# PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

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1 STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract. In this paper we introduce the sequence spaces ĉ 0 (p,f,q,s), ĉ(p,f,q,s) and ˆm(p,f,q,s) using a modulus function f and defined over a seminormed space (X,q) seminormed by q. We study some properties of these sequence spaces and obtain some inclusion relations. 1. Introduction Let m, c and c 0 be the Banach spaces of bounded, convergent and null sequences x = (x k ) with the usual norm x = sup x k. Let D be the shift operator k 0 on s, that is, Dx = (x k ) k=1, D2 x = (x k ) k=2 and so on. It may be recalled that a Banach limit (see Banach [1]) L is a nonnegative linear functional on m such that L is invariant under shift operator (that is, L (Dx) = L (x) for x m) and L (e) = 1, where e = (1, 1,...). A sequence x m is almost convergent (see Lorentz [8]) if all Banach limits of x coincide. Let ĉ denote the space of almost convergent sequences. It is proved by Lorentz [8] that ĉ = } x : lim t m,n (x) exists uniformly in n m where t m,n (x) = 1 m + 1 m D i x n, i=0 ( D 0 = 1 ). Several authors including Duran [5], King [7] and Nanda ( [12],[13]) have studied almost convergent sequences. Received by the editors: Mathematics Subject Classification. 40A05. Key words and phrases. Modulus function, seminorm, almost convergence, sequence spaces. 13

2 YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK The notion of a modulus function was introduced by Nakano [11] in We recall that a modulus f is a function from [0, ) to [0, ) such that (i) f(x) = 0 if and only if x = 0, (ii) f(x+y) f(x)+f(y), for all x 0, y 0, (iii) f is increasing, (iv) f is continuous from the right at 0. Since f (x) f (y) f ( x y ), it follows from condition (iv) that f is continuous on [0, ). Furthermore, we have f (nx) nf (x) for all n N, from condition (ii), and so f (x) = f ( nx 1 ) ( x nf, n n) hence 1 ( x ) n f (x) f n for all n N. A modulus may be bounded or unbounded. For example, f (x) = x p, (0 < p 1) is unbounded and f (x) = x 1+x [14] used a modulus function to construct some sequence spaces. is bounded. Maddox [10] and Ruckle After then some sequence spaces, defined by a modulus function, were introduced and studied by Bhardwaj [2], Bilgin [3], Connor [4], Esi [6], and many others. Definition 1.1. Let q 1, q 2 be seminorms on a vector space X. Then q 1 is said to be stronger than q 2 if whenever (x n ) is a sequence such that q 1 (x n ) 0, then also q 2 (x n ) 0. If each is stronger than the other q 1 and q 2 are said to be equivalent (one may refer to Wilansky [15]). Lemma 1.1. Let q 1 and q 2 be seminorms on a linear space X. Then q 1 is stronger than q 2 if and only if there exists a constant M such that q 2 (x) Mq 1 (x) for all x X (see for instance Wilansky [15]). Let p = (p m ) be a sequence of strictly positive real numbers and X be a seminormed space over the field C of complex numbers with the seminorm q. We 14

3 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION define the sequence spaces as follows: ĉ 0 (p, f, q, s) = ĉ (p, f, q, s) = ˆm (p, f, q, s) = where f is a modulus function. } x X : lim m m s [(f (q (t m,n (x))))] pm = 0 uniformly in n, x X : lim m m s [(f (q (t m,n (x le))))] pm = 0 for some l, uniformly in n}, } x X : sup m s [(f (q (t m,n (x))))] pm <. m,n The following inequalities will be used throughout the paper. Let p = (p m ) be a bounded sequence of strictly positive real numbers with 0 < p m sup p m = H, C = max ( 1, 2 H 1), then where a m, b m C. a m + b m pm C a m pm + b m pm }, (1.1) 2. Main results Theorem 2.1. Let p = (p m ) be a bounded sequence, then ĉ 0 (p, f, q, s), ĉ (p, f, q, s), ˆm (p, f, q, s) are linear spaces. Proof. We give the proof for ĉ 0 (p, f, q, s) only. The others can be treated similarly. Let x, y ĉ 0 (p, f, q, s). For λ, µ C, there exist positive integers M λ and N λ such that λ M λ and µ N µ. Since f is subadditive and q is a seminorm m s [f (q (t m,n (λx + µy)))] pm C (M λ ) H m s [f (q (t m,n (x)))] pm + C (N µ ) H m s [f (q (t m,n (y)))] pm 0,uniformly in n. This proves that ĉ 0 (p, f, q, s) is a linear space. Theorem 2.2. The space ĉ 0 (p, f, q, s) is a paranormed space, paranormed by g (x) = sup m s ([f (q (t m,n (x)))] pm ) 1 M, m,n where M = max (1, sup p m ). The spaces ĉ (p, f, q, s), ˆm (p, f, q, s) are paranormed by g, if inf p m > 0. 15

4 YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Proof. Omitted. Theorem 2.3. Let f be modulus function, then (i) ĉ 0 (p, f, q, s) ˆm (p, f, q, s), (ii) ĉ (p, f, q, s) ˆm (p, f, q, s). Proof. We prove the second inclusion, since the first inclusion is obvious. Let x ĉ (p, f, q, s), by definition of a modulus function (the inequality (ii)), we have m s [f (q (t m,n (x)))] pm Cm s [f (q (t m,n (x l)))] pm + Cm s [f (q (l))] pm. Then there exists an integer K l such that q (l) K l. Hence, we have m s [f (q (t m,n (x)))] pm Cm s [f (q (t m,n (x l)))] pm +Cm s max(1, [(K l ) f (1)] H ), (1) so x ˆm (p, f, q, s). Theorem 2.4. Let f, f 1, f 2 be modulus functions q, q 1, q 2 seminorms and s, s 1, s 2 0. Then (i) If s > 1 then Z (f 1, q, s) Z (f f 1, q, s), (ii) Z (p, f 1, q, s) Z (p, f 2, q, s) Z (p, f 1 + f 2, q, s), (iii) Z (p, f, q 1, s) Z (p, f, q 2, s) Z (p, f, q 1 + q 2, s), (iv) If q 1 is stronger than q 2 then Z (p, f, q 1, s) Z (p, f, q 2, s), (v) If s 1 s 2 then Z (p, f, q, s 1 ) Z (p, f, q, s 2 ), (vi) If q 1 =(equivalent to) q2, then Z (p, f, q 1, s) = Z (p, f, q 2, s), where Z = ˆm, ĉ and ĉ 0. Proof. (i) We prove this part for Z = ĉ and the rest of the cases will follow similarly. Let x ĉ (p, f, q, s), so that S m = m s [f 1 (q (t m,n (x l)))] 0. Let ε > 0 and choose δ with 0 < δ < 1 such that f (c) < ε for 0 t δ. Now we write 16

5 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION For f 1 (q (t m,n (x l))) > δ, I 1 = m N : f 1 (q (t m,n (x l))) δ} I 2 = m N : f 1 (q (t m,n (x l))) > δ}. f 1 (q (t m,n (x l))) < f 1 (q (t m,n (x l))) δ 1 < 1 + [ f1 (q (t m,n (x l))) δ 1 ] where m I 2 and [ u ] denotes the integer part of u. By the definition of f we have for f 1 (q (t m,n (x l))) > δ, f (f 1 (q (t m,n (x l)))) ( 1 + [ f 1 (q (t m,n (x l))) δ 1 ]) f (1) For f 1 (q (t m,n (x l))) δ, 2f (1) f 1 (q (t m,n (x l))) δ 1. (2.1) f (f 1 (q (t m,n (x l)))) < ε (2.2) where m I 1. By (2.1) and (2.2) we have m s [f (f 1 (q (t m,n (x l))))] m s ε + [ 2f (1) δ 1] S m,uniformly n. Hence ĉ (p, f 1, q, s) ĉ (p, f f 1, q, s). (ii) The proof follows from the following inequality 0. as m m s [(f 1 + f 2 ) (q (t m,n (x)))] pm Cm s [f 1 (q (t m,n (x)))] pm +Cm s [f 2 (q (t m,n (x)))] pm. (iii), (iv) (v)and (vi) follow easily. Corollary 2.1. Let f be a modulus function, then we have (i) If s > 1, Z (p, q, s) Z (p, f, q, s), (ii) Z (p, f, q) Z (p, f, q, s), (iii) Z (p, q) Z (p, q, s), (iv) Z (f, q) Z (f, q, s) 17

6 YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK where Z = ˆm, ĉ and ĉ 0. The proof is straightforward. Theorem 2.5. For any two sequences p = (p k ) and r = (r k ) of positive real numbers and for any two seminorms q 1 and q 2 on X we have Z (p, f, q 1, s) Z (r, f, q 2, s). Proof. The proof follows from the fact that the zero element θ belongs to each of the classes of sequences involved in the intersection. Theorem 2.6. For any two sequences p = (p m ) and r = (r m ), we have ĉ 0 (r, f, q, s) ĉ 0 (p, f, q, s) if and only if lim inf pm r m > 0. Proof. If we take y m = f (q (t m,n (x))) for all m N, then using the same technique of lemma 1 of Maddox [9], it is easy to prove the theorem. Theorem 2.7. For any two sequences p = (p m ) and r = (r m ), we have ĉ 0 (r, f, q, s) = ĉ 0 (p, f, q, s) if and only if lim inf pm r m > 0 and lim inf rm p m > 0. Theorem 2.8. Let 0 < p m r m 1. Then ˆm (r, f, q, s) is closed subspace of ˆm (p, f, q, s). Proof. Let x ˆm (r, f, q, s). Then there exists a constant B > 1 such that k s [f (t m,n (x))] rm/m B for all m, n and so k s [f (t m,n (x))] pm/m B for all m, n. Thus x ˆm (p, f, q, s). To show that ˆm (r, f, q, s) is closed, suppose that x i ˆm (r, f, q, s) and x i x ˆm (p, f, q, s). Then for every 0 < ε < 1, there exists N such that for all m, n k s [ f ( ( t m,n x i x ))] p m/m B for all i > N. Now k s [ f ( ( t m,n x i x ))] r m/m < k s [ f ( ( t m,n x i x ))] p m/m < ε for all i > N. Therefore x ˆm (r, f, q, s). This completes the proof. 18

7 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION References [1] Banach, S. : Théorie des operations linéaires, Warszawa, [2] Bhardwaj, V.K. : A generalization of a sequence space of Ruckle, Bull. Calcutta Math. Soc. 95 (5) (2003), [3] Bilgin, T : The sequence space l(p, f, q, s) on seminormed spaces. Bull. Calcutta Math. Soc. 86 (4) (1994), [4] Connor, J.S. : On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bul. 32 (2) (1989), [5] Duran, J.P. : Infinite matrices and almost convergence, Math. Zeit., 128 (1972), [6] Esi, A. : A new sequence space defined by a modulus function. J. Anal. 8 (2000), [7] King, J.P. : Almost summable sequences, Proc. Amer. Math. Soc., 16 (1966), [8] Lorentz, G.G. : A contribution the theory of divergent series, Acta Math. 80 (1948), [9] Maddox, I.J. : Spaces of strongly summable sequences, Ouart. J.Math. Oxford 2 (18) (1967), [10] Maddox, I.J. : Sequence spaces defined by a modulus, Math. Proc. Camb. Phil. Soc. 100 (1986), [11] Nakano, H. : Concave modulars, J. Math. Soc. Japan. 5 (1953), [12] Nanda, S. : Infinite matrices and almost convergence, Journal of Indian Math.Soc. 40 (1976), [13] Nanda, S. : Matrix transformations and almost boundedness, Glas. Mat., III. Ser. 14 (34) (1979), [14] Ruckle, W.H. : FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math. 25 (1973), [15] Wilansky, A. : Functional Analysis, Blaisdell Publishing Company, New York, Department of Mathematics, Firat University, 23119, Elazığ-TURKEY address: 19

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