D DAVID PUBLISHING. Banach Saks Property and Property β InCesàro Sequence Spaces. 1. Introduction. Nafisa Algorashy Mohammed 1, 2

Transcription

1 Journal of Materials Science and Engineering A 8 (1-2) (2018) doi: / / D DAVID PUBLISHING Banach Saks Property and Property β InCesàro Sequence Spaces Nafisa Algorashy Mohammed 1, 2 1. Department of Mathematics, College of Sciences and Arts, University King Khalid university, Almajardah Kingdom of Saudi Arabia Asir-Abha P. O. Box: 960-Postal Code: Mathematics Department, Faculty of Education, Nile Valley University Atbara Sudan-Postal: Khartoum PO Box 1843 Abstract: A new constant C (X) for any Banach space X is introduced. It is proved that C (X) < 2 implies the weak Banach Saks property for the space X: In particular, C( ) is found for Cesàro sequence space (1 < p < ). Moreover, it is shown that the space (1 < p < ) has property. Key words: Banach Saks property, property (), Cesàro sequence space. 1. Introduction Let and stand for the set of natural numbers and the set of reals, respectively. Let (,. ) be a real Banach space and the dual space of X. By and, we denote the closed unit ball and the unit sphere of respectively. For any subset of by conv we denote the convex hull (closed convex hull) of is the characteristic function of. Clarkson [1] introduced the concept of uniform convexity. A norm. is called UC (uniformly convex) if, for each 0, there is 0 such that, for x, y, the inequality implies A Banach space is said to have the Banach Saks (resp. weak Banach Saks) property if every bounded (resp. weakly null) sequence in admits a subsequence such that sequence of its arithmetic means 1 is convergent in norm (see Ref. [2]). Corresponding author: Yunan Cuiand Chenghui Meng, doctor of mathematics, research fields: banach-saks property, weak banach-saks property, property and cesàro sequence space. It is well known that every Banach space with the Banach Saks property is reflexive and the converse is not true (see Ref. [3, 4]) proved that any uniformly convex Banach space has the Banach Saks property. Moreover, he also proved that if is a reflexive Banach space and there is 0,2 such that, for every sequence in. Weakly convergent to zero, there are, satisfying. Then has the Banach Saks property. For a sequenc we define lim :,,. According to Kakutani s result (see Ref. [4]), we introduce the following new geometric constant connected with packing constant (see Ref. [5]) and with the Banach Saks property: C sup : is a weakly null sequence in Recall that asequence is said to be an -separated sequence if, for some >0, SeP = inf :. A Banach space is said to be nearly uniformly convex (NUC) if, for every >0, there exists 0,1such that, for every sequence.

2 26 Banach Saks Property and Property β InCesàro Sequence Spaces With sep, we have Conv 1 According to Ref. [6], for any, the dropdetermined by x is the set, A Banach space X has the drop property (D) if, for every closed set C disjoint with, there exists an element x C such that,. In Ref. [7], Rolewicz proved that if the Banach space X has the drop property, then X is reflexive. For any subset C of, we denote by its Kuratowski measure of non-compactness, i.e., the infimum of such >0 for which there is a covering of C by a finite number of sets of diameter less than. Goebel and S. ekowski [8] extended the definition of uniform convexity replacing condition (1) by a condition involving the Kuratowski measure of non-compactness, namely, that a norm. in a Banach space is UC ( -uniformly convex) if, for any >0, there is >0 such that, for each convex set contained in the such that, we have inf: 1. It is well known that UC coincides with NUC. Rolewicz [7], studying the relationships between NUC and the drop property, has defined property. A Banach space X is said to have propert y if, for any >0, there exists >0 such that,\ whenever 1<<1 +. The following result will be very helpful for our considerations (see [9]). A Banach space has property if and only if, for every >0, there exists >0 such that, for each element x and each sequence. With sep, there is an index k such that 1. 2 Denoted by the space of all real sequences x= and by the natural basisin, given 1, by a Cesàro sequence space, we mean : 1 For more details, we refer to Refs. [10-12]. 2. Results We start with the following general result: Theorem 1. Any Banach space X with C 2 has the weak Banach Saks property Proof. Take a positive number 0 such that 2.For any weakly null sequence, there exists a subsequence of such that For any, now, using Kakutani s result (see Ref. [4]), we conclude that the Banach space has the weak Banach Saks property. We will use the following lemma: Lemma 1. Let,. Then for any 0 and 0, there exists δ 0 such that When ever. Proof. Fix 0 and 0, take 2 and 2. Then for any, with and we have

3 Banach Saks Property and Property β InCesàro Sequence Spaces Replacing, by, respectively, we also conclude that Hence Theorem 2. The space has property. Proof. Suppose does not have property, then there exists >0 such that, for any 0, /1 2, there is a sequence with / and an element such that 1 2 for every n. Fix 0, /12, first, we will show that lim sup. (2) Otherwise, without loss of generality, we can assume that there exists a sequence such that as and (3). For every, let 0 be a real number corresponding to and L = 1 in Lemma 1. By absolute continuity of the norm of there exists a positive integer such that,,,,, = Take k so large that taking into account Lemma 1, convexity of the function., and Eq. (3), we have

4 28 Banach Saks Property and Property β InCesàro Sequence Spaces 1-, +, 121, This contradiction proves (2). Since, 1 1 We have for and =1,2,.. Hence, there are a subsequence of and a sequen of real numbers such that for =1, 2,. therefore, lim 1 Forn, msufficiently large. Consequently, 2 2, 1,. This contradiction shows that has property. Corollary 1 The spaces and have the Banach Saks property. Proof. It is an immediate consequence of Theorem 1 from Ref. [13]. Theorem 3 C 2 / Proof Denote K=sup :, 0, 0 Then C. Moreover, for any >0; there is a sequence with 0 such that C

5 Banach Saks Property and Property β InCesàro Sequence Spaces 29 By the definition of, there exists a subsequence.of.such that 2 (4) For any, with m take. Then, by the absolute continuity of the norm of there exists such that Putting,we have. Banach Saks Property 207. For any 1 hence by Eq. (4), we have 3C For any 1, Since 0 for =1, 2,., there exists with such that Whenever n, define, then there is such that Taking,we obtain. Hence by eq. (4), we immediately obtain 3 5C Suppose that increasing sequences, of natural numbers and a sequence of elements of are already defined and 6Cces Form,n 1,2,, k 1, mn Since there exists 0 for 1,2, there exists natural number such that Provided n, put, then there is such that

6 30 Banach Saks Property and Property β InCesàro Sequence Spaces Defining,we obtain 4 For 1,2,, 1, hence, by Eq. (4), we obtain 6 C For 1,2,, 1 using the induction principle, we can find a sequence satisfying the following conditions: (1), 0 ; (2) 6 C,, ; (3) 1 1,2, ; (4) 0 Define for each, then every and C 6 for any,,, by the arbitrariness of we have C. Let >0 be given. Take such that where Hence, for any we have 1 1, On the other hand, for mentioned above, by Lemma 1, there exists >0 such that 1

7 Banach Saks Property and Property β InCesàro Sequence Spaces 31 Whenever 1 and, take such that, Hence, for any m n we have , 2 Therefore, by the arbitrariness of we obtain C 2 / which finishes the proof of the theorem. Corollary 2. The space has the Banach Saks property. Proof. Note that, for reflexive Banach spaces, the Banach Saks property is equivalent to the weak Banach Saks property. Hence, by Theorems 1 and 3, we conclude the thesis of Corollary 2. References [1] Clarkson, J. A Uniformly Convex Spaces. Trans. Amer. Math. Soc. 40: [2] Banach, S., and Saks, S Sur la convergence forte dans les champs Lp. Studia Math. 2: [3] Diestel, J Sequences and Series in Banach Spaces, Graduate Texts in Mathematics. Vol. 92, Springer-Verlag. [4] Kottman, C. A Packing and Reflexivity in Banach Spaces. Trans. Amer. Math. Soc. 150: [5] Kutzarowa, D. N An Isomorphic Characterization of Property. Of Rolewicz, Note Mat. 10 (2): [6] Dane s, J A Geometric Theorem Useful in Non-linear Functional Analysis. Boll. Un. Mat. Ital. 6: [7] Rolewicz, S On 1-Uniform Convexity and Drop Property. Studia Math. 87: [8] Goebel, K., and Sekowski, T The Modulus of Non-compact Convexity. Ann. Univ. Maria Curie-Skłodowska, Sect. A 38 (41-8) [9] Kutzarowa, D. N An Isomorphic Characterization of Property. OfRolewicz, Note Mat. 10 (2): [10] Cui, Y. A., Zhang, Y. F., and Zhang, T. Packing Constant in Cesàro Sequence Spaces. submitted. [11] Lee, P. Y Cesàro Sequence Spaces. Math. Chronicle, New Zealand 13: [12] Shue, J. S Cesàro Sequence Spaces. Tamkang J. Math. 1: [13] Cui, Y. A., Hudzik, H., and Płuciennik, R Banach Saks Property in Some Banach Spaces. Annales Math. Polonici 65: