ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz fuctio ad examie some properties of the resultig sequece spaces. It is also show that if a sequece is λ-strogly coverget with respect to a Orlicz fuctio the it is λ-statistically coverget.. Itroductio The cocept of paraorm is closely related to liear metric spaces. It is a geeralizatio of that of absolute value. Let X be a liear space. A fuctio p : X R is called paraorm, if P. p0 0 P.2 px 0 for all x X P.3 p x = px for all x X P.4 px + y px + py for all x, y X triagle iequality P.5 if is a sequece of scalars with λ ad x is a sequece of vectors with px x 0, the p x λx 0 cotiuity of multiplicatio by scalars. A paraorm p for which px = 0 implies x = 0 is called total. It is well kow that the metric of ay liear metric space is give by some total paraorm cf. 4, Theorem 0.4.2, p.83]. Let Λ = be a o decreasig sequece of positive reals tedig to ifiity ad λ = ad + +. The geeralized de la Vallée-Poussi meas is defied by t x = x k, where I = +, ]. A sequece x = x k is said to be V, λ-summable to a umber l see 2] if t x l as. 2000 athematics Subject Classificatio: 40D05, 40A05. Key words ad phrases: sequece spaces, Orlicz fuctio, de la Vallée-Poussi meas. Received Jauary 8, 2002.
34 E. SAVAŞ, R. SAVAŞ ad We write V, λ] 0 = V, λ] = V, λ] = x = x k : lim x = x k : lim x = x k : sup x k = 0 } x k le = 0, for some l C x k < For the sets of sequeces that are strogly summable to zero, strogly summable ad strogly bouded by the de la Vallée-Poussi method. I the special case where = for =, 2, 3,..., the sets V, λ] 0, V, λ] ad V, λ] reduce to the sets ω 0, ω ad ω itroduced ad studied by addox 5]. Followig Lidestrauss ad Tzafriri 4], we recall that a Orlicz fuctio is a cotiuous, covex, o-decreasig fuctio defied for x 0 such that 0 = 0 ad x 0 for x > 0. If covexity of Orlicz fuctio is replaced by x + y x + y the this fuctio is called a modulus fuctio, defied ad discussed by Nakao 8], Ruckle 0], addox 6] ad others. Lidestrauss ad Tzafriri used the idea of Orlicz fuctio to costruct the sequece space } l = x = x k : < for some > 0. The space l with the orm x = if k= > 0 : } } k= becomes a Baach space which is called a Orlicz sequece space. For x = x p, p <, the space l coicide with the classical sequece space l p. Recetly Parashar ad Choudhary 9] have itroduced ad examied some properties of four sequece spaces defied by usig a Orlicz fuctio, which geeralized the well-kow Orlicz sequece space l ad strogly summable sequece spaces C,, p], C,, p] 0 ad C,, p]. It may be oted that the spaces of strogly summable sequeces were discussed by addox 5]. Quite recetly E. Savaş ] has also used a Orlicz fuctio to costruct some sequece spaces. I the preset paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz fuctio ad examie some properties of the resultig sequece spaces. Furthermore it is show that if a sequece is λ-strogly coverget with respect to a Orlicz fuctio the it is λ-statistically coverget. We ow itroduce the geeralizatios of the spaces of λ-strogly.. }
SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS 35 We have Defiitio. Let be a Orlicz fuctio ad p = p k be ay sequece of strictly positive real umbers. We defie the followig sequece spaces: V,, p] = V,, p] 0 = V,, p] = x = x k : lim x = x k : lim x = x k : sup ] xk l = 0 for some l ad > 0} ] = 0 for some > 0} ] } < for some > 0. We deote V,, p], V,, p] 0 ad V,, p] as V, ], V, ] 0 ad V, ] whe p k = for all k. If x V, ] we say that x is of λ-strogly coverget with respect to the Orlicz fuctio. If x = x, p k = for all k, the V,, p] = V, λ], V,, p] 0 = V, λ] 0 ad V,, p] = V, λ]. If = the, V,, p], V,, p] 0 ad V,, p] reduce the C,, p], C,, p] 0 ad C,, p] which were studied Parashar ad Choudhary 9]. 2. ai Results I this sectio we examie some topological properties of V,, p], V,, p] 0 ad V,, p] spaces. Theorem. For ay Orlicz fuctio ad ay sequece p = p k of strictly positive real umbers, V,, p], V,, p] 0 ad V,, p] are liear spaces over the set of complex umbers. Proof. We shall prove oly for V,, p] 0. The others ca be treated similarly. Let x, y V,, p] 0 ad α, β C. I order to prove the result we eed to fid some 3 > 0 such that lim ] αxk + βy k = 0. Sice x, y V,, p] 0, there exist a positive some ad 2 such that lim 3 ] = 0 ad lim ] yk = 0. 2
36 E. SAVAŞ, R. SAVAŞ Defie 3 = max 2 α, 2 β 2. Sice is o-decreasig ad covex, ] αxk + βy k αxk + βy ] k 3 K 3 2 p k ] + K 3 + + yk 2 ] yk 0 2 yk 2 ] ] as, where K = max, 2 H, H = sup p k, so that αx + βy V,, p] 0. This completes the proof. Theorem 2. For ay Orlicz fuctio ad a bouded sequece p = p k of strictly positive real umbers, V,, p] 0 is a total paraormed spaces with ] /H gx = if p/h :, =, 2, 3,.... where H = max, sup p k. Proof. Clearly gx = g x. By usig Theorem, for a α = β =, we get gx + y gx + gy. Sice 0 = 0, we get if p/h } = 0 for x = 0. Coversely, suppose gx = 0, the if p/h : ] /H = 0. This implies that for a give ε > 0, there exists some ε 0 < ε < ε such that ] /H. Thus, ε ] /H ε for each. Suppose that x m 0 for some m I. Let ε 0, the that ] /H, ε ] /H xm ε xm ε. It follows
SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS 37 which is a cotradictio. Therefore x m = 0 for each m. Fially, we prove that scalar multiplicatio is cotiuous. Let µ be ay complex umber. By defiitio ] /H µxk gµx = if p/h :, =, 2, 3,.... The gµx = if µ sp/h : where s = / µ. Sice µ p max, µ sup p, we have gµx max, µ sup p /H if sp/h : ] /H, =, 2, 3,... s ] /H, =, 2, 3,... s which coverges to zero as x coverges to zero i V,, p] 0. Now suppose µ m 0 ad x is fixed i V,, p] 0. For arbitrary ε > 0, let N be a positive iteger such that ] < ε/2 H for some > 0 ad all > N. This implies that ] < ε/2 for some > 0 ad all > N. Let 0 < µ <, usig covexity of, for > N, we get ] µxk < ] µ < ε/2 H. Sice is cotiuous everywhere i 0,, the for N ft = ] txk is cotiuous at 0. So there is > δ > 0 such that ft < ε/2 H for 0 < t < δ. Let K be such that µ m < δ for m > K the for m > K ad N ] /H µm x k < ε/2. Thus ] /H µm x k < ε
38 E. SAVAŞ, R. SAVAŞ for m > K ad all, so that gµx 0 µ 0. Defiitio 2 ]. A Orlicz fuctio is said to satisfy 2 -coditio for all values of u, if there exists a costat K > 0 such that 2u Ku, u 0. It is easy to see that always K > 2. The 2 -coditio is equivalet to the satisfactio of iequality lu Klu, for all values of u ad for l >. Theorem 3. For ay Orlicz fuctio which satisfies 2 -coditio, we have V, λ] V, ]. Proof. Let x V, λ] so that T = x k l 0 as for some l. Let ε > 0 ad choose δ with 0 < δ < such that t < ε for 0 t δ. Write y k = x k l ad cosider y k = λ + 2 where the first summatio is over y k δ ad the secod summatio over y k > δ. Sice, is cotiuous < ε ad for y k > δ we use the fact that y k < y k /δ < + y k /δ. Sice is o decreasig ad covex, it follows that y k < + δ y k < 2 2 + 2 2δ y k Sice satisfies 2 -coditio there is a costat K > 2 such that 2δ y k 2 Kδ y k 2, therefore Hece y k < 2 Kδ y k 2 + 2 Kδ y k 2 = Kδ y k 2. 2 y k Kδ 2 T which together with ε yields V, λ] V, ]. This completes proof. The method of the proof of Theorem 3 shows that for ay Orlicz fuctio which satisfies 2 -coditio; we have V, λ] 0 V, ] 0 ad V, λ] V, ]. Theorem 4. Let 0 p k q k ad q k / be bouded. The V,, q] V,, p]. The proof of Theorem 4 used the ideas similar to those used i provig Theorem 7 of Parashar ad Choudhary 9]. We ow itroduce a atural relatioship betwee strog covergece with respect to a Orlicz fuctio ad λ-statistical covergece. Recetly, ursalee 7] itroduced the cocept of statistical covergece as follows:
SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS 39 Defiitio 3. A sequece x = x k is said to be λ-statistically coverget or s λ -statistically coverget to L if for every ε > 0 lim k I : x k L ε} = 0, where the vertical bars idicate the umber of elemets i the eclosed set. I this case we write s λ lim x = L or x k Ls λ ad s λ = x : L R : s λ lim x = L}. Later o, λ-statistical covergece was geeralized by Savaş 2]. We ow establish a iclusio relatio betwee V, ] ad s λ. Theorem 5. For ay Orlicz fuctio, V, ] s λ. Proof. Let x V, ] ad ε > 0. The xk l, x k l ε xk l ε / k I : x k l ε} from which it follows that x s λ. To show that s λ strictly cotais V, ], we proceed as i 7]. We defie x = x k by x k = k if ] + k ad xk = 0 otherwise. The x / l ad for every ε 0 < ε k I : x k 0 ε} = λ ] 0 as i.e. x k 0 s λ, where ] deotes the greatest iteger fuctio. O the other had, xk 0 i.e. x k 0 V, ]. This completes the proof. Refereces ] Krasoselski,. A. ad Rutitsky, Y. B., Covex fuctios ad Orlicz Spaces, Groige, the Netherlads, 96. 2] Leidler, L., Über de la Vallee Pousische Summierbarkeit allgemeier Orthogoalreihe, Acta ath. Hug. 6 965, 375 378. 3] Lidestrauss, J., Some aspects of the theory of Baach spaces, Adv. ath. 5 970, 59 80. 4] Lidestrauss, J. ad Tzafriri, L., O Orlicz sequece spaces, Israel J. ath. 0 3 97, 379 390. 5] addox, I. J., Spaces of strogly summable sequeces, Quart. J. ath. Oxford Ser. 2, 8 967, 345 355.
40 E. SAVAŞ, R. SAVAŞ 6] addox, I. J., Sequece spaces defied by a modulus, ath. Proc. Cambridge Philos. Soc. 00 986, 6 66. 7] ursalee, λ-statistical covergece, ath. Slovaca 50, No. 2000, 5. 8] Nakao, H., Cocave modulus, J. ath. Soc. Japa 5 953, 29 49. 9] Parashar, S. D. ad Choudhary, B., Sequece spaces defied by Orlicz fuctios, Idia J. Pure Appl. ath. 25 4 994, 49 428. 0] Ruckle, W. H., FK spaces i which the sequece of coordiate vectors is bouded, Caad. J. ath. 25 973, 973 978. ] Savaş, E., Strogly almost coverget sequeces defied by Orlicz fuctios, Comm. Appl. Aal. 4 2000, 453 458. 2] Savaş, E., Strog almost covergece ad almost λ-statistical covergece, Hokkaido ath. J. 243, 2000, 53 536. 3] Savaş, E. ad Rhoades, B. E., O some ew sequece spaces of ivariat meas defied by Orlicz fuctios, ath. Iequal. Appl. 52 2002, 27 28. 4] Wilasky, A., Summability through Fuctioal Aalysis, North-Hollad ath. Stud. 85 984. Departmet of athematics, Yüzücü Yıl Üiversity Va 65080, Turkey E-mail: ekremsavas@yahoo.com