The purpose of this paper is to introduce finite convergence sequences and
functions preserving convergence of series in fuzzy n-normed spaces

Elagan, Sayed

Rahmat, Mohamad Rafi Segi

English

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

On Functions Preserving Convergence of Series in Fuzzy n-Normed Spaces

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International J.Math. Combin. Vol.3 (2009), 61-70On Functions Preserving Convergence of Series inFuzzy n-Normed SpacesSayed ElaganDepartment of Mathematics and Statistics, Faculty of Science, Taif University,Taif , El-Haweiah, P.O.Box 888, Zip Code 21974, Kingdom of Saudi Arabia (KSA).E-mail: sayed khalil2000@yahoo.comMohamad Rafi Segi RahmatSchool of Applied Mathematics, University of Nottingham Malaysia Campus,Jalan Broga, 43500 Semenyih, Selangor Darul EhsanE-mail: Mohd.Rafi@nottingham.edu.myAbstract: The purpose of this paper is to introduce finite convergence sequences andfunctions preserving convergence of series in fuzzy n-normed spaces.Keywords: Pseudo-Euclidean space, Smarandache space, fuzzy n-normed spaces, n-seminorm; function preserving convergenceAMS(2000): 34A10, 34C10§1. IntroductionA Pseudo-Euclidean space is a particular Smarandache space defined on a Euclidean space Rnsuch that a straight line passing through a point p may turn an angle θp ≥ 0. If θp > 0 , then pis called a non-Euclidean point. Otherwise, a Euclidean point. In this paper, normed spaces areconsidered to be Euclidean, i.e., every point is Euclidean. In [7], S. Ga¨hler introduced n-normson a linear space. A detailed theory of n-normed linear space can be found in [9,12,14,15].In [9], H. Gunawan and M. Mashadi gave a simple way to derive an (n − 1)- norm from then-norm in such a way that the convergence and completeness in the n-norm is related to thosein the derived (n − 1)-norm. A detailed theory of fuzzy normed linear space can be foundin [1,2,4,5,6,11,13,18]. In [16], A. Narayanan and S. Vijayabalaji have extended the n-normedlinear space to fuzzy n-normed linear space and in [20] the authors have studied the completenessof fuzzy n-normed spaces.The main purpose of this paper is to study the results concerning infinite series (see,[3,17,19,21]) in fuzzy n-normed spaces. In section 2, we quote some basic definitions of fuzzyn-normed spaces. In section 3, we consider the absolutely convergent series in fuzzy n- normedspaces and obtain some results on it. In section 4, we study the property of finite convergencesequences in fuzzy n-normed spaces. In the last section we introduce and study the concept of1Received July 8, 2009. Accepted Aug.18, 2009.62 Sayed Elagan and Mohamad Rafi Segi Rahmatfunction preserving convergence of series in fuzzy n-norm spaces and obtain some results.§2. PreliminariesLet n be a positive integer, and let X be a real vector space of dimension at least n. We recallthe definitions of an n-seminorm and a fuzzy n-norm [16].Definition 2.1 A function (x1, x2, . . . , xn) 7→ ‖x1, . . . , xn‖ from Xn to [0,∞) is called ann-seminorm on X if it has the following four properties:(S1) ‖x1, x2, . . . , xn‖ = 0 if x1, x2, . . . , xn are linearly dependent;(S2) ‖x1, x2, . . . , xn‖ is invariant under any permutation of x1, x2, . . . , xn;(S3) ‖x1, . . . , xn−1, cxn‖ = |c|‖x1, , . . . , xn−1, xn‖ for any real c;(S4) ‖x1, . . . , xn−1, y + z‖ 6 ‖x1, . . . , xn−1, y‖+ ‖x1, . . . , xn−1, z‖.An n-seminorm is called a n-norm if ‖x1, x2, . . . , xn‖ > 0 whenever x1, x2, . . . , xn arelinearly independent.Definition 2.2 A fuzzy subset N of Xn × R is called a fuzzy n-norm on X if and only if:(F1) For all t 6 0, N(x1, x2, . . . , xn, t) = 0;(F2) For all t > 0, N(x1, x2, . . . , xn, t) = 1 if and only if x1, x2, . . . , xn are linearly dependent;(F3) N(x1, x2, . . . , xn, t) is invariant under any permutation of x1, x2, . . . , xn;(F4) For all t > 0 and c ∈ R, c 6= 0,N(x1, x2, . . . , cxn, t) = N(x1, x2, . . . , xn,t|c|);(F5) For all s, t ∈ R,N(x1, . . . , xn−1, y + z, s+ t) ≥ min {N(x1, . . . , xn−1, y, s), N(x1, . . . , xn−1, z, t)} .(F6) N(x1, x2, . . . , xn, t) is a non-decreasing function of t ∈ R andlimt→∞N(x1, x2, . . . , xn, t) = 1.The pair (X,N) will be called a fuzzy n−normed space.Theorem 2.1 Let A be the family of all finite and nonempty subsets of fuzzy n-normed space(X,N) and A ∈ A. Then the system of neighborhoodsB = {B(t, r, A) : t > 0, 0 < r < 1, A ∈ A}On Functions Preserving Convergence of Series in Fuzzy n-Normed Spaces 63where B(t, r, A) = {x ∈ X : N(a1, · · · , an−1, x, t) > 1 − r, a1, · · · , an−1 ∈ A} is a base of thenull vector θ, for a linear topology on X, named N -topology generated by the fuzzy n-norm N .Proof We omit the proof since it is similar to the proof of Theorem 3.6 in [8]. Definition 2.3 A sequence {xk} in a fuzzy n− normed space (X,N) is said to convergeto x if given r > 0, t > 0, 0 < r < 1, there exists an integer n0 ∈ N such thatN(x1, x2, . . . , xn−1, xk − x, t) > 1− r for all k ≥ n0.Definition 2.4 A sequence {xk} in a fuzzy n− normed space (X,N) is said to be Cauchysequence if given ǫ > 0, t > 0, 0 < ǫ < 1, there exists an integer n0 ∈ N such thatN(x1, x2, . . . , xn−1, xm − xk, t) > 1− ǫ for all m, k ≥ n0.Theorem 2.1([13]) Let N be a fuzzy n− norm on X. Define for x1, x2, . . . , xn ∈ X andα ∈ (0, 1)‖x1, x2, . . . , xn‖α = inf {t : N(x1, x2, . . . , xn, t) ≥ α } .Then the following statements hold.(A1) for every α ∈ (0, 1), ‖•, •, . . . , •‖α is an n− seminorm on X ;(A2) If 0 < α < β < 1 and x1, x2, . . . , xn ∈ X then‖x1, x2, . . . , xn‖α ≤ ‖x1, x2, . . . , xn‖β .Example 2.3[10, Example 2.3] Let ‖•, •, . . . , •‖ be a n-norm onX . Then defineN(x1, x2, . . . , xn, t) =0 if t ≤ 0 and, for t > 0,N(x1, x2, . . . , xn, t) =tt+ ‖x1, x2, . . . , xn‖ .Then the seminorms (2.1) are given by‖x1, x2, . . . , xn‖α = α1− α‖x1, x2, . . . , xn‖.§3. Absolutely Convergent Series in Fuzzy n−Normed SpacesIn this section we introduce the notion of the absolutely convergent series in a fuzzy n-normedspace (X,N) and give some results on it.Definition 3.1 The series∞∑k=1xk is called absolutely convergent in (X,N) if∞∑k=1‖a1, ..., an−1, xk‖α <∞for all a1, ..., an−1 ∈ X and all α ∈ (0, 1).64 Sayed Elagan and Mohamad Rafi Segi RahmatUsing the definition of ‖...‖α the following lemma shows that we can express this conditiondirectly in terms of N.Lemma 3.1 The series∞∑k=1xk is absolutely convergent in (X,N) if, for every a1, ..., an−1 ∈ Xand every α ∈ (0, 1) there are tk ≥ 0 such that∞∑k=1tk < ∞ and N(a1, . . . , an−1, xk, tk) ≥ αfor all k.proof Let∞∑k=1xk be absolutely convergent in (X,N). Then∞∑k=1‖a1, ..., an−1, xk‖α <∞for every a1, ..., an−1 ∈ X and every α ∈ (0, 1). Let a1, ..., an−1 ∈ X and α ∈ (0, 1). For everyk there is tk ≥ 0 such that N(a1, . . . , an−1, xk, tk) ≥ α andtk < ‖a1, ..., an−1, xk‖α +12k.Then∞∑k=1tk <∞∑k=1‖a1, ..., an−1, xk‖α +∞∑k=112k<∞.The other direction is even easier to show. Definition 3.2 A fuzzy n−normed space (X,N) is said to be sequentially complete if everyCauchy sequence in it is convergent.Lemma 3.2 Let (X,N) be sequentially complete, then every absolutely convergent series∞∑k=1xkconverges and ∥∥∥∥∥a1, ..., an−1,∞∑k=1xk∥∥∥∥∥α≤∞∑k=1‖a1, ..., an−1, xk‖αfor every a1, ..., an−1 ∈ X and every α ∈ (0, 1).Proof Let∞∑k=1xk be an infinite series such that∞∑k=1‖a1, ..., an−1, xk‖α < ∞ for everya1, ..., an−1 ∈ X and every α ∈ (0, 1). Let yn =n∑k=1xk be a partial sum of the series. Leta1, ..., an−1 ∈ X , α ∈ (0, 1) and ǫ > 0. There is N such that∞∑k=N+1‖a1, ..., an−1, xk‖α < ǫ.On Functions Preserving Convergence of Series in Fuzzy n-Normed Spaces 65Then, for n > m ≥ N,|‖a1, ..., an−1, yn‖α − ‖a1, ..., an−1, ym‖α| ≤ ‖a1, ..., an−1, yn − ym‖α≤n∑k=m+1‖a1, ..., an−1, xk‖α≤∞∑k=N+1‖a1, ..., an−1, xk‖α< ǫ.This is shows that {yn} is a Cauchy sequence in (X,N) . But since (X,N) is sequentiallycomplete, the sequence {yn} converges and so the series∞∑k=1xk converges. Definition 3.3 Let I be any denumerable set . We say that the family (xα)α∈I of elements ina complete fuzzy n−normed space (X,N) is absolutely summable, if for a bijection Ψ of N(theset of all natural numbers) onto I the series∞∑n=1xΨ(n) is absolutely convergent.The following result may not be surprising but the proof requires some care.Theorem 3.1 Let (xα)α∈I be an absolutely summable family of elements in a sequentiallycomplete fuzzy n− normed space (X,N). Let (Bn) be an infinite sequence of a non-empty subsetof A, such that A = ∪nBn, Bi ∩ Bj = ∅ for i 6= j, then if zn =∑α∈Bnxα, the series∞∑n=0zn isabsolutely convergent and∞∑n=0zn =∑α∈Ixα.Proof It is easy to see that this is true for finite disjoint unions I =N∪n=1Bn. Now considerthe disjoint unions I =∞∪n=1Bn. By Lemma 3.2∞∑n=1‖a1, ..., an−1, zn‖α ≤∞∑n=1∑i∈Bn‖a1, ..., an−1, xi‖α=∑i∈I‖a1, ..., an−1, xi‖α <∞for every a1, ..., an−1 ∈ X, and every α ∈ (0, 1). Therefore,∞∑n=0zn is absolutely convergent.Let y =∑i∈Ixi, z =∞∑n=1zn. Let ǫ > 0, a1, ..., an−1 ∈ X and α ∈ (0, 1). There is a finite setJ ⊂ I such that∑i/∈J‖a1, ..., an−1, xi‖α <ǫ2.Choose N large enough such that B =N∪n=1Bn ⊃ J and∥∥∥∥∥a1, ..., an−1, z −N∑n=1zn∥∥∥∥∥α<ǫ2.66 Sayed Elagan and Mohamad Rafi Segi RahmatThen ∥∥∥∥∥a1, ..., an−1, y −∑i∈Bxi∥∥∥∥∥α<ǫ2.By the first part of the proofN∑n=1zn =∑i∈Bx.Therefore, ‖a1, ..., an−1, y − z‖α < ǫ. This is true for all ǫ so ‖a1, ..., an−1, y − z‖α = 0. Thisis true for all a1, ..., an−1 ∈ X , α ∈ (0, 1) and (X, N) is Hausdorff see [8, Theorem 3.1].Hence y = z. Definition 3.4 Let (X∗, N) be the dual of fuzzy n-normed space (X,N). A linear functionalf : X∗ → K where K is a scalar field of X is said to be bounded linear operator if there existsa λ > 0 such that‖a1, · · · , an−1, f(xk)‖α ≤ λ‖a1, · · · , an−1, xk‖α,for all a1, · · · , an−1 ∈ X and all α ∈ (0, 1).Definition 3.5 The series∑∞k=1 xk is said to be weakly absolutely convergent in (X,N) if∞∑k=1‖a1, · · · , an−1, f(xk)‖α <∞for all f ∈ X∗, all a1, · · · , an−1 ∈ X and all α ∈ (0, 1).Theorem 3.2 Let the series∑∞k=1 xk be weakly absolutely convergence in (X,N). Then thereexists a constant λ > 0 such that∞∑k=1‖a1, · · · , an−1, f(xk)‖α ≤ λ‖a1, · · · , an−1, f(xk)‖αProof Let {er}∞r=1 be a standard basis of the space (X,N). Define continuous operatorsSr : X∗ → X where r ∈ Z by the formula Sr(f) =∑rk=1 f(xk)ek, we have‖a1, · · · , an−1, Sr(f)‖α =r∑k=1‖a1, · · · , an−1, f(xk)ek‖α.Since for any fixed f ∈ X∗, the numbers ‖a1, · · · , an−1, Sr(f)‖α are bounded by∑∞k=1 ‖a1, · · · , an−1, f(xk)‖α,by Banach-Steinhaus theorem, we havesupr‖a1, · · · , an−1, Sr(f)‖α = λ <∞.Therefore,∞∑k=1‖a1, · · · , an−1, f(xk)‖α = supr‖a1, · · · , an−1, Sr(f)‖α≤ λ‖a1, · · · , an−1, f(xk)‖α.On Functions Preserving Convergence of Series in Fuzzy n-Normed Spaces 67§4. Finite Convergent Sequences in Fuzzy n−Normed SpacesIn this section our principal goal is to show that every sequence having finite convergent propertyis Cauchy and every Cauchy sequence has a subsequence which has finite convergent propertyin every metrizable fuzzy n−normed space (X,N).Definition 4.1 A sequence {xk} in a fuzzy n−normed space (X,N) is said to have finiteconvergent property if∞∑j=1‖a1, ..., an−1, xj − xj−1‖α <∞for all a1, ..., an−1 ∈ X and all α ∈ (0, 1).Definition 4.2 A fuzzy n− normed space (X,N) is said to be metrizable, if there is a metricd which generates the topology of the space.Theorem 4.1 Let (X,N) be a metrizable fuzzy n−normed space, then every sequence havingfinite convergent propperty is Cauchy and every Cauchy sequence has a subsequence which hasfinite convergent property.proof SinceX is metrizable, there is a sequence{‖a1,r, ..., an−1,r, x‖αr}for all a1,r, ..., an−1,r ∈X and all αr ∈ (0, 1) generating the topology of X . We choose an increasing sequence {mk,1}such that∞∑k=1∥∥a1,1, ..., an−1,1, xmk+1,1 − xmk,1∥∥α1 <∞where a1,1, ..., an−1,1 ∈ X and α1 ∈ (0, 1) . Then we choose a subsequence mk,2 of mk,1 suchthat∞∑k=1∥∥a1,2, ..., an−1,2, xmk+1,2 − xmk,2∥∥α2 <∞where a1,2, ..., an−1,2 ∈ X and α2 ∈ (0, 1) . Continuing in this way we construct recursivelysequences mk,r such that mk,r+1 is a subsequence of mk,r and such that∞∑k=1∥∥a1,r, ..., an−1,r, xmk+1,r − xmk,r∥∥αr <∞for all a1,r, ..., an−1,r ∈ X and all αr ∈ (0, 1) . Now consider the diagonal sequence mk = mk,k.Let r ∈ N. The sequence {mk}∞k=r is a subsequence of {mk,r}∞k=r . Let k ≥ r. There are pairs ofintegers (u, v) , u < v such that mk = mu,r and mk+1 = mv,r. Then by the triangle inequality∥∥a1,r, ..., an−1,r, xmk+1 − xmk∥∥αr ≤ v−1∑i=u∥∥a1,r, ..., an−1,r, xmi+1,r − xmi,r∥∥αrand therefore,∞∑k=r∥∥a1, ..., an−1, xmk+1 − xmk∥∥α ≤ ∞∑j=r∥∥a1, ..., an−1, xmj+1,r − xmj,r∥∥α68 Sayed Elagan and Mohamad Rafi Segi Rahmatfor all a1, ..., an−1 ∈ X and all α ∈ (0, 1) . The statement of the theorem follows. The above theorem shows that many Cauchy sequence has a subsequence which has finiteconvergent. Therefore, it is natural to ask for an example of Cauchy sequence has a subsequencewhich has not finite convergent property.Example 4.2 We consider the set S consisting of all convergent real sequences. Let X be thespace of all functions f : S → R equipped with the topology of pointwise convergence. Thistopology is generated by‖f1,s, ..., fn−1,s, f‖αs = |f (s)| ,for all f1,s, ..., fn−1,s, f ∈ X and all αs ∈ (0, 1) , where s ∈ S. Then consider the sequencefn ∈ X defined by fn (s) = sn where s = (sn) ∈ S. The sequence fn is a Cauchy sequence in Xbut there is no subsequence fnk such that∞∑k=1∥∥f1,s, ..., fn−1,s, fnk+1 − fnk∥∥αs <∞for all s ∈ S. We see this as follows. If n1 < n2 < n3 < ... is a sequence then define sn = (−1)k 1kfor nk ≤ n < nk+1. Then s = (sn) ∈ S but∞∑k=1∥∥f1,s, ..., fn−1,s, fnk+1 − fnk∥∥αs = ∞∑k=1∣∣snk+1 − snk ∣∣ ≥ ∞∑k=11k=∞.§5. Functions Preserving Convergence of Series in Fuzzy n−Normed SpacesIn this section we shall introduce the functions f : X → X that preserve convergence of seriesin fuzzy n−normed spaces. Our work is an extension of functions f : R → R that preserveconvergence of series studied in [19] and [3].We read in Cauchy’s condition in (X,N) as follows: the series∑∞k=1 xk converges if andonly if for every ǫ > 0 there is an N so that for all n ≥ m ≥ N ,‖a1 · · · , an−1,n∑k=mxk‖ < ǫ,where a1 · · · , an−1 ∈ X .Definition 5.1 A function f : X × X → X is said to be additive in fuzzy n-normed space(X,N) if‖a1, · · · , an−1, f(x, y)‖α = ‖a1, · · · , an−1, f(x)‖α + ‖a1, · · · , an−1, f(y)‖α,for each x, y ∈ X, a1, · · · , an−1 ∈ X and for all α ∈ (0, 1).Definition 5.2 A function f : X → X is convergence preserving (abbreviated CP) in (X,N) iffor every convergent series∑∞k=1 xk, the series∑∞k=1 f(xk) is also convergent, i.e., for everya1, · · · , an−1 ∈ X,∞∑k=1‖a1, · · · , an−1, f(xk)‖α <∞On Functions Preserving Convergence of Series in Fuzzy n-Normed Spaces 69whenever∑∞k=1 ‖a1, · · · , an−1, xk‖α <∞.Theorem 5.1 Let (X,N) be a fuzzy n-normed space and f : X → X be an additive andcontinuous function in the neighborhood B(t, r, A). Then the function f is CP of infinite seriesin (X,N).Proof Assume that f is additive and continuous inB(α, δ, A) = {x ∈ X : ‖a1, · · · , an−1, x‖α <δ}, where a1, · · · , an−1 ∈ A and δ > 0. From additivity of f in B(α, δ, A) implies that f(0) = 0.Let∑∞k=1 xk be a absolute convergent series and xk ∈ X (k = 1, 2, 3, · · · ). We show that∑∞k=1 f(xk) is also absolute convergent.By Cauchy condition for convergence of series, there exists a k ∈ N such that for everyp ∈ N‖a1, · · · , an−1,k+p∑j=k+1xj‖α < δ2.From this we have‖a1, · · · , an−1,∞∑j=k+1xj‖α < δ2.By the additivity of f in B(α, δ, A), we get‖a1, · · · , an−1, f(k+p∑j=k+1xj)‖α = ‖a1, · · · , an−1,k+p∑j=k+1f(xj)‖α < δ2.Now, let yp =∑k+pj=k+1 xj for p = 1, 2, 3, · · · and y =∑∞j=k+1 xj belong to the neighborhoodB(α, δ, A). The function f is continuous in B(α, δ, A), i.e., f(yp) → f(y) because yp → y forp→∞. Hencelimp→∞‖a1, · · · , an−1, f(k+p∑j=k+1xj)‖α = ‖a1, · · · , an−1, f(∞∑j=k+1xj)‖α.This implieslimp→∞‖a1, · · · , an−1,k+p∑j=k+1f(xj)‖α = ‖a1, · · · , an−1,∞∑j=k+1f(xj)‖αand this guarantee the convergence of the series∑∞j=k+1 f(xj) and therefore the series∑∞j=1 f(xj)must also be convergent in X , i.e., the function f is CP infinite series in (X,N). References[1] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math.,11 (2003), no. 3, 687-705.[2] S.C. Chang and J. N. Mordesen, Fuzzy linear operators and fuzzy normed linear spaces,Bull. Calcutta Math. Soc., 86 (1994), no. 5, 429-436.[3] M. Dindos, I. Martisovits and T. Salat, Remarks on infinite series in linear normed spaces,http://tatra.mat.savba.sk/Full/19/04DINDOS.ps70 Sayed Elagan and Mohamad Rafi Segi Rahmat[4] C. Felbin, Finite- dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48(1992), no. 2, 239-248.[5] C. Felbin, The completion of a fuzzy normed linear space, J. Math. Anal. 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A theorem on infinite series,

Complete fuzzy n-normed space,

Convergence preserving function: an alternative discussion,

Convergence preserving functions,

Finite dimensional fuzzy normed linear space.

Finite dimensional fuzzy normed linear spaces,

Finite- dimensional fuzzy normed linear space,

Fuzzy linear operators and fuzzy normed linear spaces,

Fuzzy n− normed linear spaces,

Fuzzy topological vector spaces.

n-inner product spaces,

On generalized fuzzy normed spaces,

On n-normed spaces,

On the completeness of fuzzy normed linear spaces,

Remarks on infinite series in linear normed spaces, http://tatra.mat.savba.sk/Full/19/04DINDOS.ps 70 Sayed Elagan and Mohamad Rafi Segi Rahmat

On Functions Preserving Convergence of Series in Fuzzy n-Normed Spaces

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