Inclusions Between the Spaces of Strongly Almost Convergent Sequences Defined by An Orlicz Function in A Seminormed Space

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1 Inclusions Between the Spaces of Strongly Alost Convergent Seuences Defined by An Orlicz Function in A Seinored Space Vinod K. Bhardwaj and Indu Bala Abstract The concept of strong alost convergence was introduced by addox in 1978 ath. Proc. Cab. Philos. Soc., ), 61-64] which has various applications. In this paper we introduce soe new seuence spaces which arise fro the notions of strong alost convergence and an Orlicz function in a seinored space. A new concept of unifor statistical convergence in a seinored space has also been introduced. Index Ters seuence space, Banach liit, strong alost convergence, Orlicz function. I. INTRODUCTION The first attepts to found a theory of seuence spaces and infinite atrices in the last two decades of the nineteenth century were otivated by probles in Fourier series, power series and systes of euations with infinitely any variables. The theory of seuence spaces has wide applications to several other branches of functional analysis, e.g., the theory of functions, suability theory, the theory of locally convex spaces. nuclear spaces and atrix transforations, and is indeed well developed to have a logic of its own. One of the typical probles concerning seuence spaces is the inclusion proble Abelian Theores) i.e. given seuence spaces S and T deterine whether S is contained in T. The ain purpose of this paper is to establish certain inclusion relations between the spaces of strongly alost convergent seuences defined by an Orlicz function in a seinored space. Let l, c and c 0 be the Banach spaces of bounded, convergent and null seuences x = x k ) with coplex ters, respectively, nored by x = sup k x k. A seuence x l is said to be alost convergent if all Banach liits of x coincide see Banach 1]). Lorentz 12] proved that x is alost convergent to a nuber l if and only if t kn = k + 1) 1 x i+n l as k uniforly in n. Let ĉ denote the space of all alost convergent seuences. Several authors including Lorentz 12], Duran 4] and King 8] have studied alost convergent seuences. addox 14] has defined x to be anuscript received arch 6, Vinod K. Bhardwaj is with the Departent of atheatics, Kurukshetra University, Kukshetra , INDIA. e-ail: vinodk bhj@rediffail.co Indu Bala is with Departent of atheatics, Governent College, Chhachhrauli Yauna Nagar) , INDIA. e-ail: bansal indu@rediffail.co strongly alost convergent to a nuber l if t kn x l ) = k + 1) 1 x i+n l as k uniforly in n. We denote the space of all strongly alost convergent seuences by ĉ] and the space of all seuences which are strongly alost convergent to zero by ĉ 0 ]. It is easy to sec that ĉ 0 ] ĉ] ĉ l. Das and Sahoo 3] extended the space ĉ] to the space w 1 ], where w 1 ] is the space defined in 3] as follows: w 1 ] = x : + 1) 1 t kn x l ) as uniforly in n, for soe l. It is obvious that ĉ] w 1 ] and ĉ] li x = w 1 ] li x = l. Lindenstrauss and Tzafriri 11] used the idea of an Orlicz function to construct the seuence space l ) of all xk seuences of scalars x k ) such that < k=1 for soe > 0. The space l euipped with the nor ) xk x = inf > 0 : 1 k=1 is a BK space 7, p. 300] usually called an Orlicz seuence space. The space l is closely related to the space l p which is an Orlicz seuence space with x) = x p, 1 p <. We recall 7], 11] that an Orlicz function is a function fro 0, ) to 0, ) which is continuous, non-decreasing and convex with 0) = 0, x) > 0 for all x > 0 and x) as x. Note that an Orlicz function is always unbounded. An Orlicz function is said to satisfy the 2 -condition for all values of u if there exists a constant K > 0 such that 2u) Ku), u 0. It is easy to see that always K > 2 10]. A siple exaple of an Orlicz function which satisfies the 2 -condition for all values of u is given by u) = a u α α > 1), since 2u) = a2 α u α = 2 α u). The Orlicz function u) = e u u 1 does not satisfy the 2 -condition. The 2 -condition is euivalent to the ineuality lu) Kl)u) which holds for all values of u, where l can be any nuber greater than unity. It is easy to see that is an Orlicz function when 1 and 2 are Orlicz functions, and that the function v v is a positive integer), the coposition of an Orlicz ISSN: Print); ISSN: Online)

2 function with itself v ties, is also an Orlicz function. If an Orlicz function satisfies the 2 -condition, then so does the coposite Orlicz function v. The following ineualities 13, p. 190] are needed throughout the paper. Let p = p i ) be a bounded seuence of positive real nubers. If H = sup i p i, then for any coplex a i and b i, a i + b i p i C a i p i + b i p i ), 1) where C = ax1, 2 H 1 ). Also for any coplex λ, λ pi ax1, λ H ). 2) Throughout the paper, X denotes a seinored space with seinor, p = p i ) is a seuence of positive real nubers, is an Orlicz function and wx) denotes the space of all X-valued seuences. We now introduce the following X-valued seuence spaces using an Orlicz function. ĉ, p, )] = x wx) : k + 1) 1 w 1, p, )] = l as k, uniforly in n, for soe l and > 0, x wx) : + 1) 1 k + 1) 1 l as, uniforly in n, for soe l and > 0, w, p, )] = x wx) : sup + 1) 1 k + 1) 1,n < for soe > 0 If we put l = 0 in w 1, p, )], then we obtain w 0, p, )]. If we take X = C, x) = x, x) = x and p i = 1 for all i, then w 1, p, )] = w 1 ] and w 0, p, )] = w 0 ]. We denote w 1, p, )], w 0, p, )] and w, p, )] as w 1 )], w 0 )] and w )] when x) = x and p i = 1 for all i. II. INCLUSION THEORES In this section we exaine soe algebraic properties of the seuence spaces defined above and investigate soe inclusion relations between these spaces. In order to discuss the properties of the seuence spaces, we assue that p i ) is bounded. Theore 2.1. ĉ, p, )], w 1, p, )], w 0, p, )] and w, p, )] are linear spaces over the coplex field C. The proof is a routine verification by using standard techniues and hence is oitted. Theore 2.2. Let, 1, 2 be Orlicz functions. Then i) if there is a positive constant β such that t) βt for all t 0, then z 1, p, )] z 0 1, p, )], ii) Z 1, p, )] Z 2, p, )] Z 1 + 2, p, )], where Z = w 0, w 1, w. Proof. We prove the theore for Z = w 0 and the other cases will follow siilarly. Let x w 0 1, p, g)] so that A n + 1) 1 1 k + 1) 1 as, uniforly in n and for soe > 0. Since t) βt for all t 0, + 1) 1 k + 1) 1 y i+n )] p i ax1 + β H ) + 1) 1 k + 1) 1 y i+n ] p i, )) where y i+n = 1 and hence x w 0 o 1, p, )]. iii) The proof is iediate using 1). Theore 2.3. Let h = inf p i > 0, be an Orlicz u/) function and if li > 0 for soe > 0, then u u/) Z, p, )] Zp, )], where Z = w 0, w 1 and w. Proof. We prove the theore for Z = w 1 and the other u/) cases will follow siilarly. If li > 0 then there u u/) exists a nuber α > 0 such that u/) αu/) for all u > 0 and soe > 0. Let x w 1, p, ))] )] so that + 1) 1 k + 1) 1 k pi l as, uniforly in n, for soe l X and > ) 1 k + 1) 1 l axα h, α H ) + 1) 1 l. k + 1) 1 Hence x w 1 p, )]. Theore 2.4. Let be an Orlicz function which satisfies 2 -condition, 1, 2 be seinors. Then i) w 0, p, 1 )] w 0, p, 2 )] w 0, p, )], ii) If there exists a constant L > 1 such that 2 x) L 1 x) for all x X, then w 0, p, 1 )] w 0, p, 2 )]. Proof. The proof of i) is straightforward using 1). ISSN: Print); ISSN: Online)

3 ii) Let x w 0, p, 1 )]. Since satisfies 2 - condition, + 1) 1 k + 1) ) 1 + 1) 1 k + 1) 1 L 1 KL) k + 1) 1 1 ax1, KL)) H ) + 1) 1 1 as, uniforly in n. k + 1) 1 Hence x w 0, p, 2 )]. Theore 2.5. For any Orlicz function, ĉ, p, )] w 1, p, g)]. ))] Proof. If k + 1) 1 k pi l as k uniforly in n, for soe l and > 0, then its arithetic ean also converges to 0 as uniforly in n. Although it sees likely that ĉ, p, )] is strictly contained in w 1, p, )], we have been unable to prove it. It is therefore an open uestion. Theore 2.6. If is an Orlicz function which satisfies 2 -condition and X is seinored algebra, then ĉ 0, p, )] is an ideal in l ), where l ) = x wx) : sup k x k ) <. Proof. Let x ĉ 0, p, )] and y l ). We show that xy ĉ 0, p, )]. Since y l ), there exists a constant T > 1 such that y k ) < T for all k. Since satisfies 2 -condition, there exists a constant K > 1 such that k + 1) 1 k + 1) 1 k + 1) 1 y i+n T ax1, KL)) H )k + 1) 1 ) )] pi y i+n ) as k, uniforly in n, for soe > 0. Thus xy ĉ 0, p, )] and the proof is coplete. III. THE SPACE OF ULTIPLIERS OF w, p, )] Suppose X, ) is seinored algebra. We define Sw, p, g)]), the space of ultipliers of w, p, )], as Sw, p, )]) = a wx) : a k x k ) w, p, )] for all x = x k ) w, p, )]. Theore 3.1. If satisfies the 2 -condition, then l ) Sw, p, )]). Proof. Let a = a k ) l ), T = sup k a k ) and x w, p, )]. Then sup,n + 1) 1 k + < for soe > 0. 1) 1 k Since satisfies the 2 -condition, there exists a constant K 1 > 1 such that + 1) 1 k + 1) 1 ai+n x i+n + 1) 1 + 1) 1 k + 1) 1 a i+n ) k + 1) T ]) K T ])) H + 1) 1 ) )] pi y i+n ) where T ] denotes the integer part of T. Hence a Sw, p, )]). k + 1) 1 IV. COPOSITE SPACE w 1 v, p, )] USING COPOSITE ORLICZ FUNCTION v Taking Orlicz function v instead of in the space w 1, p, )], we define the coposite space w 1 v, p, )] as follows: Definition 4.1. For a fixed natural nuber v, we define w 1 v, p, )] = x wx) : + 1) 1 v k + 1) 1 l as, uniforly in n, for soe l X and > 0. Theore 4.2. For any Orlicz function and v N, i) w 1 v, p, )] w 1 p, )] if there exists a constant α 1 such that t) αt for all t 0. ii) Suppose there exists a constant β, 0 < β 1 such that t) βt for all t 0 and let n, v N be ISSN: Print); ISSN: Online)

4 such that n < v, then w 1 p, )] w 1 n, p, )] w 1 v, p, )]. Proof. i) Since t) αt for all t 0 and is non-decreasing and convex, we have v t) α v t for each v N. Let x w 1 v, p, )]. Using 2), we have + 1) 1 k + 1) 1 x i+n l)] p i ax1, H ) ax1, α vh ) + 1) 1 k + 1) 1 v l and hence x w 1 p, )]. ii) Since t) βt for all t 0 and is nondecreasing and convex, we have n t) β n t for each n N. The first inclusion is easily proved by using 2). To prove the second inclusion, suppose that v n = s and let x w 1 n, p, )]. Again, using 2), we have + 1) 1 k + 1) 1 v l ax1, β sh ) + 1) 1 n k + 1) 1 l as, uniforly in n and hence x w 1 v, p, )]. Exaple t) = e t 1 t and 2 t) = t2 1 + t for all t 0 satisfy the conditions given in Theore 4.2i),ii) respectively. V. INCLUSION RELATION WITH UNIFOR STATISTICAL CONVERGENCE The idea of statistical convergence was introduced by Fast 5] and studied by various authors e.g., 2], 6], 9], 15], 21]). Although statistical convergence was introduced over nearly the last fifty years, it has becoe an active area of research in recent years. This concept has been applied in various areas such as approxiation theory 18], turnpike theory 16], 17], 20] etc. Definition 5.1 6]). The nuber seuence x = x k ) is said to be statistically convergent to l if for each ϵ > 0, li n n 1 k n : x k l ϵ = 0, where the vertical bars denote the cardinality of the set which they enclose. The set of all statistically convergent seuences is denoted by S. The concept, of unifor statistical convergence was introduced by Pehlivan and Fisher 19] as follows: Definition ]). The nuber seuence x is uniforly statistically convergent to 0 provided that for each ϵ > 0, li k + k 1) 1 ax 0 i k : x i+n ϵ = 0. The set of all uniforly statistically null seuences is denoted by S u0. We now introduce the following definition: Definition 5.3. A seuence x = x k ) in X is said to be uniforly statistically convergent to l X if for each ϵ > 0, li k + k 1) 1 ax 0 i k : x i+n l) ϵ = 0. We shall denote the set of all uniforly statistically convergent seuences by S u ). Theore 5.4. For any Orlicz function, S u ) l ) w 1, p, )] l ). Proof. Let x S u ) l ). Let y i+n = x i+n l. Since x )) l ), there exists K > 0 such that K for every > 0 and for all y i+n. Then for given ϵ > 0, + 1) 1 k + 1) 1 = + 1) 1 k + 1) 1,y i+n ) ϵ + + 1) 1 k + 1) 1,y i+n)<ϵ K H + 1) 1 k + 1) 1 ax 0 i k : y i+n) ϵ )) ) H ϵ + ax 1,. We now select N ϵ such that k + 1) 1 0 i k : y i+n ) ϵ < ϵ K H for each n and k > N ϵ. Now for k > N ϵ, + 1) 1 k + 1) 1 < K H ϵ K H + ax 1, Hence x w 1, p, )] l ). REFERENCES ϵ )) ) H. 1] S. Banach, Theorie des operations linearies, Chelsea, New York, ] J. S. Connor, The statistical and strong p-cesaro convergence of seuences, Analysis, vol. 8, pp , ] G. Das and S.K. Sahoo, On soe seuence spaces, J. ath. Anal. Appl., vol. 164, pp , ] J.P. Duran, Infinite atrices and alost convergence, ath. Zeit., vol. 128, pp , ] H. Fast, Sur la convergence statistiue, Collo. ath., vol. 2, pp , ] J. A. Fridy, On statistical convergence, Analysis, vol. 5, pp , ] P. K. Kathan and. Gupta, Seuence Spaces and Series, arcel Dekker Inc., New York, ISSN: Print); ISSN: Online)

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