Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 1. pp. 109-121, 2011 Applied Mathematics
A New Double Sequence Space Defined by a Modulus Function N.Subramanian1, S.Krishnamoorthy2, S. Balasubramanian3
1Department of Mathematics,SASTRA University, Thanjavur-613 401, India
e-mail: nsm aths@ yaho o.com
23Department of Mathematics, Govenment Arts College(Autonomus),
Kumbakonam-612 001, India
e-mail: drsk_ 01@ yaho o.com2,sbalasubram anian2009@ yaho o.com3
Received Date: June 10, 2010 Accepted Date: October 19, 2010
Abstract. The idea of single difference sequence space was introduced by Kizmaz[31] and this concept was generalized by various authors. In this paper generalized we define the double difference space 2(∆ ) on a semi normed complex linear space by using modulus function and we give various properties and some inclusion relations on this space. Further more we study some of its properties solidity, etc.
Key words: Gai sequence; analytic sequence; modulus function; semi norm; difference sequence; double sequence; duals.
2000 Mathematics Subject Classification. 40A05, 40C05, 40D05. 1.Introduction
Throughout and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively.
We write 2 for the set of all complex sequences () where ∈ N the set of positive integers. Then, 2 is a linear space under the coordinate wise addition and scalar multiplication.
Some initial works on double sequence spaces is found in Bromwich[4]. Later on, they were investigated by Hardy[6], Moricz[10], Moricz and Rhoades[11], Basarir and Solankan[2], Tripathy[18], Colak and Turkmenoglu[5], Turkmenoglu[20], and many others.
Let us define the following sets of double sequences: M() := n () ∈ 2: ∈|| ∞ o C() := n () ∈ 2: − →∞|− |= 1 ∈ C o C0() := n () ∈ 2: − →∞|| = 1 o L() := n () ∈ 2:P∞=1 P∞ =1|| ∞ o C() := C()TM() and C0() = C0()TM();
where = () is the sequence of strictly positive reals for all ∈ N and − →∞ denotes the limit in the Pringsheim’s sense. In the case = 1 for all ∈ N; M() C() C0() L() C() and C0() reduce to the sets M C C0 L Cand C0 respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. G¨khan and Colak [22,23] have proved that M() and C() C() are com-plete paranormed spaces of double sequences and gave the − − − duals of the spaces M() and C() Quite recently, in her PhD thesis, Zelter [24] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [25] have recently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Ces`ro sum-mable double sequences. Nextly, Mursaleen [26] and Mursaleen and Edely [27] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the −core for double sequences and determined those four dimensional matrices transform-ing every bounded double sequences = () into one whose core is a subset of the −core of More recently, Altay and Basar [28] have defined the spaces BS BS () CS CS CS and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M M() C C C and L respectively, and also examined some properties of those sequence spaces and determined the − duals of the spaces BS BV CSand the () − duals of the spaces CS and CS of double series. Quite recently Basar and Sever [29] have introduced the Banach space L of double sequences corresponding to the well-known space of single sequences and examined some properties of the space L Quite recently Subramanian and Misra [30] have studied the space 2
( ) of double sequences and gave some inclusion relations. We need the following inequality in the sequel of the paper. For ≥ 0 and 0 1 we have
(1) ( + )≤ +
The double series P∞=1 is called convergent if and only if the double sequence () is convergent, where =P=1( ∈ N) (see[1]).
A sequence = ()is said to be double analytic if ||1+ ∞ The vector space of all double analytic sequences will be denoted by Λ2. A sequence = () is called double gai sequence if (( + )! ||)1+ → 0 as → ∞ The double gai sequences will be denoted by 2. Let = {}
Consider a double sequence = () The ( ) section [] of the se-quence is defined by []=P
=0= for all ∈ N ; where = denotes the double sequence whose only non zero term is a 1
(+)! in the ( )
place for each ∈ N
An FK-space(or a metric space) is said to have AK property if (=) is a Schauder basis for . Or equivalently []→ .
An FDK-space is a double sequence space endowed with a complete metriz-able; locally convex topology under which the coordinate mappings = () → ()( ∈ N) are also continuous.
Orlicz[14] used the idea of Orlicz function to construct the space ¡¢ Lin-denstrauss and Tzafriri [8] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space contains a subspace iso-morphic to (1 ≤ ∞) subsequently, different classes of sequence spaces were defined by Parashar and Choudhary [15], Mursaleen et al. [12], Bektas and Altin [3], Tripathy et al. [19], Rao and Subramanian [16], and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [7]. Recalling [14] and [7], an Orlicz function is a function : [0 ∞) → [0 ∞) which is continuous, non-decreasing, and convex with (0) = 0 () 0 for 0 and () → ∞ as → ∞ If convexity of Orlicz function is replaced by subadditivity of then this function is called modulus function, defined by Nakano [13] and further discussed by Ruckle [17] and Maddox [9], and many others.
An Orlicz function is said to satisfy the ∆2− condition for all values of if there exists a constant 0 such that (2) ≤ () ( ≥ 0) The ∆2− condition is equivalent to () ≤ () for all values of and for 1
Lindenstrauss and Tzafriri [8] used the idea of Orlicz function to construct Orlicz sequence space
= n ∈ :P∞=1 ³ || ´ ∞ 0o
The space with the norm
kk = n 0 :P∞=1³||
´
≤ 1o
becomes a Banach space which is called an Orlicz sequence space. For () = (1 ≤ ∞) the spaces
coincide with the classical sequence space If is a sequence space, we give the following definitions:
(i) 0= the continuous dual of ; (ii) =© = ( ) :P∞=1|| ∞ ∈ ª ; (iii) =© = () :P∞=1 ∈ ª; (iv) =n = ( ) : ≥ 1 ¯ ¯ ¯P=1 ¯ ¯ ¯ ∞ ∈ o; (v) − ⊃ ; =n (= ) : ∈ 0o ; (vi) =n = ( ) : ||1+ ∞ ∈ o ;
are called − (¨− )dual of −( − ¨ − ) − − is de-fined by Gupta and Kamptan [21]. It is clear that ⊂ and ⊂ but ⊂ does not hold, since the sequence of partial sums of a double convergent series need not to be bounded.
The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [31] as follows
(∆) = { = () ∈ : (∆) ∈ }
for = 0 and ∞ where ∆ = − +1 for all ∈ N Here 0 and ∞ denote the classes of all, convergent,null and bounded sclar valued single sequences respectively. The above spaces are Banach spaces normed by
kk = |1| + ≥1|∆|
Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by
(∆) =© = () ∈ 2: (∆) ∈ ª where = Λ2 2and ∆ = (− +1) − (+1− +1+1) = − +1− +1+ +1+1for all ∈ N Therefore ∆0 = ( ) ; ∆= (− +1) − (+1− +1+1) Hence ∆= ∆∆−1=¡∆−1− ∆−1+1¢− ¡ ∆−1 +1− ∆−1+1+1 ¢
2. Definitions and Preliminaries
Throughout the article 2 denotes the spaces of all sequences. 2
and Λ2 denote the Pringscheims sense of double Orlicz space of gai sequences and Pringscheims sense of double Orlicz space of bounded sequences respecctively. The notion of a modulus function was introduced by Nakano [15]. We recall that a modulus is a function from [0 ∞) → [0 ∞) such that
(1) () = 0 if and only if = 0
(2) ( + ) ≤ () + () for all ≥ 0 ≥ 0 (3) is increasing,
(4) is continuous from the right at o. Since | () − ()| ≤ (| − |) it follows from condition (iv) that is continuous on [0 ∞)
Let = () be a sequence of strictly positive real numbers and ≥ 0 Let be semi normed space over the field C of complex numbers with the semi norm The symbol 2() denotes the space of all sequences defined over such that 0 for all and = ∞
Define the sets : 2 = n ∈ 2:³³((+)!||)1+ ´´ → 0 → ∞ 0o and Λ2 = n ∈ 2: ≥1 ³ ³||1+ ´´ ∞ 0o The space Λ2
is a metric space with the metric ( ) = ½ 0 : ≥1 ³ ³|−| ´´1+ ≤ 1 ¾ The space 2
is a metric space with the metric e ( ) = ½ 0 : ≥1 ³ ³(+)!|−| ´´1+ ≤ 1 ¾
We define the following sequence spaces as follows: Now we define the following sequence spaces: 2(∆ ) = n ∈ 2() : ()−³³ (( + )! |∆|)1+ ´´ → → 0 ( → ∞) ≥ 0} Λ2(∆ ) = n ∈ 2() : sup()− ³ ³ (|∆|)1+ ´´o ∞ ≥ 0}
where is a modulus function. The following inequality will be used through this article. Let = () be a sequence of positive real numbers with 0 ≤ = = ¡ 1 2−1¢ Then, for ∈ C we have (2) |+ | ≤ {||+ ||} 3. Definitions
Definition 3.1. Let p,q be semi norms on a vector space . Then p is said to be stronger that q if whenever () is a sequence such that () → 0 then also () → 0 If each is stronger than the others, the and are said to be equivalent.
Lemma 3.1. Let and be semi norms on a linear space Then is stronger than if and only if there exists a constant such that () ≤ () for all ∈
Definition 3.2.
(1) A sequence space is said to be solid or normal if () ∈ whenever () ∈ and for all sequences of scalars () with || ≤ 1 for all ∈ N
(2) Symmetric if () ∈ implies (()) ∈ where () is a permu-tation of N × N;
(3) Sequence algebra if · ∈ whenever ∈
Definition 3.3. A sequence space is said to be monotone if it contains the canonical pre-images of all its step spaces.
Remark 3.1. From the two above definitions it is clear that a sequence space is solid implies that is monotone.
Definition 3.4. A sequence is said to be convergence free if () ∈ whenever () ∈ and = 0 implies that = 0
4. Main Results
In this sectionwe will give some results on the sequence space 2(∆ ) those characterize the structure of the space 2(∆ ) .
Theorem 4.1. The sequence space 2(∆ ) is a inear space over C Proof. Let ∈ 2(∆ ) For ∈ C there exists positive integers and such that || ≤ and || ≤ Since is subadditive, is a seminorm, and ∆ is linear.
()−h³ (( + )! |∆(+ )|) 1 + ´i ≤ ³³1 || ´´ ()−h³ (( + )! |∆ |) 1 + ´i + ³³1 || ´´ ()−h³ (( + )! |∆ |) 1 + ´i → 0 as → ∞ Hence, 2(∆ ) is a linear space.
Theorem 4.2. The space 2(∆ ) is a paranormed space (not totally paranormed) paranormed by ∆() = n ()−h³ (( + )! |∆ |)1+ ´i → 0 → ∞o 1 where = ∞ and = (1 ) Proof. Clearly ∆() = 0 and ∆() = ∆(−)
where = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and is the zero element of It also
fol-lows from equation (2), Minknowski’s inequality and the definition of that ∆ is sub additive. Now for a complex number by inequality
||
≤ ³1 ||´ and the definition of modulus we have
∆() = n ()−h³³ (( + )! |∆ |)1+ ´´i → 0 → ∞o 1 ≤ (1 + [||]) · ∆()
where [||] denotes the integer part of hance → 0 → imply → and also → fixed imply →
Now suppose → 0 and is a fixed point in 2(∆ ) Given 0 let = + 1 and = + 1 be such that
n ()−h³ (( + )! |∆ |)1+ ´i → 0 as = + 1 and = + 1 → ∞} ¡2 ¢ Hence we have n ()−h³ (( + )! |∆ |)1+ ´i → 0 as = + 1 and = + 1 → ∞}1 ¡ 2 ¢ Since is continous on [0 ∞]
() =n()−h³ (( + )! |∆|)1+ ´i
→ 0 as = and = → ∞}
is continous at zero. Therefore, there exists 0 1 such that || implies n ()−h³³(( + )! |∆|)1+ ´´i → 0 as = and = → ∞} 2 for N Hence n ()−h³³(( + )! |∆|)1+ ´´i → 0 as → ∞o 1 for N Therefore () → 0 → 0 This completes the proof.
Theorem 4.3. Let 1 and 2 be modulus functions, 1 and 2 be semi-norms, and 1 and 2≥ 0 real numbers.
(1) 2(∆ 1 ) ⊆ 2(∆ ◦ 1 )
(2) 2(∆ 1 )T2(∆ 2 ) ⊆ 2(∆ 1+ 2 ) (3) 2(∆
1 )T2(∆ 2 ) ⊆ 2(∆ 1+ 2 ) (4) If 1 is stronger than 2 then 2(∆ 1 ) ⊆ 2(∆ 2 ) (5) If 1≤ 2 then 2(∆ 1) ⊆ 2(∆ 2)
Proof (1). Let () ∈ 2(∆ 1 ) Let 0 and choose with 0 1 such that () for 0 ≤ ≤ Write
= nh 1 ³ ³(( + )! |∆ |)1+ ´´i → 0 → ∞o and consider³ ()−[ ()]→ 0 → ∞ ´ = =³()−[ ()] → 0 → ∞ ´ + ³ ()−[ ()] → 0 → ∞ ´
where the first term is over ≤ and the second term over Since is continuous, we have
(3) ³
()−[ ()] → 0 → ∞ ´
max¡1 ¢ ³()−→ 0 → ∞´ and for we use the fact that
1 +£¯¯ ¯ ¯¤ By the definition of we have for
() ≤ (1) £ 1 +¡ ¢¤ ≤ 2 (1)
³ ()−[ ()]→ 0 → ∞ ´ ≤ µ 1³2 (1) ´¶ ³()−[()] → 0 → ∞ ´ By (3) and last equation we have 2(∆
1 ) ⊆ 2(∆ ◦ 1 ) Proof (2).Let = () ∈ 2(∆ 1 )T2(∆ 2 ) Then using (2) it can be shown that () ∈ 2(∆ 1+ 2 ) Hence
2(∆
1 )T2(∆ 2 ) ⊆ 2(∆ 1+ 2 )
Proof (3). The proof of (3) is similar to the proof of (2), by using the inequality ³ ()−h (1+ 2) (( + )! |∆|)1+ ´i ≤ ³()−h (1) (( + )! |∆|)1+ ´i + ³()−h (2) (( + )! |∆|)1+ ´i where = ¡1 2−1¢ Proof (4) and (5) follows easily. We get the following sequence spaces from 2(∆ ) by choosing some of the special
For () = () we get 2(∆ ) = n ∈ 2() : ()−³³ (( + )! |∆ |)1+ ´´ → 0 ( → ∞) ≥ 0o; for = 1 for all we get 2(∆ ) =
n ∈ 2() : ()−³³ (( + )! |∆ |)1+ ´´ → 0 ( → ∞) ≥ 0o; for = 0 we get 2(∆ ) = n ∈ 2() :³³ (( + )! |∆ |)1+ ´´ → 0 ( → ∞)o; for () = () and = 0 we get
2(∆ ) = n ∈ 2() :³³ (( + )! |∆ |)1+ ´´ → 0 ( → ∞)o; for = 1 for all and = 0 we get
2(∆ ) = n
∈ 2() :³³ (( + )! |∆|)1+ ´´
→ 0 ( → ∞)o; for () = = 1 for all and = 0 we gave
2(∆ ) = n ∈ 2() :³³ (( + )! |∆|)1+ ´´ → 0 ( → ∞)o Corollary 4.1.
(1) If 1 then for any modulus we have 2(∆ ) ⊆ 2(∆ ) ; (2) If 1and 2are equivalent semi norms then 2(∆ 1 ) ⊆ 2(∆ 2 ) ; (3) 2(∆ ) ⊆ 2(∆ ) ;
(4) 2(∆ ) ⊆ 2(∆ ) ; (5) 2(∆ ) ⊆ 2(∆ )
Proof (1). If 1() = in Theorem 4.3(1), then the result follows easily. Proof (2). It follows from Theorem 4.3(4).
Proof (3). If we take 1= 0 and 2 = in the Theorem 4.3(5), then we get 2(∆ ) ⊆ 2(∆ )
Proof (4). If we take 1 = 0 2= and () = in Theorem 4.3(5), then we get 2(∆ ) ⊆ 2(∆ )
Proof (5). If we take 1= 0 2= and = 1 for all in Theorem 4.3 (5), then 2(∆ ) ⊆ 2(∆ )
Theorem 4.4.2¡∆−1 ¢⊂ 2(∆ ) for ≥ 1 and the inclusion is strict.
Proof. Let ∈ 2¡∆−1 ¢ Then we have (4) ()−³³¡( + )!¯¯∆−1
¯
¯¢1+´´
→ 0 ( → ∞)
Since ( + 1 + 1)−≤ ()−≤ 2( + 1 + 1)− for all ∈ N we get the following inequality ()−³³¡( + )!¯∆¯ −1+1+1¯¯¢1+´´ ≤ 2( + 1 + 1)−³³¡( + )!¯¯∆−1
+1+1 ¯
¯¢1+´´ From (4) and last equation together imply that
(5) ()−³³¡( + )!¯¯∆−1+1+1 ¯
¯¢1+´´
→ 0 ( → ∞)
Since is increasing, ( + ) ≤ () + () and is a semi norm, from (4) and (5) we get
()−³³ (( + )! |∆ |)1+ ´´ = ()−¡¡¡( + )!¯¯∆−1− ∆−1+1− ∆−1+1+ +∆−1+1+1¯¯1+´´ ≤ ()−³³¡( + )!¯¯∆−1 ¯ ¯¢1+´´ + + ()−³³¡( + )!¯¯∆−1+1 ¯ ¯¢1+´´ + + ()−³³¡( + )!¯∆¯ −1+1¯¯¢1+´´+ + ()−³³¡( + )!¯¯∆−1 +1+1 ¯ ¯¢1+´´ → 0 → ∞ Thus 2¡∆−1 ¢⊂ 2(∆ ) This completes the proof. Theorem 4.5.
(1) Let 0 ≤ ≤ ∞ for each ∈ N Then 2(∆ ) ⊆ 2(∆ )
(2) 2(∆ ) ⊆ 2(∆ ) (3) 2(∆ ) ⊆ 2(∆ )
Proof (1). If ∈ 2(∆ ) then for all sufficiently large ³ ³ (( + )! |∆|)1+ ´´ ≤ 1 and so ³ ³ (( + )! |∆|)1+ ´´ ≤³³ (( + )! |∆|)1+ ´´ This completes the proof.
Theorem 4.6.
(1) If 0 ≤ 1 for each ∈ N then 2(∆ ) ⊆ 2(∆ ) ; (2) If ≥ 1 for all ∈ N then 2(∆ ) ⊆ 2(∆ )
Proof (1). If we take = and = 1 for all ∈ N in Theorem 4.5 (i), then
2(∆ ) ⊆ 2(∆ )
Proof (2). If we take = and = 1 for all ∈ N in Theorem 4.5 (2), then
This completes the proof
Theorem 4.7. 2(∆ ) are not solid for 0
Proof. Tp prove that the space are not solid in general, consider the following example.
Example. Let = C () = () = || = (−1)+ = 0 = 1 for all ∈ N Then
(( + )! ||)
1
+ = () ∈ 2(∆ ) but
() ∈ 2(∆ )
Theorem 4.8. 2(∆ ) are not sequence algebra. Proof. Let () = || () = = 0 = 1 for all ∈ N Consider (( + )! ||) 1 + = ()−1and (( + )! | |) 1 + = ()−1 then ∈ 2(∆ ) and · ∈ 2(∆ ) References
1. T.Apostol, Mathematical Analysis, Addison-wesley , London, 1978.
2. M.Basarir and O.Solancan, On some double sequence spaces, J. Indian Acad. Math., 21(2) (1999), 193-200.
3. C.Bektas and Y.Altin, The sequence space ( ) on seminormed spaces,
Indian J. Pure Appl. Math., 34(4) (2003), 529-534.
4. T.J.I’A.Bromwich, An introduction to the theory of infinite series Macmillan and Co.Ltd. ,New York, (1965).
5. R.Colak and A.Turkmenoglu, The double sequence spaces2∞() 20()and2(), (to appear).
6. G.H.Hardy, On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19 (1917), 86-95.
7. M.A.Krasnoselskii and Y.B.Rutickii, Convex functions and Orlicz spaces, Gornin-gen, Netherlands, 1961.
8. J.Lindenstrauss and L.Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390.
9. I.J.Maddox, Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 100(1) (1986), 161-166.
10. F.Moricz, Extentions of the spaces and 0 from single to double sequences,
Acta. Math. Hungerica, 57(1-2), (1991), 129-136.
11. F.Moricz and B.E.Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104, (1988), 283-294.
12. M.Mursaleen,M.A.Khan and Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio Math. , Vol. XXXII (1999), 145-150.
13. H.Nakano, Concave modulars, J. Math. Soc. Japan, 5(1953), 29-49.
14. W.Orlicz,Uber Raume¡¢Bull. Int. Acad. Polon. Sci. A, (1936), 93-107.
15. S.D.Parashar and B.Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math. , 25(4)(1994), 419-428.
16. K.Chandrasekhara Rao and N.Subramanian, The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68(2004), 3755-3764.
17. W.H.Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25(1973), 973-978.
18. B.C.Tripathy, On statistically convergent double sequences, Tamkang J. Math., 34(3), (2003), 231-237.
19. B.C.Tripathy,M.Et and Y.Altin, Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Analysis and Applications, 1(3)(2003), 175-192.
20. A.Turkmenoglu, Matrix transformation between some classes of double sequences, Jour. Inst. of math. and Comp. Sci. (Math. Seri. ), 12(1), (1999), 23-31.
21. P.K.Kamthan and M.Gupta, Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York , 1981.
22. A.Gökhan and R.Colak, The double sequence spaces
2 ()and 2 (), Appl.
Math. Comput., 157(2), (2004), 491-501.
23. A.Gökhan and R.Colak, Double sequence spaces ∞
2 , ibid., 160(1), (2005),
147-153.
24. M.Zeltser, Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001. 25. M.Mursaleen and O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1), (2003), 223-231.
26. M.Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), (2004), 523-531.
27. M.Mursaleen and O.H.H. Edely,Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), (2004), 532-540.
28. B.Altay and F.Basar, Some new spaces of double sequences, J. Math. Anal. Appl., 309(1), (2005), 70-90.
29. F.Basar and Y.Sever, The spaceof double sequences, Math. J. Okayama Univ,
51, (2009), 149-157.
30. N.Subramanian and U.K.Misra, The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46, (2010).
