Research Article | Open Access
A New Difference Sequence Set of Order and Its Geometrical Properties
We introduce a new class of sequences named as and, for this space, we study some inclusion relations, topological properties, and geometrical properties such as order continuous, the Fatou property, and the Banach-Saks property of type .
1. Introduction, Definitions, and Preliminaries
By we denote the space of all complex (or real) sequences. If , then we simply write instead of . We will write , , and for the sequence spaces of all bounded, convergent, and null sequences, respectively. Also by and we denote the spaces of all absolutely summable and -absolutely summable series, respectively.
The notion of difference sequence spaces was generalized by Et and Çolak  such as , for , , and . They showed that these sequence spaces are -spaces with the norm where , , , , and . Recently difference sequences and related concepts have been studied in ([2–13]) and by many others.
Let be a sequence space. Then is called(i)solid (or normal) if for all sequences of scalars with for all , whenever ,(ii)symmetric if implies , where is a permutation of ,(iii)monotone provided contains the canaonical preimages of all its step spaces,(iv)sequence algebra if , whenever .
It is well known that if is normal then is monotone.
Throughout this paper denotes the class of all subsets of ; those do not contain more than elements. Let be a nondecreasing sequence of positive numbers such that for all . The class of all sequences is denoted by . The sequence space was introduced by Sargent  and he studied some of its properties and obtained some relations with the space . Later on it was investigated by Tripathy and Sen  and Tripathy and Mahanta .
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 Fréchet space 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 -space; 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 .
We say that is order continuous if for any sequence in such that for each and () we have that holds.
A Köthe sequence space is said to be order continuous 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 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
2. Inclusion and Topological Properties of the Space
In this section we introduce a new class of sequences and establish some inclusion relations. Also we show that this space is not perfect and normal.
Let be a fixed positive integer, any real number, and a positive real number such that . Now we define the sequence space as In the case , we will write instead of and in the special case and we will write instead of .
The proof of each of the following results is straightforward, so we choose to state these results without proof.
Theorem 1. Let , , and let be a positive real number such that . Then the sequence space is a -space normed by
Theorem 2. Let , , and let be a positive real number such that ; then .
Theorem 3. Let and be fixed real numbers such that and a positive real number such that ; then .
Theorem 4. Let and be fixed real numbers such that and a positive real number such that . For any two sequences and of real numbers such that . Then if and only if .
Proof. Let and . Then
and there exists a positive number such that and so that for all . Therefore for all we have
Now taking supremum over and we get
and so .
Conversely let and suppose that . Then there exists an increasing sequence of naturals numbers such that . Let , where is the set of positive real numbers; then there exists such that for all . Hence and so . Then we can write for all . Now taking supremum over and we get Since (9) holds for all (we may take the number sufficiently large), we have when with Hence . This contradicts to . Hence .
The following results are derivable easily from Theorem 4.
Corollary 5. Let and be fixed real numbers such that and a positive real number such that . For any two sequences and of real numbers such that . Then one has(i) if and only if ,(ii) if and only if ,(iii) if and only if .
Theorem 6. Consider and the inclusion is strict.
Proof. It follows from Minkowski’s inequality. To show the inclusion is strict, let for all , , , and ; then .
Theorem 7. The sequence space is solid and hence monotone, but the sequence space is neither solid nor symmetric and sequence algebra for .
Proof. Let and be sequences such that for each . Then we get Hence is solid and hence monotone. To show the space is normal, let , for all , , , and , then ; but when for all . Hence is not solid. The other cases can be proved on considering similar examples.
Theorem 8. Consider
Proof. It is omitted.
Theorem 9. If , then .
Proof. Proof follows from the following inequality:
3. Geometrical Properties of the Space
In this section, we study some geometrical properties of the space . Some of these geometrical properties are the order continuous, the Fatou property, and the Banach-Saks property of type . Let us start with the following theorem.
Theorem 10. The space is order continuous.
Proof. To prove this theorem, we have to show that is an -space. It is easy to see that contains which is the space of real sequences which have only a finite number of nonzero coordinates. By using definition of -properties, we have that has a unique representation ; that is, as , which means that has . Hence, since -space containing has -property, the space is order continuous.
Theorem 11. The space has the Fatou property.
Proof. Let be any real sequence from and any nondecreasing sequence of nonnegative elements from such that as coordinatewisely and .
Let us denote . Then, since the supremum is homogeneous, we have Moreover, by the assumptions that is nondecreasing 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 nondecreasing, we obtain that the sequence is bounded from above by . As a result, , which together with the opposite inequality proved already yields that .
Theorem 12. The space has the Banach-Saks property of the type .
Proof. It can be proved with standard technic.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2014 Mikail Et and Vatan Karakaya. 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.