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Banach-Saks Type and Gurariǐ Modulus of Convexity of Some Banach Sequence Spaces
Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from above for the spaces of one type among them.
Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In the literature, there are many papers concerning geometric properties of various Banach sequence spaces. For example, geometry of Orlicz spaces and of Musielak-Orlicz spaces has been studied in [1–25]. Several authors including Cui and Hudzik [26–29], Cui and Meng , Suantai , and Lee  investigated the geometric properties of Cesàro sequence space . Also Cesàro-Orlicz sequence spaces equipped with Luxemburg norm have been studied in [5, 33–36]. Additionally, geometry of Orlicz-Lorentz sequence spaces and of generalized Orlicz-Lorentz sequence space were studied in [37, 38]. Furthermore, Mursaleen et al.  studied some geometric properties of normed Euler sequence space. Additionally, Hudzik and Narloch  have studied relationships between monotonicity and complex rotundity properties with some consequences. Besides, some geometrical properties of Calderon-Lozanovskii sequence spaces have been investigated in [41–43].
Quite recently, Karakaya  defined a new sequence space involving lacunary sequence space equipped with the Luxemburg norm and studied Kadec-Klee and rotundity properties of these spaces. Further information on topological and geometric properties of sequence spaces can be found in [39, 44–62].
Let be a real Banach space and and be the unit sphere and the unit ball of , respectively. Let , , , , and be the spaces of all real sequences, null, convergent, and bounded sequences and absolutely convergent series, respectively, and let be the space of those real sequences which have only a finite number of nonzero coordinates and .
Note that , , and are Banach spaces with the sup-norm and are Banach spaces with the norm , while is not a Banach space with respect to any norm.
Let us recall that a sequence in a Banach space is called of (or for short) if for each there exists a unique sequence of scalars such that , that is, .
A sequence space with a linear topology is called a - if each of the projection maps defined by for is continuous for each natural . A is a complete metric linear space and the metric is generated by an -norm and a Fréchet space which is a -space is called an -; that is, a -space is called an -space if is a complete linear metric space. In other words, is an -space if is a Fréchet space with continuous coordinate projections. All the sequence spaces mentioned above are spaces except the space .
An -space which contains the space is said to have the if, for every sequence , , where . The spaces equipped with the sup-norm equipped with the norm and endowed with the metric have property , while and do not have property . Banach spaces with continuous coordinate projections are called -spaces.
We say that is order continuous if, for any sequence in (the positive cone in ) such that and for each , we have holds.
A Köthe sequence space is said to be order continuous that if all sequences in are order continuous. It is easy to see that is order continuous if and only if as .
A Köthe sequence space is said to have the if, for any real sequence and any in such that coordinatewisely and , we have the fact that and .
A Banach space is said to have the - if every bounded sequence in admits a subsequence such that the sequence is convergent in with respect to the norm, where A Banach space is said to have the whenever given any weakly null sequence in there exists its subsequence such that the sequence converges to zero strongly.
Given any , we say that a Banach space has the if there exists a constant such that every weakly null sequence has a subsequence such that (see ) The Banach-Saks property of type and the weak Banach-Saks property for Cesro sequence spaces have been considered in . These properties and stronger property for Musielak-Orlicz and Nakano sequence spaces have been studied in .
We say that a Banach space has the if every nonexpansive self-mapping defined on a nonempty weakly compact convex subset of has a fixed point in .
Clarkson of a normed space is defined (see Clarkson  and Day ) by the formula for any . The inequality for all characterizes the uniform convexity of and the equality characterizes strict convexity (=rotundity) of .
The Gurariǐ modulus of convexity of a normed space is defined (see [68, 69]) by for any . It is obvious that for any Banach space and any . It is also known that for any and that is rotund if and only if and as well as that is uniformly convex if and only if for any . Gurariǐ  proved that if is renormed by the norm then for any and .
Now, we will define Köthe sequence spaces and that will be considered in this paper.
Let denote the set whose elements are finite sets of distinct positive integers. Given any element of , we denote by the sequence such that for , and otherwise. Further, we define that is, is the set of those whose support has cardinality at most . Let us define where .
Given any , we define the following sequence space, introduced in : Sargent  established the relationship of this space to the space and characterized some matrix transformations. In , matrix classes have been characterized, where is assumed to be any -space.
It has been proved in  that, for , is a Banach space if it is endowed with the norm and that one has the following.(i)If , for all , then . Moreover, .(ii)If , then . Also, for any .(iii) if and only if .
It is easy to see that is a Köthe sequence space, indeed a -space with respect to its natural norm (see ). Note that throughout the present paper we will study the space except the case , for which it is reduced to the space .
Now we will introduce the space . Let and let be arbitrary real sequences with all coordinates and different from zero and let, for any , It is obvious that this is a linear space. It is known (see ) that the functional is a norm in and that the couple is a Banach space. The space is a generalization of three spaces. Namely, one has the following.(i)If and , then is the Cesáro sequence space of nonabsolute type (see ) and .(ii)Let be a real sequence with and for all . If and , where for any . Then we obtain that is the Riesz sequence space of nonabsolute type denoted by and that (see ).(iii)Let be a real sequence with and , for all , . If and , where for any . Then reduces to the Nörlund sequence space of nonabsolute type denoted by and (see ).
2. Banach-Saks Type of Sequence Space
In this section, we investigate some properties of the space such as the Fatou property, the Banach-Saks property of type , and the weak fixed point property. Let us start with the following lemma.
Lemma 1. If a -space containing has the , then it is order continuous, that is, as for any .
Proof. From the definition of property , we have that every has the unique representation as . Hence as , that is, as , which means that is order continuous.
Corollary 2. The space is order continuous.
Proof. It is easy to see that contains and that every has the unique representation as , which means that has the property . Hence, from the above lemma, is order continuous.
Theorem 3. The space has the Fatou property.
Proof. Let x be any real sequence from and be any non-decreasing sequence of non-negative elements from such that as coordinatewisely and .
Let us denote . Then, since the supremum is homogeneous, we have Moreover, by the assumptions that is non-decreasing and convergent to coordinatewisely and by the Beppo-Levi theorem, we have whence Therefore, . On the other hand, since for any natural number and the sequence is non-decreasing, we obtain that the sequence is bounded from above by . In consequence , which together with the opposite inequality proved already, yields that .
Theorem 4. The space has the Banach-Saks property of the type .
Proof. Let be a sequence of positive numbers with . Let be a weakly null sequence in . Let us set and . Then, there exists such that
where consist of the elements of which exceed . Since implies that coordinatewise, there is such that
for all . Set . Then there exists such that
where consist of all elements of which exceed . Using again the fact that coordinatewise, there exists such that
for all .
Continuing this process, we can find two increasing sequences and such that where and consist of the elements of which exceed . Since , we have for all and . Hence By using the norm of the space , we have Therefore, This completes the proof of the theorem.
Theorem 5. For , the space has the weak fixed point property, if , where .
Proof. If , for all , it follows that Hence where stands for the Garcia-Falset coefficient of . Therefore, has in this case the weak fixed point property.
3. Banach-Saks Type and Gurariǐ Modulus of Sequence Spaces
Theorem 6. The space has the Banach-Saks property of the type .
Proof. Let be a sequence of positive numbers for which . Let be a weakly null sequence in . Set and . Then there exists such that Since the fact that is a weakly null sequence implies that , coordinatewise, there is an such that for all . Set . Then there exists an such that By using the fact that coordinatewise, there exists an such that for all . Continuing this process, we can find by induction two increasing subsequences and of natural numbers such that for all and where . Hence, On the other hand, since , it can be easily seen that . Therefore, and Hence we obtain By using the inequality for all and , we have Therefore, the space has the Banach-Saks type , which completes the proof of the theorem.
Let us define the matrix by for all , where depends only on and depends only on . The matrix is called generalized weighted mean or factorable matrix. By , we denote the inverse of the matrix as follows:
Theorem 7. For , by (39), one has the fact that the Gurariǐ modulus of convexity for the normed space satisfies the inequality for any .
Proof. Let . By using (39), we have Let . Then using (40), let us consider the following sequences: Since and , we have By using the sequences given above, we obtain the following equalities: To complete the upper estimate of the Gurariǐ modulus of convexity, it remains to calculate the infimum of for . We have Consequently, we get for the inequality which is the desired result.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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