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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 509613, 8 pages
On a New Space of Double Sequences
Department of Mathematics, Art and Science Faculty, Ondokuz Mayıs University, Kurupelit Campus, Samsun, Turkey
Received 3 May 2013; Accepted 17 June 2013
Academic Editor: Józef Banaś
Copyright © 2013 Cenap Duyar and Oğuz Oğur. 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.
We introduce a new space of double sequences related to -absolute convergent double sequence space, combining an Orlicz function and an infinity double matrix. We study some properties of and obtain some inclusion relations involving .
Throughout this work, and denote the set of positive integers and complex numbers, respectively. A complex double sequence is a function from into and briefly denoted by . Throughout this work, and denote the spaces of single complex sequences and double complex sequences, respectively. If, for all , there is such that where and , then a double sequence is said to be converging (in terms of Pringsheim) to . A real double sequence is nondecreasing, if for . A double series is infinity sum , and its convergence implies the convergence by of partial sums sequence , where (see [1–4]).
A double sequence space is said to be solid if for all double sequences of scalars such that for all whenever .
Now let be a family of subsets having most elements in . Also denotes the class of subsets in such that the element numbers of and are most and , respectively. Besides is taken as a nondecreasing double sequence of the positive real numbers such that An Orlicz function is a function which is continuous, nondecreasing, and convex with , for , and as .
An Orlicz function can always be represented in the following integral form: , where is known as the kernel of , is right differentiable for , , for , is nondecreasing, and as .
An Orlicz function is said to be satisfied -condition for all values of , if there exists a constant such that for all .
Lindenstrauss and Tzafriri  used the idea of Orlicz functions to construct Orlicz sequence space The sequence space is a Banach space according to the norm defined by This space is called an Orlicz sequence space. The space is closely related to the space , which is an Orlicz sequence space with for .
The space , introduced by Sargent in , is in the form Sargent studied some properties of this space and examined relationship between this space and -space. Similar sequence classes were studied by many mathematicians using Orlicz functions (see [15–17]).
Later on, this space was investigated from sequence space point of view by Rath , Rath and Tripathy , Tripathy and Sen , Tripathy and Mahanta , and others. Recently Altun and Bilgin  introduced and studied the following sequence space .
Let be an infinite matrix of complex numbers, an Orlicz space, and a bounded sequence of positive real numbers such that . Then the space is defined by where if converges for each .
Let be a double sequence. A set is defined by
If for all , then is said to be symmetric.
In this work, we introduce the following sequence space.
Let be an infinite double matrix of complex numbers, an Orlicz function and bounded double sequence of positive real numbers such that . Then the space is defined by where if converges for each .
Also, we introduce and investigate the following space:
In this work, we also use the following sequence spaces:
The following inequality will be used throughout this paper: where and.
2. Main Results
Definition 1. Let be a set of increasing positive integer binaries, namely, if and only if and , and be a double sequence space. A -set space is a double sequence space, defined by The canonical preimage of a double sequence is a double sequence with The canonical preimage of a set space is a set of canonical preimages of all elements in .
Definition 2. If a double sequence space contains the canonical preimages of all set spaces, then is said to be monotone.
The following lemma is an easy result of the definitions.
Lemma 3. If a double sequence space is solid, then is monotone.
Proposition 4. The space is a -linear space.
Proof. Let , be in and , in . Then there exist positive numbers and such that Let . Using that is nondecreasing convex function, we have Thus we can write This shows that . Hence is a linear space.
Proposition 5. The space is a paranormed space with the paranorm
Proof. It is clear that and if . If there are and such that then where . This shows that . Using this triangle inequality we can write Separately, we obtain Hence where and as . Consequently is a paranom on .
Proposition 6. The class of double sequences is solid.
Proof. Let be a double sequence of scalars such that and . Then we can write
This implies that , and hence the class is solid.
Corollary 7. The space is monotone.
Theorem 8. Let be another double sequence like . Then if and only if .
Proof. Let . Then for all . If , then
for some . Thus
for some , and hence . This shows that .
Conversely, let . We say for all and suppose . Then there exists a subsequence of such that . If , then we have
This is a contradiction as . This completes the proof.
Corollary 9. if and only if and .
Theorem 10. Let , , and be Orlicz functions satisfying -condition. Then (a),(b).
Proof. (a) Let . Then there exists such that
By the continuity of , we select a number with such that , whenever , for arbitrary . Now let . We can write
By the properties of , we have
Again we can write for . If we use that satisfies -condition, then we find and so Hence we get
Finally, we have , and hence .(b) Let . Then there exists such that By the inequality we have .
Theorem 11. (a). (b) if and only if .(c) if and only if .
Proof. (a) Let , and let a set be defined as follows:
Since is a nondecreasing double sequence, is a nonincreasing double sequence. So we obtain for all and hence Thus we have .
Conversely if , then it is clear that for all , and hence This shows that if , then . Thus we have .(b) It is clear that where for all . Then we can write . By Theorem 8, we have , and according to alternative (a) (c) Firstly we show that if for all . Let . Then for and , we can find some such that This gives the inclusion . Conversely let . Then for , we can find some such that This shows that . By Theorem 8 and alternative (a), we can write if and only if . This completes the proof.
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