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Journal of Operators
Volume 2013 (2013), Article ID 904838, 7 pages
Some Properties of the Sequence Space
School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir 182 320, India
Received 29 March 2013; Revised 6 June 2013; Accepted 6 June 2013
Academic Editor: Palle E. Jorgensen
Copyright © 2013 Kuldip Raj and Sunil K. Sharma. 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 the sequence space defined by a Musielak-Orlicz function . We also study some topological properties and prove some inclusion relations involving this space.
1. Introduction and Preliminaries
The concept of 2-normed spaces was initially developed by Gähler  in the mid-1960s, while one can see that of -normed spaces in Misiak . Since then, many others have studied this concept and obtained various results; see Gunawan [3, 4] and Gunawan and Mashadi . Let and be a linear space over the field , where is the field of real or complex numbers of dimension , where . A real valued function on satisfying the following four conditions: (1) if and only if are linearly dependent in ; (2) is invariant under permutation; (3) for any ; and (4)is called an -norm on , and the pair is called an -normed space over the field .
For example, we may take being equipped with the -norm = the volume of the -dimensional parallelopiped spanned by the vectors which may be given explicitly by the formula where for each . Let be an -normed space of dimension and be linearly independent set in . Then the following function on defined by defines an -norm on with respect to .
A sequence in an -normed space is said to converge to some if
A sequence in an -normed space is said to be Cauchy if If every Cauchy sequence in converges to some , then is said to be complete with respect to the -norm. Any complete -normed space is said to be -Banach space.
An Orlicz function is a function which is continuous, nondecreasing, and convex with , for and as .
Lindenstrauss and Tzafriri  used the idea of Orlicz function to define the following sequence space. Let be the space of all real or complex sequences ; then which is called an Orlicz sequence space. The space is a Banach space with the norm
It is shown in  that every Orlicz sequence space contains a subspace isomorphic to . The -condition is equivalent to for all values of and for . A sequence of Orlicz function is called a Musielak-Orlicz function; see [7, 8]. A sequence defined by is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz function , the Musielak-Orlicz sequence space and its subspace are defined as follows: where is a convex modular defined by
We consider equipped with the Luxemburg norm or equipped with the Orlicz norm
Let be a linear metric space. A function : is called paranorm if (1) for all , (2) for all , (3) for all , (4)if is a sequence of scalars with as , and is a sequence of vectors with as ; then as .
A paranorm for which implies is called total paranorm, and the pair is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [9, Theorem , p. 183]). For more details about sequence spaces, see [10–23] and the references therein.
Let be an infinite series with sequence of partial sums and be any positive real number. The Euler transform of the sequence is defined by
The series is said to be summable to the number if and is said to be absolutely summable or summable if
Let be a sequence of scalars; we denote , where is defined by (12). After applications of Abel's transform, we have where
Note that, for any sequences , and scalar , we have and .
Let be a Musielak-Orlicz function and be a bounded sequence of positive real numbers. We define the following sequence space in the present paper: If we take for all , we have If we take , we get
The following inequality will be used throughout the paper. If , , then for all and . Also, for all .
In this paper, we study some topological properties and inclusion relations between the previously defined sequence spaces.
2. Main Results
Theorem 1. Let be a Musielak-Orlicz function and be a bounded sequence of positive real numbers; the space is linear over the complex field .
Proof. Let , and . Then, there exist positive integers and such that Let . Since is nondecreasing, convex function and by using inequality (20), we have Therefore, . Hence, is a linear space.
Theorem 2. Let be a Musielak-Orlicz function and be a bounded sequence of positive real numbers. Then, the space is a paranormed space with where .
Proof. Clearly, and . Since , we get for . Finally, we prove that multiplication is continuous. Let be any number such that
Thus, we have
where . Since , then . Hence,
and therefore converges to zero when converges to zero in . Now, suppose that as and in . For arbitrary , let be a positive integer such that
for some . This implies that
Let ; then, using convexity of , we get
Since is continuous everywhere in , then is continuous at . So there is such that for . Let be such that for ; we have
Hence, as . This completes the proof of the theorem.
Theorem 3. If and are two Musielak-Orlicz functions and are nonnegative real numbers, then one has(i), (ii)If , then ,(iii)If and are equivalent, then .
Proof. It is obvious, so we omit the details.
Theorem 4. Suppose that for each . Then .
Proof. Let . Then there exists some such that
This implies that for sufficiently large value of , say , for some fixed . Since is nondecreasing, we get
Theorem 5. (i) If for each , then .
(ii) If for all , then .
Proof. It is easy to prove by using Theorem 4; so we omit the details.
3. Statistical Convergence
The notion of statistical convergence was introduced by Fast  and Schoenberg  independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, and number theory. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy , Connor , Šalát , Mursaleen , Işik , Savaş , Malkowsky and Savas , Kolk , Maddox , Mohiuddine and Aiyub , and many others. In the recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Cech compactification of natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.
Let be the sequence of positive integers such that , and as . Then is called a lacunary sequence. The intervals determined by will be denoted by , and the ratio will be denoted by (see ).
Let be a lacunary sequence; then the sequence is said to be -lacunary statistically convergent to the number provided that, for every , In this case, we write or .
Let be a lacunary sequence, be a Musielak-Orlicz function, be a sequence of strictly positive real numbers, and be a bounded sequence of positive real numbers. A sequence is said to be strongly -lacunary convergent to the number with respect to the Musielak-Orlicz function provided that
The set of all strongly -lacunary convergent sequences to the number with respect to the Musielak-Orlicz function is denoted by . In this case, we write . In the special case , for all , we shall write instead of .
In this section, we give some results about -lacunary statistical convergence and give some relations between the set of -lacunary statistical convergence sequences and other spaces which are defined with respect to Musielak-Orlicz function .
Theorem 6. Let be a lacunary sequence.(i)If a sequence is strongly -lacunary convergent to , then it is -lacunary statistically convergent to .(ii)If a bounded sequence is -lacunary statistically convergent to , then it is strongly -lacunary convergent to .
Proof. (i) Let and . Then, we can write
(ii) Suppose that , and let . Let be given and take such that for all and set , where . Now, for all , we have
Thus, . This completes the proof of the theorem.
Theorem 7. For any lacunary sequence , if , then .
Proof. If , then there exists a such that for sufficiently large . Since , we have . Let . Then, for every , we have
Theorem 8. Let be a lacunary sequence, be a Musielak-Orlicz function, and . Then, .
Proof. Let . Then there exists a number such that
Then, given , we have
Hence, . This completes the proof of the theorem.
Theorem 9. Let be a lacunary sequence, be a Musielak-Orlicz function, and be a bounded sequence; then .
Proof. Let with . Since , there is a constant such that , and given , we have
The authors thank the referees for their valuable comments and suggestions.
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