Let X and Y be weak hypervector spaces and Lw (X,Y ) be the set of all weak linear operators from X into Y . We prove some algebraic properties of Lw (X,Y )

Hosseinzadeh, Roja

Taghavi, Ali

English

APAV - Academy of Sciences, Letters, Arts and Technology (E-Journals)

RATIO MATHEMATICA 22 (2012) 37-43 ISSN:1592-7415Operators on Weak HypervectorSpacesAli Taghavi and Roja HosseinzadehDepartment of Mathematics, Faculty of Basic Sciences,University of Mazandaran, P. O. Box 47416-1468, Babolsar, Iran.Taghavi@umz.ac.ir, ro.hosseinzadeh@umz.ac.irAbstractLet X and Y be weak hypervector spaces and Lw(X;Y ) be the setof all weak linear operators from X into Y . We prove some algebraicproperties of Lw(X;Y ).Key words: weak hypervector space, weak subhypervector space,normal weak hypervector space, weak linear operator2000 AMS subject classications: 46J10, 47B48.1 IntroductionThe concept of hyperstructure was rst introduced by Marty [3] in 1934and has attracted attention of many authors in last decades and has con-structed some other structures such as hyperrings, hypergroups, hypermod-ules, hyperelds, and hypervector spaces. These constructions has been ap-plied to many disciplines such as geometry, hypergraphs, binary relations,combinatorics, codes, cryptography, probability and etc. A wealth of appli-cations of this concepts are given in [1  2] and [12].In 1988 the concept of hypervector space was rst introduced by Scafati-Tallini. She studied more properties of this new structure in [11]. In [11],Tallini introduced the concept of norm on weak hypervector spaces. We usedthis denition to extend some theorems of analysis from classic vector spacesto hypervector spaces. For example see [5 7; 9]. Moreover, in [4] we denedthe concept of dimension of weak hypervector spaces and also authors in [8]introduce the new concept hyperalgebra and quotient hyperalgebra.37A. Taghavi and R. HosseinzadehNow we want to use some of our dened concepts and prove some algebraicproperties of Lw(X; Y ), where Lw(X; Y ) is the set of all weak linear operatorsfrom the weak hypervector space X into the weak hypervector space Y . Notethat the hypervector spaces used in this paper are the special case wherethere is only one hyperoperation, the external one, all the others are ordinaryoperations. The general hypervector spaces have all operations multivaluedalso in the hypereld (see [12]).2 PreliminariesWe need some Preliminary denitions for to state our results. In thissection we state them.Denition 2.1. [11] A weak or weakly distributive hypervector space over aeld F is a quadruple (X,+,o,F ) such that (X,+) is an abelian group ando : F X  ! P(X) is a multivalued product such that(i) 8a 2 F; 8x; y 2 X; [ao(x+ y)] \ [aox+ aoy] 6= ;;(ii) 8a; b 2 F; 8x 2 X; [(a+ b)ox] \ [aox+ box] 6= ;;(iii) 8a; b 2 F; 8x 2 X; ao(box) = (ab)ox;(iv) 8a 2 F; 8x 2 X; ao( x) = ( a)ox =  (aox);(v) 8x 2 X; x 2 1ox:We call (i) and (ii) weak right and left distributive laws, respectively.Note that the set ao(box) in (3) is of the form [y2boxaoy.Denition 2.2. [11] Let (X;+; o; F ) be a weak hypervector space over a eldF , that is the eld of real or complex numbers. We dene a pseudonorm inX as a mapping k:k : X  ! R, of X into the reals such that:(i) k0k = 0,(ii) 8x; y 2 X; kx+ yk  kxk+ kyk,(iii) 8a 2 F; 8x 2 X; sup kaoxk = jajkxk.Denition 2.3. Let X and Y be hypervector spaces over F . A map T :X  ! Y is called(i) linear if and only ifT (x+ y) = T (x) + T (y); T (aox)  aoT (x); 8x; y 2 X; a 2 F38Operators on Weak Hypervector Spaces(ii) antilinear if and only ifT (x+ y) = T (x) + T (y); T (aox)  aoT (x); 8x; y 2 X; a 2 F(iii) strong linear if and only ifT (x+ y) = T (x) + T (y); T (aox) = aoT (x); 8x; y 2 X; a 2 F:3 Main resultsBefore to state our results we describe some fundamental concepts andlemmas from [4]. For more details see [4]. By Lemma 3.1 in [4] we havethe following denition. Throughout paper, suppose that X and Y are weakhypervector spaces over a eld F .Denition 3.1. [4] If a 2 F and x 2 X, then zaox for 0 6= a is that elementof aox such that x 2 a 1ozaox and for a = 0, we dene zaox = 0.As the descriptions in [4], zaox is not unique, necessarily. So the set ofall these elements denoted by Zaox. In the mentioned paper we introduced acertain category of weak hypervector spaces. These weak hypervector spaceshave been called "normal". We proved that Zaox is singleton in a normalweak hypervector space.Denition 3.2. [4] Suppose X satisfy the following conditions:(i) (Za1ox + Za2ox) \ Z(a1+a2)ox 6= ;; 8x 2 X; 8a1; a2 2 F ,(ii) (Zaox1 + Zaox2) \ Zao(x1+x2) 6= ;; 8x1; x2 2 X; 8a 2 F .Then X is called a normal weak hypervector space.Lemma 3.1. [4] If a 2 F , 0 6= b 2 F and x 2 X, then the followingproperties hold:(i) x 2 Z1ox;(ii) aoZbox = abox;(iii) Z aox =  Zaox;(iv) If X is normal, then Zaox is singleton.In [4], the following lemma stated a criterion for normality of a weakhypervector space.39A. Taghavi and R. HosseinzadehLemma 3.2. [4] X is normal if and only if(i) za1ox + za2ox = z(a1+a2)ox; 8x 2 X; 8a1; a2 2 F ,(ii) zaox1 + zaox2 = zao(x1+x2); 8x1; x2 2 X; 8a 2 F .Denition 3.3. [6] Let T : X  ! Y be an operator. T is said to be boundedif there exists a positive real number K such that we havekTxk  Kkxk (8x 2 X):Denition 3.4. [9] A map T : X  ! Y is called weak linear operator if Tis additive and satisesT (Zaox)  aoTx; (a 2 F; x 2 X):Denote the set of all weak linear operators and the set of all bounded weaklinear operators from X into Y by Lw(X; Y ) and Bw(X; Y ), respectively.Theorem 3.1. [4] Let X be normal. Then X with the same dened sum andthe following scalar product is a classical vector space:a:x = zaox; 8a 2 F; x 2 X:Lemma 3.3. Let Y be normal, T 2 Lw(X; Y ) and a 2 F . DeneaT : X ! Yx 7! a:TxThen aT is a weak linear operator, where the operation '.' is the dened scalarproduct in Theorem 3.1. Moreover, for all a; b 2 F and T; S 2 Lw(X;Y ) wehavea(T + S) = aT + aS;(a+ b)T = aT + bT:Proof. Let a:u = zaou, where u 2 Y . From Theorem ?, we know that Ywith this scalar product is a classical vector space. Let x; y 2 X and b 2 F .By Lemma 3.1 we have(aT )(zbox) = a:T (zbox)  a:(boTx)= fa:u : u 2 boTxg= fzaou : u 2 boTxg= zao(boTx) = zaboTx= bozaoTx = bo(a:Tx) = bo(aT )x40Operators on Weak Hypervector Spacesand also from the normality of Y , we obtain(aT )(x+ y) = a:T (x+ y)  a:(Tx+ Ty)= zao(Tx+Ty)= zaoTx + zaoTy= a:Tx+ a:Ty= (aT )x+ (aT )y:Hence aT is a weak linear operator. Now let T; S 2 Lw(X; Y ) and x 2 X.The normality of Y yields[a(T + S)]x = a:(T + S)x = zao(T+S)x = zao(Tx+Sx)= zaoTx + zaoSx= a:Tx+ a:Sx= (aT )x+ (aS)x= (aT + aS)xwhich implies thata(T + S) = aT + aS:The second relation is proved in a similar way.Theorem 3.2. Let Y be normal. Then Lw(X;Y ) with the following sum andproduct is a weak hypervector space over F .(T + S)x = Tx+ Sx (T; S 2 Lw(X; Y ); x 2 X)aoT = fS 2 Lw(X; Y ) : Sx 2 aoTx; 8x 2 Xg (a 2 F; T 2 Lw(X; Y )):Proof. First we show that aoT is a nonempty subset of Lw(X;Y ). ByLemma 3.3, aT 2 Lw(X; Y ) and for any x 2 X we have(aT )x = a:Tx = zaoTx 2 aoTxwhich imply that aT 2 aoT . It is easy to check that (Lw(X; Y );+) is anabelian group. We show the correctness the rst property of scalar product,the rest properties are obtained in a similar way. By Lemma 3.3, for alla 2 F and T; S 2 Lw(X;Y ) we havea(T + S) = aT + aS:This together witha(T + S) 2 ao(T + S); aT 2 aoT ; aS 2 aoS41A. Taghavi and R. Hosseinzadehimply that[ao(T + S)] \ [aoT + aoS] 6= ;and this completes the proof.Theorem 3.3. Let Y be normal. Then the following statements are hold.(i) For all a 2 F and T 2 Lw(X; Y ) we have zaoT = aT .(ii) Lw(X; Y ) is a normal weak hypervector space.Proof. (i) By Denition 3.1 we havezaoT 2 aoT ; T 2 a 1ozaoTwhich for all x 2 X implieszaoTx 2 (aoT )x; Tx 2 (a 1ozaoT )x:Since by Theorem 3.2 we have(aoT )x  aoTx; (a 1ozaoT )x  a 1ozaoTx;we obtainzaoTx 2 aoTx; Tx 2 a 1ozaoTx:These relations, by Denition 3.1 yield that zaoTx = zaoTx. So we obtainzaoTx = (aT )x, for all x 2 X and hence zaoT = aT .(ii) The normality of Lw(X; Y ) can be concluded from Lemma 3.3 andpart (i).Theorem 3.4. Let Y be normal. Then Bw(X;Y ) with the dened sum andscalar product in Theorem ? is a subhypervector space of Lw(X;Y ).Proof. It is enough to show that T + S; aoT 2 Bw(X;Y ) for any a 2 Fand T; S 2 Bw(X; Y ) it is easy to check that T+S 2 Bw(X;Y ). Let S 2 aoT .Hence Sx 2 aoTx and sokSxk  jajkTxk  jajkTkkxk:This completes the proof.Acknowledgements. This research is partially supported by the Re-search Center in Algebraic Hyperstructures and Fuzzy Mathematics, Univer-sity of Mazandaran, Babolsar, Iran.42Operators on Weak Hypervector SpacesReferences[1] P. Corsini, Prolegomena of hypergroup theory, Aviani editore, (1993).[2] P. Corsini and V. Leoreanu, Applications of Hyperstructure theory,Kluwer Academic Publishers, Advances in Mathematics (Dordrecht),(2003).[3] F. Marty, Sur nue generalizeation de la notion de group, 8th congress ofthe Scandinavic Mathematics, Stockholm, (1934), 45-49.[4] A. Taghavi and R. Hosseinzadeh, A note on dimension of weak hyper-vector spaces, Italian J. of Pure and Appl. Math, To appear.[5] A. Taghavi and R. Hosseinzadeh, Hahn-Banach Theorem for function-als on hypervector spaces, The Journal of Mathematics and ComputerScience, Vol .2 No.4 (2011) 682-690.[6] A. Taghavi and R. Hosseinzadeh, Operators on normed hypervectorspaces, Southeast Asian Bulletin of Mathematics, (2011) 35: 367-372.[7] A. Taghavi and R. Hosseinzadeh, Uniform Boundedness Principle foroperators on hypervector spaces, Iranian Journal of Mathematical Sci-ences and Informatics, Accepted.[8] A. Taghavi and R. Parvinianzadeh, Hyperalgebras and Quotient Hyper-algebras, Italian J. of Pure and Appl. Math, No. 26 (2009) 17-24.[9] A. Taghavi, T. Vougiouklis, R. Hosseinzadeh, A note on Operators onNormed Finite Dimensional Weak Hypervector Spaces , Scientic Bul-letin, Accepted.[10] M.Scafati-Tallini, Characterization of remarkable Hypervector space, Al-gebraic Hyperstructures and Aplications, Samotraki, Greece, (2002),Spanidis Press, Xanthi, (2003), 231-237.[11] M.Scafati-Tallini, Weak Hypervector space and norms in such spaces,Algebraic Hyperstructures and Applications, Jasi, Rumania, HadronicPress. (1994), 199{206.[12] T. Vougiouklis, Hyperstructures and their representations, HadronicPress, (1994).43

Operators on Weak Hypervector Spaces

Ali Taghavi

Roja Hosseinzadeh

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RATIO MATHEMATICA 22 (2012) 37-43 ISSN:1592-7415Operators on Weak HypervectorSpacesAli Taghavi and Roja HosseinzadehDepartment of Mathematics, Faculty of Basic Sciences,University of Mazandaran, P. O. Box 47416-1468, Babolsar, Iran.Taghavi@umz.ac.ir, ro.hosseinzadeh@umz.ac.irAbstractLet X and Y be weak hypervector spaces and Lw(X,Y ) be the setof all weak linear operators from X into Y . We prove some algebraicproperties of Lw(X,Y ).Key words: weak hypervector space, weak subhypervector space,normal weak hypervector space, weak linear operator2000 AMS subject classifications: 46J10, 47B48.1 IntroductionThe concept of hyperstructure was first introduced by Marty [3] in 1934and has attracted attention of many authors in last decades and has con-structed some other structures such as hyperrings, hypergroups, hypermod-ules, hyperfields, and hypervector spaces. These constructions has been ap-plied to many disciplines such as geometry, hypergraphs, binary relations,combinatorics, codes, cryptography, probability and etc. A wealth of appli-cations of this concepts are given in [1− 2] and [12].In 1988 the concept of hypervector space was first introduced by Scafati-Tallini. She studied more properties of this new structure in [11]. In [11],Tallini introduced the concept of norm on weak hypervector spaces. We usedthis definition to extend some theorems of analysis from classic vector spacesto hypervector spaces. For example see [5−7, 9]. Moreover, in [4] we definedthe concept of dimension of weak hypervector spaces and also authors in [8]introduce the new concept hyperalgebra and quotient hyperalgebra.37A. Taghavi and R. HosseinzadehNow we want to use some of our defined concepts and prove some algebraicproperties of Lw(X, Y ), where Lw(X, Y ) is the set of all weak linear operatorsfrom the weak hypervector space X into the weak hypervector space Y . Notethat the hypervector spaces used in this paper are the special case wherethere is only one hyperoperation, the external one, all the others are ordinaryoperations. The general hypervector spaces have all operations multivaluedalso in the hyperfield (see [12]).2 PreliminariesWe need some Preliminary definitions for to state our results. In thissection we state them.Definition 2.1. [11] A weak or weakly distributive hypervector space over afield F is a quadruple (X,+,o,F ) such that (X,+) is an abelian group ando : F ×X −→ P∗(X) is a multivalued product such that(i) ∀a ∈ F, ∀x, y ∈ X, [ao(x+ y)] ∩ [aox+ aoy] ̸= ∅,(ii) ∀a, b ∈ F, ∀x ∈ X, [(a+ b)ox] ∩ [aox+ box] ̸= ∅,(iii) ∀a, b ∈ F, ∀x ∈ X, ao(box) = (ab)ox,(iv) ∀a ∈ F, ∀x ∈ X, ao(−x) = (−a)ox = −(aox),(v) ∀x ∈ X, x ∈ 1ox.We call (i) and (ii) weak right and left distributive laws, respectively.Note that the set ao(box) in (3) is of the form ∪y∈boxaoy.Definition 2.2. [11] Let (X,+, o, F ) be a weak hypervector space over a fieldF , that is the field of real or complex numbers. We define a pseudonorm inX as a mapping ∥.∥ : X −→ R, of X into the reals such that:(i) ∥0∥ = 0,(ii) ∀x, y ∈ X, ∥x+ y∥ ≤ ∥x∥+ ∥y∥,(iii) ∀a ∈ F, ∀x ∈ X, sup ∥aox∥ = |a|∥x∥.Definition 2.3. Let X and Y be hypervector spaces over F . A map T :X −→ Y is called(i) linear if and only ifT (x+ y) = T (x) + T (y), T (aox) ⊆ aoT (x), ∀x, y ∈ X, a ∈ F38Operators on Weak Hypervector Spaces(ii) antilinear if and only ifT (x+ y) = T (x) + T (y), T (aox) ⊇ aoT (x), ∀x, y ∈ X, a ∈ F(iii) strong linear if and only ifT (x+ y) = T (x) + T (y), T (aox) = aoT (x), ∀x, y ∈ X, a ∈ F.3 Main resultsBefore to state our results we describe some fundamental concepts andlemmas from [4]. For more details see [4]. By Lemma 3.1 in [4] we havethe following definition. Throughout paper, suppose that X and Y are weakhypervector spaces over a field F .Definition 3.1. [4] If a ∈ F and x ∈ X, then zaox for 0 ̸= a is that elementof aox such that x ∈ a−1ozaox and for a = 0, we define zaox = 0.As the descriptions in [4], zaox is not unique, necessarily. So the set ofall these elements denoted by Zaox. In the mentioned paper we introduced acertain category of weak hypervector spaces. These weak hypervector spaceshave been called ”normal”. We proved that Zaox is singleton in a normalweak hypervector space.Definition 3.2. [4] Suppose X satisfy the following conditions:(i) (Za1ox + Za2ox) ∩ Z(a1+a2)ox ̸= ∅, ∀x ∈ X, ∀a1, a2 ∈ F ,(ii) (Zaox1 + Zaox2) ∩ Zao(x1+x2) ̸= ∅, ∀x1, x2 ∈ X, ∀a ∈ F .Then X is called a normal weak hypervector space.Lemma 3.1. [4] If a ∈ F , 0 ̸= b ∈ F and x ∈ X, then the followingproperties hold:(i) x ∈ Z1ox;(ii) aoZbox = abox;(iii) Z−aox = −Zaox;(iv) If X is normal, then Zaox is singleton.In [4], the following lemma stated a criterion for normality of a weakhypervector space.39A. Taghavi and R. HosseinzadehLemma 3.2. [4] X is normal if and only if(i) za1ox + za2ox = z(a1+a2)ox, ∀x ∈ X, ∀a1, a2 ∈ F ,(ii) zaox1 + zaox2 = zao(x1+x2), ∀x1, x2 ∈ X, ∀a ∈ F .Definition 3.3. [6] Let T : X −→ Y be an operator. T is said to be boundedif there exists a positive real number K such that we have∥Tx∥ ≤ K∥x∥ (∀x ∈ X).Definition 3.4. [9] A map T : X −→ Y is called weak linear operator if Tis additive and satisfiesT (Zaox) ⊆ aoTx, (a ∈ F, x ∈ X).Denote the set of all weak linear operators and the set of all bounded weaklinear operators from X into Y by Lw(X, Y ) and Bw(X, Y ), respectively.Theorem 3.1. [4] Let X be normal. Then X with the same defined sum andthe following scalar product is a classical vector space:a.x = zaox, ∀a ∈ F, x ∈ X.Lemma 3.3. Let Y be normal, T ∈ Lw(X, Y ) and a ∈ F . DefineaT : X → Yx 7→ a.TxThen aT is a weak linear operator, where the operation ’.’ is the defined scalarproduct in Theorem 3.1. Moreover, for all a, b ∈ F and T, S ∈ Lw(X,Y ) wehavea(T + S) = aT + aS,(a+ b)T = aT + bT.Proof. Let a.u = zaou, where u ∈ Y . From Theorem ?, we know that Ywith this scalar product is a classical vector space. Let x, y ∈ X and b ∈ F .By Lemma 3.1 we have(aT )(zbox) = a.T (zbox) ⊆ a.(boTx)= {a.u : u ∈ boTx}= {zaou : u ∈ boTx}= zao(boTx) = zaboTx= bozaoTx = bo(a.Tx) = bo(aT )x40Operators on Weak Hypervector Spacesand also from the normality of Y , we obtain(aT )(x+ y) = a.T (x+ y) ⊆ a.(Tx+ Ty)= zao(Tx+Ty)= zaoTx + zaoTy= a.Tx+ a.Ty= (aT )x+ (aT )y.Hence aT is a weak linear operator. Now let T, S ∈ Lw(X, Y ) and x ∈ X.The normality of Y yields[a(T + S)]x = a.(T + S)x = zao(T+S)x = zao(Tx+Sx)= zaoTx + zaoSx= a.Tx+ a.Sx= (aT )x+ (aS)x= (aT + aS)xwhich implies thata(T + S) = aT + aS.The second relation is proved in a similar way.Theorem 3.2. Let Y be normal. Then Lw(X,Y ) with the following sum andproduct is a weak hypervector space over F .(T + S)x = Tx+ Sx (T, S ∈ Lw(X, Y ), x ∈ X)aoT = {S ∈ Lw(X, Y ) : Sx ∈ aoTx, ∀x ∈ X} (a ∈ F, T ∈ Lw(X, Y )).Proof. First we show that aoT is a nonempty subset of Lw(X,Y ). ByLemma 3.3, aT ∈ Lw(X, Y ) and for any x ∈ X we have(aT )x = a.Tx = zaoTx ∈ aoTxwhich imply that aT ∈ aoT . It is easy to check that (Lw(X, Y ),+) is anabelian group. We show the correctness the first property of scalar product,the rest properties are obtained in a similar way. By Lemma 3.3, for alla ∈ F and T, S ∈ Lw(X,Y ) we havea(T + S) = aT + aS.This together witha(T + S) ∈ ao(T + S), aT ∈ aoT , aS ∈ aoS41A. Taghavi and R. Hosseinzadehimply that[ao(T + S)] ∩ [aoT + aoS] ̸= ∅and this completes the proof.Theorem 3.3. Let Y be normal. Then the following statements are hold.(i) For all a ∈ F and T ∈ Lw(X, Y ) we have zaoT = aT .(ii) Lw(X, Y ) is a normal weak hypervector space.Proof. (i) By Definition 3.1 we havezaoT ∈ aoT , T ∈ a−1ozaoTwhich for all x ∈ X implieszaoTx ∈ (aoT )x, Tx ∈ (a−1ozaoT )x.Since by Theorem 3.2 we have(aoT )x ⊆ aoTx, (a−1ozaoT )x ⊆ a−1ozaoTx,we obtainzaoTx ∈ aoTx, Tx ∈ a−1ozaoTx.These relations, by Definition 3.1 yield that zaoTx = zaoTx. So we obtainzaoTx = (aT )x, for all x ∈ X and hence zaoT = aT .(ii) The normality of Lw(X, Y ) can be concluded from Lemma 3.3 andpart (i).Theorem 3.4. Let Y be normal. Then Bw(X,Y ) with the defined sum andscalar product in Theorem ? is a subhypervector space of Lw(X,Y ).Proof. It is enough to show that T + S, aoT ∈ Bw(X,Y ) for any a ∈ Fand T, S ∈ Bw(X, Y ) it is easy to check that T+S ∈ Bw(X,Y ). Let S ∈ aoT .Hence Sx ∈ aoTx and so∥Sx∥ ≤ |a|∥Tx∥ ≤ |a|∥T∥∥x∥.This completes the proof.Acknowledgements. This research is partially supported by the Re-search Center in Algebraic Hyperstructures and Fuzzy Mathematics, Univer-sity of Mazandaran, Babolsar, Iran.42Operators on Weak Hypervector SpacesReferences[1] P. Corsini, Prolegomena of hypergroup theory, Aviani editore, (1993).[2] P. Corsini and V. Leoreanu, Applications of Hyperstructure theory,Kluwer Academic Publishers, Advances in Mathematics (Dordrecht),(2003).[3] F. Marty, Sur nue generalizeation de la notion de group, 8th congress ofthe Scandinavic Mathematics, Stockholm, (1934), 45-49.[4] A. Taghavi and R. Hosseinzadeh, A note on dimension of weak hyper-vector spaces, Italian J. of Pure and Appl. Math, To appear.[5] A. Taghavi and R. Hosseinzadeh, Hahn-Banach Theorem for function-als on hypervector spaces, The Journal of Mathematics and ComputerScience, Vol .2 No.4 (2011) 682-690.[6] A. Taghavi and R. Hosseinzadeh, Operators on normed hypervectorspaces, Southeast Asian Bulletin of Mathematics, (2011) 35: 367-372.[7] A. Taghavi and R. Hosseinzadeh, Uniform Boundedness Principle foroperators on hypervector spaces, Iranian Journal of Mathematical Sci-ences and Informatics, Accepted.[8] A. Taghavi and R. Parvinianzadeh, Hyperalgebras and Quotient Hyper-algebras, Italian J. of Pure and Appl. Math, No. 26 (2009) 17-24.[9] A. Taghavi, T. Vougiouklis, R. Hosseinzadeh, A note on Operators onNormed Finite Dimensional Weak Hypervector Spaces , Scientific Bul-letin, Accepted.[10] M.Scafati-Tallini, Characterization of remarkable Hypervector space, Al-gebraic Hyperstructures and Aplications, Samotraki, Greece, (2002),Spanidis Press, Xanthi, (2003), 231-237.[11] M.Scafati-Tallini, Weak Hypervector space and norms in such spaces,Algebraic Hyperstructures and Applications, Jasi, Rumania, HadronicPress. (1994), 199–206.[12] T. Vougiouklis, Hyperstructures and their representations, HadronicPress, (1994).43

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Operators on Weak Hypervector Spaces

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