On Locally Convex Topological Vector Space Valued Null Function Space c 0 (S,T, Φ, ξ, u) Defined by Semi Norm and Orlicz Function

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1 Jounal of Intitute of Science and Technology, 204, 9(): 62-68, Intitute of Science and Technology, T.U. On Locally Convex Topological Vecto Space Valued Null Function Space c 0 (S,T, Φ, ξ, u) Defined by Semi Nom and Olicz Function Naayan Paad Pahai Cental Depatment of Mathematic Tibhuvan Univeity, Kitipu, Kathmandu, Nepal nppahai@gmail.com ABSTRACT The aim of thi pape i to intoduce and tudy a new cla c 0 (S, T, Φ, ξ, u) of locally convex pace T- valued function uing Olicz function Φ a a genealization of ome of the well known equence pace and function pace. Beide the invetigation petaining to the tuctue of the cla c 0 (S, T, Φ, ξ, u), ou pimaily inteet i to exploe ome of the containment elation of the cla c 0 (S, T, Φ, ξ, u) in tem of diffeent ξ and u o that uch a cla of function i contained in o equal to anothe cla of imila natue. Keywod: Solid pace, Olicz function, Olicz pace, locally convex topological vecto pace, eminom INTRODUCTION We begin with ecalling ome notation and baic definition that ae ued in thi pape. Definition : A equence pace S i aid to be olid if ξ = < ξ k > S and γ = < γ k > a equence of cala with γ k, fo all k, then γ ξ = < γ k ξ k > S. So fa, a good numbe of eeach wok have been done on vaiou type of equence pace and function pace. Definition 2: By an Olicz function we mean a continuou, non deceaing and convex function Φ: [0, ) [0, ) atifying Φ(0) = 0, Φ() > 0 fo > 0 and Φ () a. It i noted that an Olicz function i alway unbounded and an Olicz function atifie the inequality Φ(γ) γ Φ (), 0 < γ < (Kanoel'kiî & Rutickiî, 96). An Olicz function Φ can be epeented in the following integal fom ξ Φ (ξ) = 0 q(t) dt whee q, known a the kenel of Φ, i ightdiffeentiable fo t 0, q(0) =0, q(t) > 0 fo t > 0, q i non deceaing,and q(t) a t (Kanoel'kiî & Rutickiî, 96). Definition 3: Lindentau and Tzafii (97) ued the notion of Olicz function to contuct the equence pace l Φ of cala < ξ k > uch that k =.Φ ξ k < fo ome > 0. They poved that the pace l Φ equipped with the nom defined by ξ Φ = inf > 0:. Φ ξ k k = Thi become a Banach pace, which i called an Olicz equence pace. The pace l Φ (S) i cloely elated to the pace l p which i an Olicz equence pace with Φ () = p : p <. Subequently, Kamthan and Gupta (98), Rao and Ren (99), Paaha and Choudhay (994), Chen (996), Ghoh and Sivatava (999), Rao and Subemanina (2004), Sava and Patteon (2005), Bhadwaj and Bala (2007), Khan (2008), Baaiv and Altundag (2009), Kolk (20), Pahai (203), Sivatava and Pahai (20 & 203) and many othe have been intoduced and tudied the algebaic and topological popetie of vaiou equence pace uing Olicz function a a genealization of eveal well known equence pace. Definition 4: A topological linea pace S i a vecto pace S ove a topological field K (mot often the complex numbe C with thei tandad topologie) which i endowed with a topology I uch that if, 2 S, α K; the mapping: 62

2 Naayan Paad Pahai (i) vecto addition S SS uch that (, 2 ) + 2 and (ii) cala multiplication K S S uch that (α, ) α ae continuou. Thi topology I i called a vecto topology o a linea topology on S. If I i given by ome metic then the topological vecto pace i called a linea metic pace. All nomed pace o inne poduct pace endowed with the topology defined by it nom o inne poduct ae well-known example of topological vecto pace. A local bae of topological vecto pace S i a collection B of neighbouhood θ uch that evey neighbouhood of θ contain the membe of B. A et S in a topological vecto pace S i aid to be abobing if fo evey S thee exit an α > 0 uch that ν S fo all ν C uch that ν α ; and balanced if ν S S fo evey ν C uch that ν. It i called convex in S if fo evey α 0, we have α S + (-α) S S; and abolutely convex in S if it i both balanced and convex. Definition 5: The gauge o Minkowki functional of a et A in a vecto pace X i a map x q A (x) fom X into the extended et R + { } of non-negative eal numbe defined a follow: inf if thee exit > 0 uch that x A and q A (x) = if x A fo all > 0. Definition 6: A eminom (peudonom) on a linea pace S ove the cala C with zeo element θ i a ubadditive function p : S R + atifying p(α) = α p(), fo all α C and S. Clealy, if p() = 0 implie = θ,then p i a nom. If S i a vecto pace equipped with a family {p i : i I} of eminom then thee exit a unique locally convex topology I on S uch that each p i i I - continuou (Rudin, 99 & Pak, 2005). The cla c 0 (S, T, Φ, ξ, u) of locally convex pace valued function Let S be an abitay non empty et (not neceaily countable) and F(S) be the collection of all finite ubet of S. Let (T, I) be a Haudoff locally convex topological vecto pace (lctvs) ove the field of complex numbe C and T * be the topological dual of T. Let U(T) denote the fundamental ytem of balanced, convex and obeving neighbouhood of zeo vecto θ of T. p U will denote gauge o Minkowki functional of U (T). Thu, D = {p U : U (T)}i the collection of all continuou eminom geneating the topology I of T. Let u and w be any function on S R +, the et of poitive eal numbe, and l (S, R + ) = { u : S R + uch that up < }. Futhe, we wite ξ, η fo function on S C {0}, and the collection of all uch function will be denoted by (S, C {0}). We intoduce the following new cla of locally convex topological vecto pace valued function: c 0 (S, T, Φ, ξ, u) = {φ : S T : fo evey ε > 0 and p U D, thee exit J (S) uch that fo ome > 0, Φ [ p U (ξ () φ () )] < ε fo each S J }; When ξ : S C {0} i a function uch that ξ () = fo all. Then, c 0 (S, T, Φ, ξ, u) will be denoted by c 0 (S,T,Φ,u) and when u : S R + i a function uch that = fo all, then c 0 (S, T, Φ, ξ, u) will be denoted by c 0 (S, T, Φ, ξ). In fact, thee clae ae the genealization of the familia equence and function pace, tudied in Sivatava (996), Tiwai et al. (2008 & 200), Pahai (20), Sivatava and Pahai ( 20) uing nom. RESULTS We exploe the tuctue of the cla c 0 (S, T, Φ, ξ, u) of lc TVS T valued function by invetigating the condition in tem of diffeent u and ξ o that a cla i contained in o equal to anothe imila cla and theeby deive the condition of thei equality. We hall denote the zeo element of thi cla by θ,which we hall mean the function of θ : S T uch that θ () = 0, fo all S. Moeove, we hall fequently ue the notation L = up and A [α] = max (, α ), fo cala α. But when the function and w() occu, then to ditinguih L, we ue the notation L(u) and L(w) epectively. Theoem : The cla c 0 (S, T, Φ, ξ, u) fom a olid. Poof: Let φ c 0 (S, T, Φ, ξ, u), > 0 be aociated with φ and ε > 0. Then fo p U D, thee exit a J (S) uch that Φ [ p U (ξ() φ())] < ε fo evey S J. 63

3 On Locally Convex Topological Vecto Space Valued Null Function Space c 0 (S,T, Φ, ξ, u) Defined by Semi Nom and... Now, if we take cala α(), S uch that α(), then Φ [ p U (α() ξ() φ())] Φ α() [ p U (ξ() φ ())] Φ [ p U (ξ() φ())] < ε. Thi how that α φ c 0 (S, T, Φ, ξ, u) and hence c 0 (S, T, Φ, ξ, u) i olid. Theoem 2: If u l (S, R + ) and ξ, η (S, C {0}), then c 0 (S, T, Φ, ξ, u) c 0 (S, T, Φ, η, u) if lim inf ξ() > 0. η() Poof: Aume that lim inf ξ() > 0. η() Then thee exit m > 0 uch that, m η() < ξ () fo all but finitely many S. Let φ c 0 (S, T, Φ, ξ, u), > 0 be aociated with φ and ε > 0. Then fo p U D, thee exit J (S) uch that Φ [ p U (ξ () φ())] < ε fo each S J. Let u chooe uch that < m. Then fo uch, uing non deceaing popety of Φ, we have Φ Φ [ p U (η() φ())] [ ξ () p U (φ())] m = Φ [ η() p U (φ())] Φ [ p U (ξ () φ())] < ε, fo each S J. Since p U D i abitay in the above dicuion, theefoe we eaily get φ c 0 (S, T, Φ, η, u). Thi pove that c 0 (S, T, Φ, ξ, u) c 0 (S, T, Φ, η, u). Theoem 3: If u l (S, R + ),ξ, η (S, C {0}) and c 0 (S, T, Φ, ξ, u) c 0 (S, T, Φ, η, u), then lim inf ξ() η() > 0. Poof: Aume that c 0 (S, T, Φ, ξ, u) c 0 (S, T, Φ, η, u) but lim inf ξ() η() = 0. Then we can find a equence < k > of ditinct point in S uch that fo evey k, k ξ( k ) u( k ) < η( k ) u( k ).... () We now chooe t T and p V D uch that p V (t) = and define φ : S T by φ()= (ξ()) k / t fo = k k = 2 and......(2) θ othewie. Let > 0.Then fo each p U D and each k, we have Φ [ p U (ξ ( k ) φ( k ))] u( k ) [ p U (k / u( k ) t)] u( k ) = Φ k [ p U (t)] u( k ) = Φ k Φ A [(p U (t)) L(u) ] 0,a k and Φ [ p U (ξ () φ())] = 0, fo k, k. Thu fo a given ε > 0, we can find a finite ubet J of S atifying Φ [ p U (ξ () φ())] < ε fo all S J. Thi clealy how that φ c 0 (S, T, Φ, ξ, u). But fo each k, in view of equation () and (2), we have Φ [ p V (η( k ) φ( k ) )] u( k ) =Φ [ p V (η( k ) (ξ( k )) k /u( k) t)] u( k ) u( k ) =Φ k η( k ) [p λ( k ) V (t)] u( k ) Φ, which i independent of k. Thi how that φ c 0 (S, T, Φ, η, u), a contadiction. Thi complete the poof. On combining the Theoem 2 and 3, we get: Theoem 4: If u l (S, R + ) and ξ, η (S, C {0}), then c 0 (S, T, Φ, ξ, u) c 0 (S, T, Φ, η, u) if and only if lim inf ξ() > 0. η() 64

4 Naayan Paad Pahai Theoem 5:Let u l (S, R + ). Then fo any ξ, η (S, C {0}), c 0 (S, T, Φ, η, u) c 0 (S, T, Φ, ξ, u), if lim up ξ() <. η() Poof: Aume that lim up ξ() <. η() Then thee exit a contant d > 0 uch that ξ() < d η() fo all but finitely many S. Let φ c 0 (S, T, Φ, η, u), > 0 be aociated with φ and ε > 0. Then fo p U D, thee exit J (S) uch that Φ [ p U (η() φ ())] < ε fo each S J. Let u chooe uch that d <. Then fo uch, uing non deceaing popety of Φ, we have Φ [ p U (ξ() φ())] = Φ [ ξ() p U (φ())] Φ d η() [ p U (φ ())] Φ [ p U (η() φ ())] < ε fo all S J. Since p U D i an abitay, it clealy how that φ c 0 (S, T, Φ, ξ, u). Thi pove that c 0 (S, T, Φ, η, u) c 0 (S, T, Φ, ξ, u). Theoem 6:Let u l (S, R + ). Fo any ξ, η (S, C {0}) uch that c 0 (S, T, Φ, η, u) c 0 (S, T, Φ, ξ, u), then lim up ξ() <. η() Poof: Aume that c 0 (S, T, Φ, η, u) c 0 (S, T, Φ, ξ, u) but lim up ξ() =. η() Then thee exit a equence < k > of ditinct point in S uch that fo each k, ξ ( k ) u( k ) > k η( k ) u( k )..(3) Now, we chooe t T and p V D with p V (t) = and define φ : S T by φ () = (η()) k / t fo = k k = 2 and...(4) θ othewie. Let > 0. Then fo each p U D and each k, we have [ p U (η( k ) φ( k ))] u( k ) Φ [ p U (k /u(k) t)] u( k ) =Φ =Φ k [ p U (t)] u( k ) k Φ [ p U (t)] u( k ) k Φ A [( p U (t)) L(u) ] 0, a k and Φ [ p U (η() φ())] = 0, fo k, k. Thu fo given ε > 0, we can find a finite ubet J of S uch that Φ [ p U (η() φ())] < ε fo all S J. Thi how that φ c 0 (S, T, Φ, η, u). But on the othe hand fo each k and in view of equation (3) and (4), we have Φ [ p V (ξ( k ) φ( k ))] u( k ) =Φ [ p V (ξ( k ) (η( k )) k / u( k ) t)] u( k ) u( k ) = Φ k ξ( k ) [ p η( k ) V (t)] u( k ) Φ, which i independent of k. Thi how that φ c 0 (S, T, Φ, ξ, u), a contadiction. Thi complete the poof. On combining the Theoem 5 and 6, we get: Theoem 7: Let u l (S, R + ). Then fo any ξ, η (S, C {0}), c 0 (S, T, Φ, η, u) c 0 (S, T, Φ, ξ, u) if and only if lim up ξ() <. η() When Theoem 4 and 7 ae combined, we get: Theoem 8: Let u l (S, R + ). Then fo any ξ, η (S, C {0}), c 0 (S, T, Φ, ξ, u) = c 0 (S, T, Φ, η, u) if and only if 0 < lim inf ξ() lim up ξ() η() <. η() Coollay 9: Fo u l (S, + ) and ξ (S, C {0}). Then (i) c 0 (S, T, Φ, ξ, u) c 0 (S, T, Φ, u) if and only if lim inf ξ () > 0; 65

5 On Locally Convex Topological Vecto Space Valued Null Function Space c 0 (S,T, Φ, ξ, u) Defined by Semi Nom and... (ii) c 0 (S, T, Φ, u) c 0 (S, T, Φ, ξ, u) if and only if lim up ξ () < ; and (iii) c 0 (S, T, Φ, ξ, u) = c 0 (S, T, Φ, u) if and only if 0<lim inf ξ () lim up ξ () <. Poof: If we conide, η : S C {0} uch that η ()= fo each, in Theoem 4, 7 and 8, we eaily obtain the aetion (i), (ii) and (iii) epectively. Theoem 0: If u l (S, R + ), w : S R + and ξ ( S, C {0}), then c 0 ( S, T, Φ, ξ, u) c 0 (S, T, Φ, ξ, w) if lim inf w() > 0. Poof: Aume that lim inf w() > 0. Then thee exit m > 0 uch that w() > m fo all but finitely many S. Let φ c 0 (S, T, Φ, ξ, u), > 0 be aociated with φ and ε > 0. Then fo 0 < ρ < with ρ m Φ thee exit J (S) atifying Φ < ε and p U D, [ p U (ξ () φ())] < Φ ρ fo each S J. Since Φ i non deceaing, theefoe, [ p U (ξ () φ () )] < ρ and o [ p U (ξ () φ())] w() [(p U (ξ () φ())) ] m < ρ m Hence uing convexity of Φ, we have Φ [ p U (ξ() φ())] w() ρ m Φ Φ ρ m < ε, fo each S J. Since p U D i abitay, we eaily get: φ c 0 (S, T, Φ, ξ, w). Hence, c 0 (S, T, Φ, ξ, u) c 0 (S, T, Φ, ξ, w). Theoem : If u l (S, R + ), w : S R +, ξ ( S, C {0}) uch that c 0 ( S, T, Φ, ξ, u) c 0 (S, T, Φ, ξ, w), then lim inf w() > 0. Poof: Aume that the incluion, c 0 (S, T, Φ, ξ, u) c 0 (S, T, Φ, ξ, w) hold but lim inf w() = 0. Then thee exit a equence < k > of ditinct point in S uch that fo each k, k w( k ) < u( k ). (5) Now, taking p V D and t T with p V (t) = define φ : S T by the elation φ()= (ξ ()) k -/ t fo = k k = 2 3 and...(6) θ othewie. Let > 0.Then fo each p U D and k, we have Φ [ p U (ξ( k ) φ( k ))] u( k ) [ p U (k / u( k ) t)] u( k ) = Φ k Φ [ p U (t)] u( k ) k Φ and fo k, k, Φ Thi how that φ c 0 (S, T, Φ, ξ, u). A [(p U (t)) L(u) ] [ p U (ξ() φ())] = 0. On the othe hand fo each k and in view of equation (5) and (6), we have Φ [ p V (ξ( k ) φ( k ))] w( k ) [ p V (k / u( k ) t)] w( k ) =Φ Φ k /k Φ. e Thi how that φ c 0 (S, T, Φ, ξ, w), a contadiction. Thi complete the poof. On combining the Theoem 0 and, one can obtain: Theoem 2: If u l (S, R + ), w : S R + and ξ ( S, C {0}), then c 0 ( S, T, Φ, ξ, u) c 0 (S, T, Φ, ξ, w)if and only if lim inf w() > 0. Theoem 3: Let u : S R +, w l (S, R + ) and ξ (S, C {0}), then c 0 (S, T, Φ, ξ, w) c 0 (S, T, Φ, ξ, u) if lim up w() <. Poof: Aume that lim up w() <. Then thee exit a contant d > 0 uch that w() < d fo all but finitely many S. 66

6 Naayan Paad Pahai Let φ c 0 (S, T, Φ, ξ, w), > 0 be aociated with φ and ε > 0. Then fo 0 < ρ < with ρ /d Φ thee exit J (S) atifying Φ < ε and p U D, [ p U (ξ () φ())] w() < Φ ρ fo each S J. Since Φ i non deceaing, theefoe [ p U (ξ () φ () )] w() < ρ < and o [ p U (ξ () φ())] [( p U (ξ () φ())) w() ] /d < ρ /d. Hence uing the convexity of Φ, we have Φ [ p U (ξ () φ())] Φ η /d Φ ρ /d < ε fo each S J. Since p U D i abitay, thi clealy implie that φ c 0 (S, T, Φ, ξ, u) and hence c 0 (S, T, Φ, ξ, w) c 0 (S, T, Φ, ξ, u). Thi complete the poof. Theoem 4: Let u : S R +, w l (S, R + ), ξ (S, C {0}) and c 0 (S, T, Φ, ξ, w) c 0 (S, T, Φ,ξ, u), then lim up w() <. Poof: Suppoe that c 0 (S, T, Φ, ξ, w) c 0 (S, T, Φ, ξ, u) but lim up w() =. Then thee exit a equence < k > in S of ditinct point uch that w( k ) > k u( k ) fo each k.. (7) Now, taking p V D and t T with p V (t) =. We define φ : S T by φ() = (ξ ()) k -/w() t fo = k k...(8 θ othewie. Then analogou to the poof of Theoem and in view of equation (7) and (8), we can how that φ c 0 (S, T, Φ, ξ, w) and φ c 0 (S, T, Φ, ξ, u), a contadiction. The poof i now complete. On combining the Theoem 3 and 4, one can obtain: Theoem 5: Let u : S R +, w l (S, R + ) and ξ (S, C {0}), then c 0 (S, T, Φ, ξ, w) c 0 (S, T, Φ, ξ, u) if and only if lim up w() <. CONCLUSION Peent pape examined ome condition that chaacteize the linea pace tuctue and containment elation on the locally convex topological vecto pace valued null function defined by emi nom and Olicz function. In fact, thee eult can be ued fo futhe genealization to invetigate othe popetie of the function pace uing Olicz function. REFERENCES Baaiv, M. and Altundag, S On genealized paanomed tatitically convegent equence pace defined by Olicz function. Handawi. Pub. Co., J. of Inequality and Application. 2009: -3. Bhadwaj,V.N. and Bala, I Banach pace valued equence pace M (X, p). Int. J. of Pue and Appl. Math. 4(5): Chen, S.T Geomety of Olicz Space, Dietation Math. The Intitute of Math, Polih Academy of Science. Ghoh, D. and Sivatava P. D On Some Vecto Valued Sequence Space uing Olicz Function. Glanik Matematicki 34(54): Kamthan, P. K. and Gupta, M. 98. Sequence Space and Seie, Lectue Note in Pue and Applied Mathematic. Macel Dekke, New Yok, 65. Khan, V.A On a new equence pace defined by Olicz function. Common. Fac. Sci.Univ. Ank-eie 57(2): Kolk, E. 20. Topologie in genealized Olicz equence pace. Filomat. 25(4):9-2. Kanoel'kiî, M. A. and Rutickiî, Y. B. 96. Convex Function and Olicz Space, P. Noodhoff Ltd- Goningen-The Netheland. Lindentau, J. and Tzafii, L Claical Banachpace; Spinge-Velag, New Yok/ Belin. Pahai, N. P. 20. On Banach Space Valued Sequence Space l (X,M, λ, p, L) Defined by Olicz Function. Nepal Jou. of Science and Tech. 2: Pahai,N.P On Cetain Topological Stuctue of Paanomed Olicz Space (S ((X,. ),Φ, α, u ), F ) of Vecto Valued Sequence, Intenational Jou. of Mathematical Achive. 4 (): Pak, J. A Diect Poof of Ekeland Pinciple in Locally Convex Haudoff Topological Vecto 67

7 On Locally Convex Topological Vecto Space Valued Null Function Space c 0 (S,T, Φ, ξ, u) Defined by Semi Nom and... Space. Kangweon-Kyungki Math. Jou. 3(): Paaha, S. D. and Choudhay, B Sequence pace defined by Olicz function. Indian J. Pue Appl. Math 25(4): Rao, K. C. & Subemanina, N The Olicz Space of Entie Sequence. IJMa 68: Rao, M. M. and Ren, Z. D. 99. Theoy of Olicz pace. Macel Dekke Inc., New Yok. Rudin, W. 99. Functional Analyi, McGaw-Hill. Ruckle. W.H. 98. Sequence pace. Pitman Advanced Publihing Pogamme. Sava E. and Patteon, F An Olicz Extenion of Some New Sequence Space. Rend. Intit. Mat. Univ. Tiete 37: Sivatava, B.K On cetain genealized equence pace, Ph.D. Deetation, Goakhpu Univeity. Sivatava, J.K. and Pahai, N.P. 20. On Banach pace valued equence pace l M (X, λ, p, L) defined by Olicz function. South Eat Aian J.Math. & Math.Sc. 0() : Sivatava, J. K. and Pahai, N. P. 20. On Banach pace valued equence pace c 0 (X,M, λ, p, L) defined by Olicz function. Jou. of Rajathan Academy of Phyical Sc. (2): Sivatava, J.K. and Pahai,N.P On 2-Banach pace valued paanomed equence pace c 0 (X, M,.,., λ, p ) defined by Olicz function. Jou. of Rajathan Academy of Phyical Sc. 2(3): Tiwai, R. K. and Sivatava, J. K On cetain Banach pace valued function pace- I, Math. Foum, 20: 4-3. Tiwai, R. K and Sivatava, J. K On cetain Banach pace valued function pace- II. Math. Foum 22: