Research Article | Open Access
Hongwei Jiao, Yunrui Guo, "Some Inequalities Concerning the Weakly Convergent Sequence Coefficient in Banach Spaces", Abstract and Applied Analysis, vol. 2008, Article ID 840387, 8 pages, 2008. https://doi.org/10.1155/2008/840387
Some Inequalities Concerning the Weakly Convergent Sequence Coefficient in Banach Spaces
We establish two inequalities concerning the weakly convergent sequence coefficient and other parameters, which enable us to obtain some sufficient conditions for normal structure.
Throughout this paper, we denote by and the unit ball and the unit sphere of a Banach space , respectively. Let be a nonempty bounded subset of . The numbers and are, respectively, called the diameter and the Chebyshev radius of A Banach space is said to have normal structure (resp., weak normal structure) iffor every closed bounded (resp., weakly compact) convex subset of with . A Banach space is said to have uniform normal structure (resp., weak uniform normal structure) ifwhere the infimum is taken over all bounded closed (resp., weakly compact) convex subsets of with It is clear that normal structure and weak normal structure coincide when is reflexive.
Throughout this paper, we assume that does not have the Schur property. The weakly convergent sequence coefficient , introduced by Bynum , is reformulated by Sims and Smyth  as the following equivalent form:Obviously and it is known that implies that has weak normal structure.
There are many geometric conditions implying a Banach space to have normal structure (see, e.g., [2–8]), among themHere, the coefficient  is defined as the infimum of the set of real numbers such thatfor all and all weakly null sequence in . 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.
2. Generalized James Constant
The James constant, or the nonsquare constantis introduced by Casini  and Gao and Lau  and generalized by Dhompongsa et al.  in the following sense:where is a nonnegative parameter. Obviously, , since it is known that in the definition of , can be replaced by .
Theorem 2.1. Let and let be a Banach space without the Schur Property. Then
Proof. . If ,
our estimate is trivial since and
Suppose that . Then is uniformly nonsquare and therefore reflexive (see ). Let be a weakly null sequence in . Assume that exists and consider a normalized functional sequence such that . Note that the reflexivity of guarantees, by passing to a subsequence, if necessary, that there exists such that Let and choose large enough so that andfor all By the definition of ,Then we can choose large enough such that
(1)(2)(3) HenceLet us put (for short ). It follows that andAlsoThis together with the definition of gives thatSince the sequence and are arbitrary, we get
Moreover, if we put It follows that andAlsoThis together with the definition of gives thatSince the sequence and are arbitrary, we getAdding up (2.11) and (2.16) yields (2.3) as desired.
Corollary 2.2. Let be a Banach space withfor some Then has normal structure.
Corollary 2.3. Let be a Banach space withThen has normal structure.
3. Generalized Von Neumann-Jordan Constant
The von Neumann-Jordan constant is introduced by Clarkson  and reformulated by Kato et al.  in the following way:The generalized version of this constant is given by Dhompongsa et al.  aswhere is a nonnegative parameter. Obviously,
Theorem 3.1. Let and let be a Banach space without the Schur Property. Then
Proof. . If ,
then (3.3) is trivial.
Suppose that . Then is uniformly nonsquare and therefore reflexive (see ). Let be a weakly null sequence in and assume that exists and let be chosen as in Theorem 2.1.
Now let us put ,(for short ). It is easy to check that : and also thatBy the definition of Since the sequence and are arbitrary, we get
Moreover, if we put ,it follows that andand also thatThis together with the definition of gives thatSince the sequence and are arbitrary, we getAdding up (3.8) and (3.13) yields the inequality (3.3) as desired.
Corollary 3.2. Let be a Banach space withfor some Then has normal structure.
Corollary 3.3. Let be a Banach space withThen has normal structure.
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 (10671057), Basic Research Foundation for Department of Education of Henan Province (2008A110003), and the Natural Science Foundation of Henan Institute of Science and Technology (06055).
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