Some vector-valued statistical convergent sequence spaces

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1 Malaya J. Mat. 32)205) 6 67 Some vector-valued statistical coverget sequece spaces Kuldip Raj a, ad Suruchi Padoh b a School of Mathematics, Shri Mata Vaisho Devi Uiversity, Katra-82320, J&K, Idia. b School of Mathematics, Shri Mata Vaisho Devi Uiversity, Katra-82320, J&K, Idia. Abstract I the preset paper we itroduce some vector-valued statistical coverget sequece spaces defied by a sequece of modulus fuctios associated with multiplier sequeces ad we also mae a effort to study some topological properties ad iclusio relatio betwee these spaces. Keywords: Modulus fuctio, paraorm space, differece sequece space, statistical covergece. 200 MSC: 40A05, 46A45, 40C05. c 202 MJM. All rights reserved. Itroductio ad Prelimiaries The study o vector-valued sequece spaces was exploited by Kamtha ], Ratha ad Srivastava 8], Leoard 4], Gupta 9], Tripathy ad Se 26] ad may others. The scope for the studies o sequece spaces was exteded o itroducig the otio of associated multiplier sequeces. Goes ad Goes 8] defied the differetiated sequece space de ad itegrated sequece space E for a give sequece space E, with the help of multiplier sequeces ) ad ) respectively. Kamtha used the multiplier sequece!) see ]. The study o multiplier sequece spaces were carried out by Cola 2], Cola et al. 3], Srivastava ad Srivastava 25], Tripathy ad Mahata 28] ad may others. Let w be the set of all sequeces of real or complex umbers ad let l, c ad c 0 be the Baach spaces of bouded, coverget ad ull sequeces x = x ) respectively with the usual orm x = sup x, where N, is the set of positive itegers. Throughout the paper, for all N, E are semiormed spaces semiormed by q ad X is a semiormed space semiormed by q. By we ), ce ), l E ) ad l p E ) we deote the spaces of all, coverget, bouded ad p-absoluetly summable E -valued sequeces. I the case E = C the field of complex umbers) for all N, oe has the scalar valued sequece spaces respectively. The zero elemet of E is deoted by θ ad the zero sequece is deoted by θ = θ ). The otio of differece sequece spaces was itroduced by Kizmaz 2], who studied the differece sequece spaces l ), c ) ad c 0 ). The otio was further geeralized by Et ad Cola 4] by itroducig the spaces l ), c ) ad c 0 ). Let w be the space of all complex or real sequeces x = x ) ad let m, s be o-egative itegers, the for Z = l, c, c 0 we have sequece spaces where m s x = m s x ) = m s followig biomial represetatio Z m s ) = x = x ) w : m s x ) Z}, x m s x + ) ad 0 s x = x for all N, which is equivalet to the m s x = m v=0 ) v m v ) x +sv. Taig s =, we get the spaces which were studied by Et ad Cola 4]. Taig m = s =, we get the spaces which were itroduced ad studied by Kizmaz 2]. Correspodig author. address: uldipraj68@gmail.com Kuldip Raj), suruchi.padoh87@gmail.com Suruchi Padoh).

2 62 K. Raj et al. / Some vector-valued statistical coverget sequece spaces Defiitio.. A modulus fuctio is a fuctio f : 0, ) 0, ) such that. f x) = 0 if ad oly if x = 0, 2. f x + y) f x) + f y) for all x 0, y 0, 3. f is icreasig, 4. f is cotiuous from right at 0. It follows that f must be cotiuous everywhere o 0, ). The modulus fuctio may be bouded or ubouded. For example, if we tae f x) = x+ x, the f x) is bouded. If f x) = xp, 0 < p <, the the modulus f x) is ubouded. Subsequetly, modulus fuctio has bee discussed i ], 6], 9], 20], 23]) ad may others. Defiitio.2. Let X be a liear metric space. A fuctio p : X R is called paraorm, if. px) 0, for all x X, 2. p x) = px), for all x X, 3. px + y) px) + py), 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 px x) 0 as, the pλ x λx) 0 as. A paraorm p for which px) = 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 29], Theorem 0.4.2, P-83). Let p = p ) be a bouded sequece of positive real umbers, let F = f ) be a sequece of modulus fuctio. Also let t = t = p ad suppose u = u ) is a sequece of strictly positive real umbers. I this paper we defie the followig sequece spaces: W 0 m s, F, Q, p, u, t) = x ) : x E for all N ad there exists r > 0 such that t = f q p u m s x r ))] p 0 as }, W m s, F, Q, p, u, t) = x ) : x E for all N ad there exists r > 0 such that t = f q p u m s x r l ))] p } 0 as, l E ad W m s, F, Q, p, u, t) = x ) : x E for all N ad there exists r > 0 such that sup t = f q p u m s x r ))] p < }. I the case f = f ad q = q for all N, we write W 0 m s, f, q, p, u, t), W m s, f, q, p, u, t) ad W m s, f, q, p, u, t) istead of W 0 m s, F, Q, p, u, t), W m s, F, Q, p, u, t) ad W m s, F, Q, p, u, t) respectively. Throughout the paper Z deotes ay of the values 0, ad. If x = x ) W m s, f, q, p, u, t), we say that x is strogly u q,t Cesaro summable with respect to the modulus fuctio f ad write x l W m s, f, q, p, u, t); l is called the u q,t limit of x with respect to the modulus fuctio f. The mai aim of this paper is to itroduced the sequece spaces W Z m s, F, Q, p, u, t), Z = 0, ad. We also mae a effort to study some topological properties ad iclusio relatios betwee these spaces.

3 K. Raj et al. / Some vector-valued statistical coverget sequece spaces 63 2 Mai Results Theorem 2.. Let F = f ) be a sequece of modulus fuctios ad p = p ) be a bouded sequece of positive real umbers. The the spaces W Z m s, F, Q, p, u, t), Z = 0,, are liear spaces over the complex field C. Proof. We shall prove the result for Z = 0. Let x = x ) W 0 m s, F, Q, p, u, t). The there exists r > 0 such that t f q p u m s x r ))] p 0 as. Let λ C. Without loss of geerality we ca tae λ = 0. Let ρ = r λ ) > 0, the we have t f q p u m ) ) ] p t s λx ρ = f q p u m s x r ))] p 0 as. Therefore λx W 0 m s, F, Q, p, u, t), for all λ C ad for all x = x ) W 0 m s, F, Q, p, u, t). Next, suppose that x = x ), y = y ) W 0 m s, F, Q, p, u, t). The there exists r, r 2 > 0 such that ad = Thus give ε > 0, there exists, 2 > 0 such that ad = t f q p u m )) ] p s x r 0 as t f q p u m )) ] p s y r 2 0 as. = t f q p u m )) ] p s x r < εp, for all = t f q p u m )) ] p s y r 2 < εp, for all 2. = Let r = r r 2 r + r 2 ) ad 0 = max, 2 ). The we have for all 0, t f q p u m s x + y )r ))] p t f q p u m ) s x r r2 r + r 2 ) t + f q p u m ) s y r 2 r r + r 2 ) ] p < εp. = Hece x + y W 0 m s, F, Q, p, u, t). Thus W 0 m s, F, Q, p, u, t) is a liear space. Similarly we ca prove that W m s, F, Q, p, u, t) ad W m s, F, Q, p, u, t) are liear spaces. Theorem 2.2. Let F = f ) be a sequece of modulus fuctios ad p = p ) be a bouded sequece of positive real umbers. The the space W 0 m s, F, Q, p, u, t) is a complete paraormed space with paraorm defied by where M = max, sup p }. gx) = sup f q p t u m s x r ))] p ) M, = Proof. Let x i) ) be a Cauchy sequece i W 0 m s, F, Q, p, u, t). The for a give ε > 0, there exists 0 such that gx i x j ) < ε, for all i, j 0. Thus, we have = ] t f q p u m s x i xj )r))) p M < ε, for all i, j 0. 2.) = t f q p u m s x i xj )r))) < ε, for all i, j 0. = m s x i xj ) < ε, for all i, j 0, for all N.

4 64 K. Raj et al. / Some vector-valued statistical coverget sequece spaces Hece x i ) i= is a Cauchy sequece i E, for each N. Sice E s are complete for each N, so x i ) i= coverges i E, for each N. O taig limit as j i 2.), we have = f q p t u m s x i x )r ))) p ] M < ε, for all i 0. = m s x i x) W 0 m s, F, Q, p, u, t). Sice W 0 m s, F, Q, p, u, t) is a liear space, so we have x = x i) x i) x) W 0 m s, F, Q, p, u, t). Thus W 0 m s, F, Q, p, u, t) is a complete paraormed space. This completes the proof of the theorem. Theorem 2.3. Let F = f ) be a sequece of modulus fuctios ad p = p ) be a bouded sequece of positive real umbers. The W 0 m s, F, Q, p, u, t) W m s, F, Q, p, u, t) W m s, F, Q, p, u, t). Proof. It is easy to prove so we omit the details. Theorem 2.4. Let F = f ) ad G = g ) be ay two sequeces of modulus fuctios. For ay bouded sequeces p = p ) ad t = t ) of strictly positive real umbers ad ay two sequeces of semiorms Q = q ), V = v ), the followig are true: i) W Z m s, f, Q, u, t) W Z m s, f g, Q, u, t), ii) W Z m s, F, Q, p, u, t) W Z m s, F, V, p, u, t) W Z m s, F, Q + V, p, u, t), iii) W Z m s, F, Q, p, u, t) W Z m s, G, Q, p, u, t) W Z m s, F + G, Q, p, u, t), iv) if q is stroger tha v, the W Z m s, F, Q, p, u, t) W Z m s, F, V, p, u, t), v) if q is equivalet v, the W Z m s, F, Q, p, u, t) = W Z m s, F, V, p, u, t), vi) W Z m s, F, Q, p, u, t) W Z m s, F, V, p, u, t) = ϕ. Proof. We shall prove i) for the case Z = 0. Let ε > 0. We choose δ, 0 < δ <, such that f t) < ε for 0 t δ ad all N. We write y = g t q p u m s x r )) ad cosider = f y ) ] = f y ) ] + f y ) ], 2 where the first summatio is over y δ ad the secod summatio is over y > δ. Sice f is cotiuous, we have f y ) ] < ε. 2.2) By the defiitio of f, we have the followig relatio for y > δ: Hece 2 f y ) < 2 f ) y δ. f y ) ] 2δ f ) y. 2.3) = It follows from 2.2) ad 2.3) that W Z m s, f, Q, u, t) W Z m s, f g, Q, u, t). Similarly, we ca prove the result for other cases. Theorem 2.5. Let f be a modulus fuctio. The W Z m s, Q, u, t) W Z m s, f, Q, u, t). Proof. It is easy to prove i view of Theorem 2.4i). ) Theorem 2.6. Let 0 < p < r ad r p be bouded. The W Z m s, F, Q, r, u, t) W Z m s, F, Q, p, u, t). t Proof. By taig y = f q p u m s x r ))] r for all ad usig the same techique as i Theorem 5 of Maddox 5], oe ca easily prove the theorem. Theorem 2.7. Let f be a modulus fuctio. If lim m f m) m = β > 0, the W m s, Q, p, u, t) W m s, f, Q, p, u, t). Proof. It is easy to prove so we omit the details.

5 K. Raj et al. / Some vector-valued statistical coverget sequece spaces 65 3 u q,t -Statistical Covergece The otio of statistical covergece was itroduced by Fast 6] ad Schoeberg 24] idepedetly. Over the years ad uder differet ames, statistical covergece has bee discussed i the theory of Fourier aalysis, ergodic theory ad umber theory. Later o, it was further ivestigated from the sequece space poit of view ad lied with summability theory by Fridy 7], Coor 5], Salat 2], Murasalee 7], Isi 0], Savas 22], Malowsy ad Savas 6], Kol 3], Maddox 5], Tripathy ad Se 27] ad may others. I recet years, geeralizatios of statistical covergece have appeared i the study of strog itegral summability ad the structure of ideals of bouded cotiuous fuctios o locally compact spaces. Statistical covergece ad its geeralizatios are also coected with subsets of the Stoe-Cech compactificatio of atural umbers. Moreover, statistical covergece is closely related to the cocept of covergece i probability. The otio depeds o the desity of subsets of the set N of atural umbers. Defiitio 3.3. A subset E of N is said to have the atural desity δe) if the followig limit exists: δe) = lim χ E ), = where χ E is the characteristic fuctio of E. It is clear that ay fiite subset of N has zero atural desity ad δe c ) = δe). Defiitio 3.4. A sequece x = x ) is said to be u q,t -statistical coverget to l if for every ε > 0, δ N : q p t u m s x r l ) ε }) = 0. I this case we write x l Su,t) q. The set of all uq,t -statistical coverget sequeces is deoted by S q u,t. By S, we deote the set of all statistically coverget sequeces. If qx) = x, u = p = t = for all N ad r =, the S q u,t is same as S. I case l = 0 we write Sq 0u,t istead of S q u,t. Theorem 3.8. Let p = p ) be a bouded sequece ad 0 < h = if p p sup p = H < ad let f be a modulus fuctio. The = W m s, f, q, p, u, t) S q u,t. Proof. Let x W m s, f, q, p, u, t) ad let ε > 0 be give. Let ad 2 deote the sums over with qp t u m s x r l) ε ad qp t u m s x r l) < ε, respectively. The f q p t u m s x r l ))] p f q p t u m s x r l ))] p Hece, x S q u,t. ] p f ε) ] h, ] ) H mi f ε) f ε) : qp t u m ] h, ] ) H s x l) ε} mi f ε) f ε). Theorem 3.9. Let f be a bouded modulus fuctio. The S q u,t W m s, f, q, p, u, t). Proof. Suppose that f is bouded. Let ɛ > 0 ad let ad 2 be the sums itroduced i the Theorem 3.. Sice f is bouded, there exists a iteger K such that f x) < K for all x 0. The

6 66 K. Raj et al. / Some vector-valued statistical coverget sequece spaces = f q p t u m s x r l ))] p Hece, x W m s, f, q, p, u, t). f q p t u m s x r l ))] p + f q p t u m s x r l ))] p ) 2 ] p f ε) maxk h, K H ) + 2 maxk h, K H ) : qp t u m s x l) ε} + max f ε) h, f ε) H ). Theorem 3.0. S q u,t = W m s, f, q, p, u, t) if ad oly if f is bouded. Proof. Let f be bouded. By Theorems 3. ad 3.2, we have S q u,t = W m s, f, q, p, u, t). Coversely, suppose that f is ubouded. The there exists a sequece t ) of positive umbers with f t ) = 2 for =, 2,. If we choose The we have p t u i m t, i = 2, =, 2, s x i r = 0, otherwise. : p t u m s x r ɛ} for all, ad so x S q u,t but x / W m s, f, q, p, u, t) for X = C, qx) = x ad p = for all N. This cotradicts the assumptio that S q u,t = W m s, f, q, p, u, t). Refereces ] Altio H., Alti Y., Isi M. The Sequece Space Bv σ M, P, Q, S) O Semiormed Spaces, Idia J. Pure Appl. Math., ), ] Cola R. O ivariat sequece spaces, Erc. Uiv. J. Sci., 5 989), ] Cola R. O certai sequece spaces ad their Kothe-Toeplitz duals, Red. Mat. Appl., Ser, 3 993), ] Et M., Cola R. O some geeralized differece sequece spaces, Soochow J. Math., 2 995), ] Coor J. S. A topological ad fuctioal aalytic approach to statistical covergece, Appl. Numer. Harmoic Aal.,999), ] Fast H. Sur la covergece statistique, Colloq. Math., 2 95), ] Fridy J. A. O the statistical covergece, Aalysis, 5 985), ] Goes G., Goes S. Sequece of bouded variatio ad sequeces of Fourier coefficiets, Math. Zeit, 8 970), ] Gupta M. The geeralized spaces l X) ad m 0 X), J. Math. Aal. Appl., ), ] Isi M. O statistical covergece of geeralized differece sequece spaces, Soochow J. Math., ), ] Kamtha P. K. Bases i a certai class of Frechet spaces, Tamag J. Math., 7 976), ] Kizmaz H. O certai sequece spaces, Cad. Math. Bull., 24 98), ] Kol E. The statistical covergece i Baach spaces, Acta. Commet. Uiv. Tartu, ), ] Leoard I. E. Baach sequece spaces, J. Math. Aal. Appl., ),

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