebyshev radius of C. A Banach space X is said to have normal structure resp., weak normal structure if diam C r C >. for every closed bounded resp., weakly compact convex subset C of X with diam C > 0. A Banach space X is said to have uniform normal structure resp., weak uniform normal structure if { } diam C inf >,. r C where the infimum is taken over all bounded closed resp., weakly compact convex subsets C of X with diam C > 0. It is clear that normal structure and weak normal structure coincide when X is reflexive.

2 Abstract and Applied Analysis Throughout this paper, we assume that X does not have the Schur property. The weakly convergent sequence coefficient WCS X, introduced by Bynum, is reformulated by Sims and Smyth as the following equivalent form: { WCS X inf lim n,m,n / m xn x m w : xn 0, xn, lim n,m,n / m xn x m exists }..3 Obviously WCS X and it is known that WCS X > implies that X has weak normal structure. There are many geometric conditions implying a Banach space to have normal structure see, e.g., 8, among them C NJ X < or J X < μ X μ X.. Here, the coefficient μ X 9 is defined as the infimum of the set of real numbers r>0 such that lim sup x xn rlim sup x xn n n.5 for all x X and all weakly null sequence x n in X. The aim of this paper is to state some estimates concerning the weakly convergent sequence coefficient. By these estimates, we get sufficient conditions for normal structure in terms of the generalized James and von Neumann-Jordan constants and thus generalize the above results.. Generalized James constant The James constant, or the nonsquare constant J X sup { x y x y : x, y S X }. is introduced by Casini 0 and Gao and Lau and generalized by Dhompongsa et al. in the following sense: J a, X sup { x y x z : x, y, z B X, y z a x },. where a is a nonnegative parameter. Obviously, J 0,X J X, since it is known that in the definition of J X, S X can be replaced by B X. Theorem.. Let 0 a and let X be a Banach space without the Schur Property. Then WCS X a / min,μ X a J a, X..3

3 H. Jiao and Y. Guo 3 Proof. If J a, X, our estimate is trivial since WCS X andμ X. Suppose that J a, X <. Then X is uniformly nonsquare and therefore reflexive see.let{x n } be a weakly null sequence in S X. Assume that d lim n,m,n / m x n x m exists and consider a normalized functional sequence {xn} such that xn x n. Note that the reflexivity of X guarantees, by passing to a subsequence, if necessary, that there exists x X such that w x. Let 0 <ɛ<and choose N large enough so that x x N <ɛ/and x n d ɛ< xn x m <d ɛ. for all m>n.by the definition of μ X, lim sup x m x N d ɛ μ X lim sup x N x m d ɛ μ X. m m Then we can choose M>Nlarge enough such that x N x M <ɛ; x M x x N <ɛ/; 3 x N x M / d ɛ μ X ɛ. Hence x M xn x M x x N x x N <ɛ..5.6 Let us put x x N x M / d ɛ, y a x N x M d ɛ, z x N a x M d ɛ for short μ μ X. It follows that x, y, z B X, y z a x and d ɛ x y a a a ɛ. x N x N xn x M x N xm.7.8 Also d ɛ x z a a a ɛ. x M x M xm x N x M xn.9

4 Abstract and Applied Analysis This together with the definition of J a, X gives that d ɛ J a, X a ɛ..0 Since the sequence {x n } and ɛ are arbitrary, we get WCS X μ a J a, X μ a.. Moreover, if we put x x N x M / d ɛ, y a x N a x M d ɛ, z a x N a x M.. d ɛ It follows that x, y, z B X, y z a x and Also d ɛ x y a a a x N x N ɛ. d ɛ x z a a a a x M xn a x M x M ɛ. a x N xm a x N xm x M xn.3. This together with the definition of J a, X gives that d ɛ J a, X a ɛ..5 Since the sequence {x n } and ɛ are arbitrary, we get WCS X 3 a J a, X..6 Adding up. and.6 yields.3 as desired.

5 H. Jiao and Y. Guo 5 Corollary.. Let X be a Banach space with J a, X < a min,μ X a,.7 for some 0 a. Then X has normal structure. Corollary.3. Let X be a Banach space with J X < min,μ X..8 Then X has normal structure. Remark.. Corollary.3 includes 5, Theorem. 3. Generalized von Neumann-Jordan constant The von Neumann-Jordan constant is introduced by Clarkson and reformulated by Kato et al. 6 in the following way: { x y x y } C NJ X sup x y : x, y X, x y / The generalized version of this constant is given by Dhompongsa et al. 3 as { x y x z } C NJ a, X sup x y z : x y z / 0, y z a x, 3. where a is a nonnegative parameter. Obviously, C NJ X C NJ 0,X. Theorem 3.. Let 0 a and let X be a Banach space without the Schur Property. Then [ ] a /min μ X a, WCS X. 3.3 C NJ a, X Proof. If C NJ a, X, then 3.3 is trivial. Suppose that C NJ a, X <. Then X is uniformly nonsquare and therefore reflexive see 3. Let{x n } be a weakly null sequence in S X and assume that d lim n,m,n / m x n x m exists and let x N,x M,x N,x M be chosen as in Theorem.. Now let us put x x N x M / d ɛ, y a a x N x M a xn a x M, z d ɛ d ɛ 3.

6 6 Abstract and Applied Analysis for short μ μ X. It is easy to check that y z a x, x B X : y a, z a, 3.5 andalsothat a d ɛ x y x N a x N a ɛ, a d ɛ x z x M a x M a ɛ. xn xm a a x M a a x N x N xm x M xn 3.6 By the definition of C NJ a, X, C NJ a, X ɛ d ɛ a. 3.7 Since the sequence {x n } and ɛ are arbitrary, we get WCS X C NJ a, X a. 3.8 μ a Moreover, if we put x x N x M / d ɛ, y a a x N a x M d ɛ, z a a x N a x M d ɛ, 3.9 it follows that x B X, y z a x and y a d ɛ a a, z d ɛ a, 3.0

7 H. Jiao and Y. Guo 7 andalsothat d ɛ x y a a a d ɛ x z a a a x N x N ɛ, a x M xn a x M x M ɛ. a x N xm a x N xm x M xn 3. This together with the definition of C NJ a, X gives that C NJ a, X a ɛ. 3. d ɛ Since the sequence {x n } and ɛ are arbitrary, we get [ ] CNJ a WCS X a, X. 3.3 Adding up 3.8 and 3.3 yields the inequality 3.3 as desired. Corollary 3.. Let X be a Banach space with a C NJ a, X < min μ X a, 3. for some a 0,. Then X has normal structure. Corollary 3.3. Let X be a Banach space with C NJ X < min,μ X. 3.5 Then X has normal structure. Remark 3.. Corollary 3.3 includes 5, Theorem.

8 8 Abstract and Applied Analysis Acknowledgments The author would like to express his sincere thanks to the anonymous referees for their careful review of the manuscript and the valuable comments. This paper is supported by the National Natural Science Foundation of China , Basic Research Foundation for Department of Education of Henan Province 008A0003, and the Natural Science Foundation of Henan Institute of Science and Technology References W. L. Bynum, Normal structure coefficients for Banach spaces, Pacific Journal of Mathematics, vol. 86, no., pp. 7 36, 980. B. Sims and M. A. Smyth, On some Banach space properties sufficient for weak normal structure and their permanence properties, Transactions of the American Mathematical Society, vol. 35, no., pp , S. Dhompongsa, P. Piraisangjun, and S. Saejung, Generalised Jordan-von Neumann constants and uniform normal structure, Bulletin of the Australian Mathematical Society, vol. 67, no., pp. 5 0, 003. S. Dhompongsa, A. Kaewkhao, and S. Tasena, On a generalized James constant, Journal of Mathematical Analysis and Applications, vol. 85, no., pp. 9 35, A. Jiménez-Melado, E. Llorens-Fuster, and S. Saejung, The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces, Proceedings of the American Mathematical Society, vol. 3, no., pp , M. Kato, L. Maligranda, and Y. Takahashi, On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces, Studia Mathematica, vol., no. 3, pp , S. Saejung, On James and von Neumann-Jordan constants and sufficient conditions for the fixed point property, Journal of Mathematical Analysis and Applications, vol. 33, no., pp. 08 0, F. Wang and C. Yang, Some sufficient conditions for uniform normal structure in Banach spaces, Acta Mathematica Sinica, vol. 5, no., pp , A. Jiménez-Melado and E. Llorens-Fuster, The fixed point property for some uniformly nonsquare Banach spaces, Bollettino della Unione Matemàtica Italiana. Serie A, vol. 0, no. 3, pp , E. Casini, About some parameters of normed linear spaces, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Fisiche. Serie VIII, vol. 80, no. -, pp. 5, 986. J. Gao and K.-S. Lau, On the geometry of spheres in normed linear spaces, Journal of Australian Mathematical Society. Series A, vol. 8, no., pp. 0, 990. J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue spaces, Annals of Mathematics, vol. 38, no., pp. 5, 937.