J. Aa. Num. Theor. 2, No. 2, 69-76 (2014) 69 Joural of Aalysis & Number Theory A Iteratioal Joural http://dx.doi.org/10.12785/jat/020208 O Some New Etire Sequece Spaces Kuldip Raj 1, ad Ayha Esi 2, 1 School of athematics, Shri ata Vaisho Devi Uiversity, Katra J & K, Idia 2 Departmet of athematics, Sciece ad Art Faculty, Adiyama Uiversity, 02040 Adiyama, Turey Received: 24 Feb. 2014, Revised: 26 Apr. 2014, Accepted: 29 Apr. 2014 Published olie: 1 Jul. 2014 Abstract: I this paper we itroduce etire sequece spaces ad aalytic sequece spaces o semiormed spaces defied by a usiela-orlicz fuctio ad study some toplological properties ad iclusio relatios betwee these spaces. We also mae a effort to study these sequece spaces over -ormed spaces. Keywords: paraorm space, Orlicz fuctio, usiela-orlicz fuctio, solid, mootoe, etire sequece space, aalytic sequece space. subjclass[2000] 40A05, 40C05, 40D05. 1 Itroductio A Orlicz fuctio : [0, ) [0, ) is a cotiuous, o-decreasig ad covex fuctio such that (0) = 0, (x) > 0 for x > 0 ad (x) as x. Lidestrauss ad Tzafriri [17] used the idea of Orlicz fuctio to defie the followig sequece space. Let w be the space of all real or complex sequeces x=(x ), the l = x w: ( x ) < which is called a Orlicz sequece space. Also l is a Baach space with the orm x =if > 0 : ( x ) 1. Also, it was show i [17] that every Orlicz sequece space l cotais a subspace isomorphic to l p (p 1). The 2 coditio is equivalet to (Lx) L(x), for all L with 0<L<1. A Orlicz fuctio ca always be represeted i the followig itegral form x (x)= η(t)dt 0 where η is ow as the erel of, is right differetiable for t 0, η(0) = 0, η(t) > 0, η is o-decreasig ad η(t) as t. A sequece = ( ) of Orlicz fuctios is called a usiela-orlicz fuctio see ([18],[20]). A sequece N =(N ) of Orlicz fuctios defied by N (v)=sup v u (u) : u 0,,2,... is called the complemetary fuctio of the usiela-orlicz fuctio. For a give usiela-orlicz fuctio, the usiela-orlicz sequece space t ad its subspace h are defied as follows t = x w:i (cx)<, for some c>0, h = x w:i (cx)<, for all c>0, where I is a covex modular defied by I (x)= (x ),x=(x ) t. We cosider t equipped with the Luxemburg orm ( x ) x =if >0:I 1 or equipped with the Orlicz orm 1 ) x 0 = if 1+I (x) : >0. ( Correspodig author e-mail: uldipraj68@gmail.com, aesi23@hotmail.com
70 Kuldip Raj, Ayha Esi: O Some New Etire Sequece Spaces Let X be a liear metric space. A fuctio p : X R is called paraorm, if 1.p(x) 0, for all x X, 2.p( x)= p(x), for all x X, 3.p(x+y) p(x)+ p(y), for all x,y X, 4.if (λ ) is a sequece of scalars with λ λ as ad (x ) is a sequece of vectors with p(x x) 0 as, the p(λ x λ x) 0 as. A paraorm p for which p(x) = 0 implies x = 0 is called total paraorm ad the pair (X, p) is called a total paraormed space. It is well ow that the metric of ay liear metric space is give by some total paraorm (see [28], Theorem 10.4.2, P-183). For more details about sequece spaces see( [1], [3], [5], [15], [16], [21], [22], [23], [24], [25], [26], [27]). A complex sequece, whose th term is x is deoted by (x ). Let ϕ be the set of all fiite sequeces. A sequece x=(x ) is said to be aalytic if sup x 1 <. The vector space of all aalytic sequeces will be deoted by Λ. A sequece x is called etire sequece if lim x 1 = 0. The vector space of all etire sequeces will be deoted by Γ. Let σ be a oe-oe mappig of the set of positive itegers ito itself such that σ m ()=σ(σ m 1 ()),m=1,2,3,. A cotiuous liear fuctioal φ o Λ is said to be a ivariat mea or a σ mea if ad oly if 1.φ(x) 0whe the sequece x=(x ) has x 0 for all, 2.φ(e)=1 where e=(1,1,1, ) ad 3.φ(x σ() )=φ(x ) for all x Λ. For certai ids of mappigs σ, every ivariat mea φ exteds the limit fuctioal o the space C of all coverget sequeces i the sese that φ(x)= limx for all x C. Cosequetly C V σ, where V σ is the set of aalytic sequeces all of those σ meas are equal. If x=(x ), set T x=(tx) 1 =(x σ() ). It ca be show that V σ = x=(x ) : lim m t m (x ) 1 = L uiformly i, L=σ lim (x ) 1, where t m (x)= (x + Tx + + T m x ) 1 m+1 Give a sequece x =x its th sectio is the sequece x () =x 1,x 2, x,0,0,, δ () =(0,0,,1,0,0, ), i the th place ad zeros elsewhere. The space ( cosistig ) of all those sequeces x i w such that x 1/ 0 as for some arbitrary fixed > 0 is deoted by Γ ad is ow as usiela-orlicz space of etire sequeces. The space Γ is a metric space ( x y 1/ ) with the metric d(x, y) = sup for all x=x ad y=y i Γ. The space cosistig of all those sequeces x i w such. ( ( x that sup ( 1/ ))) < for some arbitrarily fixed > 0 is deoted by Λ ad is ow as usiela-orlicz space of aalytic sequeces. A sequece space E is said to be solid or ormal if (α x ) E wheever (x ) E ad for all sequeces of scalars (α ) with α 1 (see [20]). The followig iequality will be used throughout the paper. Let p=(p ) be a sequece of positive real umbers with 0 p sup p = G, K = max(1,2 G 1 ) the a + b p K a p + b p for all ad a,b C. (1.1) Also a p max(1, a G ) for all a C. Let =( ) be a usiela-orlicz fuctio, X be locally covex Hausdorff topological liear space whose topology is determied by a set of cotiuous semiorms q. The symbol Λ(X), Γ(X) deotes the space of all aalytic ad etire sequeces recpectively defied over X. I this paper we defie the followig classes of sequeces: Λ (p,σ,q,s)= x Λ(x) : sup s[ ( xσ () 1 ))] p < uiformly i, 0, s 0 ad for some > 0, Γ (p,σ,q,s)= x Γ(x) : s[ ( xσ () 1 ))] p 0 as uiformly i 0, s 0 ad for some > 0. If we tae p=(p )=1, we get Λ (σ,q,s)= x Λ(x) : sup s[ ( xσ () 1, Γ (σ,q,s)= x Γ(x) : 0, s 0 ad for some > 0, ))] < uiformly i s[ ( xσ () 1 ))] 0 as. uiformly i 0, s 0 ad for some > 0 The mai purpose of this paper is to study some etire ad aalytic sequece spaces o semiormed spaces defied by a usiela-orlicz fuctio = ( ). We study some topological properties ad iclusio relatios betwee the spaces Λ (p,σ,q,s) ad Γ (p,σ,q,s) i the secod sectio of this paper. I the third sectio we mae a effort to study some properties of these sequece spaces over -ormed spaces. 2 Some topological properties of spaces Λ (p,σ,q,s) ad Γ (p,σ,q,s) Theorem 2.1Let = ( ) be a usiela-orlicz fuctio ad p = (p ) be a sequece of strictly positive
J. Aa. Num. Theor. 2, No. 2, 69-76 (2014) / www.aturalspublishig.com/jourals.asp 71 real umbers. The the spaces Γ (p,σ,q,s) ad Λ (p,σ,q,s) are liear spaces over the field of complex umbers C. Proof. Let x=(x ), y=(y ) Γ (p,σ,q,s). The there exist positive umbers 1 ad 2 such that ad s[ ( xσ () 1 ))] p 0 as 1 s[ ( yσ () 1 ))] p 0 as. 2 Let 3 = max(2 α 1,2 β 2 ). Sice = ( ) is o decreasig, covex ad q is a semiorm so by usig iequality(1.1), we have s[ ( αx σ () + β y σ () 1 3 )) K + K ] p s[ ( αxσ () + β y σ () ) 1 )] p 3 3 1 ( 2 p s[ ( xσ q () 1 )) 1 ( yσ + () 1 ))] p 2 s[ ( xσ () 1 )) ( yσ + () 1 ))] p 1 2 0 as. s[ ( xσ () 1 ))] p 1 s[ ( yσ () 1 ))] p 2 Thus αx+β y Γ (p,σ,q,s). Hece Γ (p,σ,q,s) is a liear space. Similarly, we ca show that Λ (p,σ,q,s) is a liear space. Theorem 2.2Suppose = ( ) is usiela-orlicz fuctio ad p = (p ) be a sequece of strictly positive real umbers. The the space Γ (p,σ,q,s) is a paraormed space with the paraorm defied by g(x) = if pm : sup s[ ( x σ () 1 ] p )) 1 1, uiformly i > 0, > 0, where = max(1,sup p ). Proof. Clearly g(x) 0,g(x) = g( x) ad g(θ) = 0, where θ is the zero sequece of X. Let (x ), (y ) Γ (p,σ,q,s). Let 1, 2 > 0 be such that sup s[ ( xσ () 1 ))] p 1 1 ad sup s[ ( yσ () 1 ))] p 1. 1 Let = 1 + 2 ad by usig iowsi s iequality, we have sup s[ ( x σ () + y σ () 1 ))] p 1 1 Hece g(x+y) + 1. 1 + 2 sup 1 if ( 1 + 2 ) pm : sup s[ ( x σ () + y σ () 1 1 1 + 2 if ( 1 ) pm : sup s[ ( x σ () 1 1 1 + if ( 2 ) pm : sup s[ ( y σ () 1 1 2 s[ ( x σ () 1 ))] p 1 2 sup s[ ( y σ () 1 ))] p 1 + 2 1 2 ))] p 1, 1, 2 > 0,m N ))] p 1, 1 > 0, m N ))] p 1, 2 > 0,m N. Thus we have g(x+y)g(x)+g(y). Hece g satisfies the triagle iequality. Now g(λx) = if () pm = if (r λ ) pm : sup s[ ( x σ () 1 ))] p 1, > 0, m N 1 : sup s[ ( x σ () 1 ))] p 1, r> 0, m N, 1 where r= λ. Hece Γ (p,σ,q,s) is a paraormed space. Theorem 2.3Let = ( ) be a usiela-orlicz fuctio. The Γ (p,σ,q,s) Λ (p,σ,q,s) Γ (p,σ,q,s). Proof. The proof is trivial so we omit. Theorem 2.4Γ (p,σ,q,s) Λ (p,σ,q,s). Proof. The proof is trivial so we omit. Theorem 2.5Let 0 p r ad let r p be bouded. The Γ (r,σ,q,s) Γ (p,σ,q,s). Proof. Let x Γ (r,σ,q,s). The s[ ( xσ () 1 ))] r 0 as. (2.1) Let t = s[ ( xσ () 1 ))] q ad λ = p r. Sice p r, we have 0λ 1. Tae 0<λ < λ. Defie t, if t 1 u = 0, if t < 1
72 Kuldip Raj, Ayha Esi: O Some New Etire Sequece Spaces ad t = u +v, v λ 0, if t 1 v = t, if t < 1 t λ v λ. Sice tλ = u λ +vλ. It follows that uλ = u λ + vλ, the tλ u t, t + v λ. Now s[[ ( x σ () 1 ))] r ] λ s[ ( ( x σ () 1 ))] r q = s[ ( x σ () 1 )) r ] p /r s[ ( ( x σ () 1 ))] r q But = s[ ( x σ () 1 ))] p s[ ( ( x σ () 1 ))] r. q s[ ( xσ () 1 ))] r 0 as (by(2.1)). Therefore s[ ( xσ () 1 ))] p 0 as. Hece x Γ (p,σ,q,s). From (2.1), we get Γ (r,σ,q,s) Γ (p,σ,q,s). Theorem 2.6(i) Let 0 < if p p 1. The Γ (p,σ,q,s) Γ (σ,q,s), (ii) let 1 p sup p <. The Γ (σ,q,s) Γ (p,σ,q,s). Proof.(i) Let x Γ (p,σ,q,s). The s[ ( xσ () 1 ))] p 0 as. (2.2) Sice 0<if p p 1, s[ ( xσ () 1 ))] s[ ( xσ () 1 ))] p 0 as. (2.3) From (2.2) ad (2.3) it follows that, x Γ (σ,q,s). Thus Γ (p,σ,q,s) Γ (σ,q,s). (ii) Let p 1 for each ad sup p < ad let x Γ (σ,q,s). The s[ ( xσ () 1 ))] 0 as. (2.4) Sice 1 p sup p <, we have s[ ( xσ () 1 ))] p s[ ( ( xσ q () 1 ))] s[ ( xσ () 1 ))] p 0 as. This implies that x Γ (p,σ,q,s). Therefore Γ (σ,q,s) Γ (p,σ,q,s). Theorem 2.7Suppose s[ ( xσ () 1 ))] p x 1/, the Γ Γ (p,σ,q,s). Proof. Let x Γ. The we have, x 1/ 0 as. (2.5) But s[ ( xσ () 1 ))] p x 1/, by our assumptio, implies that s[ ( xσ () 1 ))] p 0 as. by(2.5) The x Γ (p,σ,q,s) ad Γ Γ (p,σ,q,s). Theorem 2.8Γ (p,σ,q,s) is solid. Proof. Let x y ad let y = (y ) Γ (p,σ,q,s), because =( ) is o-decreasig s[ ( xσ () 1 ))] p s[ ( ( yσ q () 1 ))] p. Sice y Γ (p,σ,q,s). Therefore, ad hece s[ ( ( yσ () 1 ))] p 0 as s[ ( xσ () 1 ))] p 0 as. Therefore x=x Γ (p,σ,q,s). Theorem 2.9Γ (p,σ,q,s) is mootoe. Proof. The proof is trivial. 3 Sequece spaces over - ormed spaces The cocept of 2-ormed spaces was iitially developed by Gähler[11] i the mid of 1960 s, while that of -ormed spaces oe ca see i isia[19]. Sice the, may others have studied this cocept ad obtaied various results, see Guawa ([12,[13]) ad Guawa ad ashadi [14]. Let N ad X be a liear space over the field R, where R is field of reals of dimesio d, where d 2. A real valued fuctio,, o X satisfyig the followig four coditios: 1. x 1,x 2,,x = 0 if ad oly if x 1,x 2,,x are liearly depedet i X;
J. Aa. Num. Theor. 2, No. 2, 69-76 (2014) / www.aturalspublishig.com/jourals.asp 73 2. x 1,x 2,,x is ivariat uder permutatio; 3. αx 1,x 2,,x = α x 1,x 2,,x for ay α R, ad 4. x+x,x 2,,x x,x 2,,x + x,x 2,,x is called a -orm o X, ad the pair (X,,, ) is called a -ormed space over the field R. For example, we may tae X = R beig equipped with the -orm x 1,x 2,,x E = the volume of the -dimesioal parallelopiped spaed by the vectors x 1,x 2,,x which may be give explicitly by the formula x 1,x 2,,x E = det(x i j ), where x i = (x i1,x i2,,x i ) R for each i = 1,2,,. Let(X,,, ) be a -ormed space of dimesio d 2 ada 1,a 2,,a be liearly idepedet set i X. The the fuctio,, o X 1 defied by If we tae p=(p )=1, we get x 1,x 2,,x 1 = max x 1,x 2,,x 1,a i : i=1,2,, Λ (σ,q,s,.,,. ) = is ow as a ( 1)-orm o X with respect to a 1,a 2,,a. Let N ad X be a real vector space of dimesio d, where 2 d. Let β 1 be the collectio of liearly idepedet sets B with 1 elemets. For B β 1, let us defie q B (x 1 )= x 1,x 2, x, x 1 X. The q B is a semiorm o X ad the family q=q B : B β 1 of semiorms geerates a locally covex topology o X. The semiorms q B have the followig properties: 1.er B )=the liear spa of B. 2.For B β 1, y Bad x X\the liear spa of B we have q B x\y (y)=q B (x). See ([10]) A sequece(x ) i a -ormed space(x,,, ) is said to coverge to some L X if lim x L,z 1,,z 1 =0 for every z 1,,z 1 X. A sequece(x ) i a -ormed space(x,,, ) is said to be Cauchy if lim x x p,z 1,,z 1 =0 for every z 1,,z 1 X.,p If every Cauchy sequece i X coverges to some L X, the X is said to be complete with respect to the -orm. Ay complete -ormed space is said to be -Baach space. For more details about ormed spaces oe ca see ([2], [4], [6], [7], [8], [9]) ad refereces therei. Let = ( ) be a usiela-orlicz fuctio, X be locally covex Hausdorff topological real liear ormed space whose topology is determied by a set of cotiuous semiorms q. The symbol Λ(X), Γ(X) deotes the space of all aalytic ad etire sequeces respectively defied over X. I this sectio, for each z 1,,z 1 X we defie the followig classes of sequeces: Λ (p,σ,q,s,.,,. )= x Λ(x) : sup s[ ( (x σ () ) 1, Γ (p,σ,q,s,.,,. ) = s[ ( (x σ () ) 1 ))] p,z 1,,z 1 < uiformly i 0, s 0 for some > 0, x Γ(x) : ))] p,z 1,,z 1 0 as uiformly i 0, s 0 for some > 0. sup s[ ( (x σ () ) 1 ))],z 1,,z 1, 0 for some > 0, Γ (σ,q,s,.,,. ) = s[ ( (x σ () ) 1 x Λ(x) : < uiformly i 0, s x Γ(x) : ))],z 1,,z 1 0 as uiformly i 0, s 0 for some > 0. I the preset sectio we study some topological properties of the spaces Λ (p,σ,q,s,.,,. ) ad Γ (p,σ,q,s,.,,. ) ad also examie some iclusio relatio betwee these spaces. Theorem 3.1Let = ( ) be a usiela-orlicz fuctio ad p = (p ) be a sequece of strictly positive real umbers. The the spaces Γ (p,σ,q,s,.,,. ) ad Λ (p,σ,q,s,.,,. ) are liear space over the field of real umbersr. Proof. Let x = (x ), y = (y ) Γ (p,σ,q,s,.,,. ). The there exist positive umbers 1 ad 2 such that ad s[ ( (x σ () ) 1 ))] p,z 1,,z 1 0 as 1 s[ ( (y σ () ) 1 ))] p,z 1,,z 1 0 as. 2 Let 3 = max(2 α 1,2 β 2 ). Sice = ( ) is o decreasig, covex ad q is a semiorm ad by usig
74 Kuldip Raj, Ayha Esi: O Some New Etire Sequece Spaces iequality(1.1), we have s[ ( (αxσ () + β y σ () ) 1 ) 3 s[ ( α(xσ () ) + (y σ () ) 3 3 ) 1 1 ( 2 p s[ q ( (x σ () ) 1 )),z 1,,z 1 1 + ( (y σ () ) 1 ))] p,z 1,,z 1 2 s[ ( (x σ () ) 1 )),z 1,,z 1 1 + ( (y σ () ) 1 ))] p,z 1,,z 1 2 ] p,z 1,,z 1 K s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1 + K s[ ( (y σ () ) 1 ))] p,z 1,,z 1 2 0 as. )] p,z 1,,z 1 Thus αx + β y Γ (p,σ,q,s,.,,. ). Hece Γ (p,σ,q,s,.,,. ) is a liear space. Similarly, we ca prove Λ (p,σ,q,s,.,,. ) is a liear space. Theorem 3.2Suppose = ( ) is usiela-orlicz fuctio ad p = (p ) be a sequece of strictly positive real umbers. The the space Γ (p,σ,q,s,.,,. ) is a paraormed space with the paraorm defied by g(x) = if pm : sup s[ ( (xσ ()) 1 )] p,z 1,,z 1 ) 1, 1 uiformly i > 0, > 0, where = max(1,sup p ). Proof. Clearly g(x) 0,g(x) = g( x) ad g(θ) = 0, where θ is the zero sequece of X. Let (x ), (y ) Γ (p,σ,q,s,.,,. ). Let 1, 2 > 0 be such that ad sup s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1 1 sup s[ ( (y σ () ) 1 ))] p,z 1,,z 1 1. 1 The sup s[ ( (x σ () + y σ () ) 1 1 1 ))] p,z 1,,z 1 1 sup s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1 + 2 1 + 2 sup s[ ( (y σ () ) 1 ))] p,z 1,,z 1 1 + 2 2 1. Hece g(x+y) if 1 ( 1 + 2 ) pm : sup 1 1, 2 > 0, m N if ( 1 ) pm : sup 1 1 > 0, m N + if ( 2 ) pm : sup 1. 2 > 0, m N s[ ( (x σ () + y σ () ) 1 ))] p,z 1,,z 1 1, 1 + 2 s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1, s[ ( (y σ () ) 1 ))] p,z 1,,z 1 1, Thus we have g(x+y)g(x)+g(y). Hece g satisfies the triagle iequality. Now g(λ x) = if () pm = if (r λ ) pm : sup s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1, > 0, m N 1 : sup s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1,r > 0,m N, 1 where r = λ. Hece Γ (p,σ,q,s,.,,. ) is paraormed space. Theorem 3.3Let = ( ) be a usiela-orlicz fuctio. The Γ (p,σ,q,s,.,,. ) Λ (p,σ,q,s,.,,. ) Γ (p,σ,q,s,.,,. ). Proof. It is easy to prove so we omit the proof. Theorem 3.4 Γ (p,σ,q,s,.,,. ) Λ (p,σ,q,s,.,,. ). Proof. It is easy to prove so we omit the proof. Theorem 3.5Γ (p,σ,q,s,.,,. ) is solid. Proof. Let x y ad let y = (y ) Γ (p,σ,q,s,.,,. ), sice = ( ) is o-decreasig, so s[ ( (x σ ()) 1 ))] p,z 1,,z 1 s[ ( (y σ ()) 1 p.,z 1,,z 1 ))] a
J. Aa. Num. Theor. 2, No. 2, 69-76 (2014) / www.aturalspublishig.com/jourals.asp 75 Sice y Γ (p,σ,q,s,.,,. ). Therefore, s[ ( (y σ () ) 1 p,z 1,,z 1 ))] 0 as. So that s[ ( (x σ () ) 1 p,z 1,,z 1 ))] 0 as. Therefore x = (x ) Γ (p,σ,q,s,.,,. ). Hece Γ (p,σ,q,s,.,,. ) is solid. Theorem 3.6Γ (p,σ,q,s,.,,. ) is mootoe. Proof. The proof is trivial so we omit it. Refereces [1] H. Dutta ad B.S. Reddy, O some sequece spaces, Tamsui Oxford Joural of Iformatio ad athematical Scieces, 28 (1) (2012), pp. 1-12. [2] H. Dutta, A Orlicz extesio of differece sequeces o real liear -ormed spaces, Joural of Iequalities ad Applicatios, 2013 (2013), art. o. 232. [3] H. Dutta ad F. Başar, A geeralizatio of Orlicz sequece spaces by Ces`sro mea of order oe, Acta athematica Uiversitatis Comeiaae, 80(2) (2011), pp. 185-200. [4] H. Dutta ad B.S. Reddy, O o-stadard -orm o some sequece spaces, It. J. Pure Appl. ath., 68(1) (2011), pp. 1-11. [5] H. Dutta ad T. Bilgi, Strogly (V λ,a, vm, p)-summable sequece spaces defied by a Orlicz fuctio, Applied athematics Letters, 24(7) (2011), pp. 1057-1062. [6] H. Dutta, B.S. Reddy ad S.S. Cheg, Strogly summable sequeces defied over real ormed spaces, Applied athematics E - Notes, 10(2010), pp. 199-209. [7] H. Dutta, O -ormed liear space valued strogly (C, 1)- summable differece sequeces, Asia-Europea Joural of athematics, 3(4) (2010), pp. 565-575. [8] H. Dutta, O sequece spaces with elemets i a sequece of real liear -ormed spaces, Applied athematics Letters, 23(9) (2010), pp. 1109-1113. [9] H. Dutta, A applicatio of lacuary summability method to -orm, Iteratioal Joural of Applied athematics ad Statistics, 15(09) (2009), pp. 89-97. [10] H. Dutta, O sequece spaces with elemets i a sequece of real liear -ormed spaces, Applied athematics Letters, 23(9) (2010), pp. 1109-1113. [11] S. Gahler, Liear 2-ormietre Rume, ath. Nachr., 28 (1965), pp. 1-43. [12] H. Guawa, O -Ier Product, -Norms, ad the Cauchy-Schwartz Iequality, Sci. ath. Jap., 5 (2001), pp. 47-54. [13] H. Guawa, The space of p-summable sequece ad its atural -orm, Bull. Aust. ath. Soc., 64 (2001), pp. 137-147. [14] H. Guawa ad., ashadi,o -ormed spaces, It. J. ath. ath. Sci., 27 (2001), pp. 631-639. [15] P. K. Kamtha ad. Gupta, Sequece spaces ad series, Lecture Notes i Pure ad Applied athematics, 65 arcel Deer, Ic., New Yor,(1981). [16] V. Karaaya ad H. Dutta, O some vector valued geeralized differece modular sequece spaces, Filomat, 25(3) (2011), pp. 15-27. [17] J. Lidestrauss ad L. Tzafriri, O Orlicz sequece spaces, Israel J. ath; 10, 379-390 (1971). [18] L. aligrada, Orlicz spaces ad iterpolatio, Semiars i athematics 5, Polish Academy of Sciece, 1989. [19] A. isia, -ier product spaces, ath. Nachr., 140 (1989), pp. 299-319. [20] J. usiela, Orlicz spaces ad modular spaces, Lecture Notes i athematics, 1034,(1983). [21] S. D. Prashar ad B. Choudhary, Sequece spaces defied by Orlicz fuctios, Idia J. Pure Appl. ath. 25(14) (1994), 419-428. [22] K. Raj, A. K. Sharma ad S. K. Sharma, A Sequece space defied by usiela-orlicz fuctios, It. J. Pure Appl. ath., 67 (2011), 475-484. [23] K. Raj, S. K. Sharma ad A. K. Sharma, Differece sequece spaces i -ormed spaces defied by usiela-orlicz fuctios, Arme. J ath., 3 (2010), pp. 127-141. [24] K. Raj ad S. K. Sharma, Geeralized differece sequece spaces defied by usiela-orlicz fuctio, Iteratioal J. of ath. Sci. & Egg. Appls., 5 (2011), pp. 337-351. [25] K. Raj ad S. K. Sharma, Some differece sequece spaces defied by sequece of modulus fuctio, It. Joural of athematical Archive, 2 (2011), pp. 236-240. [26] B. C. Tripathy ad H. Dutta, Some differece paraormed sequece spaces defied by Orlicz fuctios, Fasciculi athematici, Nr 42 (2009), 121-131. [27] B.C. Tripathy, ad H. Dutta, O some lacuary differece sequece spaces defied by a sequece of orlicz fuctios ad q-lacuary m statistical Covergece, Aalele Stiitifice ale Uiversitatii Ovidius Costata, Seria atematica, 20(1) (2012), pp. 417-430. [28] A. Wilasy, Summability through Fuctioal Aalysis, North- Hollad ath. Stud. (1984). Kuldip Raj ia a assistat professor, wors atschool of athematics, Shri ata Vaisho Devi Uiversity, Katra-182320, J&K, Idia. His qualificatio is Ph. D. (1999, athematics) i the area of Fuctioal aalysis, Operator theory, Sequece, Series ad Summability.His total teachig experiece is 15 years. His research experiece is 20 years. He has 100 published papers.
76 Kuldip Raj, Ayha Esi: O Some New Etire Sequece Spaces Ayha Esi was bor i Istabul, Turey, o arch 5, 1965. Ayha Esi got his B.Sc. from Iou Uiversity i 1987 ad. Sc. ad Ph.D. degree i pure mathematics from Elazig Uiversity, Turey i 1990 ad 1995, respectively. His research iterests iclude Summability Theory, Sequeces ad Series i Aalysis ad Fuctioal Aalysis. I 2000, Esi was appoited to Educatio Faculty i Gaziatep Uiversity. I 2002, Esi was appoited as the head of Departmet of athematics i Sciece ad Art Faculty i Adiyama of the Iou Uiversity. I 2006, Esi joied the Departmet of athematics of Adiyama Uiversity.