@FMI c Kyung Moon Sa Co.

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1 Annals of Fuzzy Mahemacs and Informacs Volume 8, No. 2, (Augus 2014), pp ISSN: (prn verson) ISSN: (elecronc verson) c Kyung Moon Sa Co. hp:// Fne dmensonal nuonsc fuzzy normed lnear spaces T. Bag, S. K. Samana Receved 30 Sepember 2013; Revsed 13 January 2014; Acceped 21 January 2014 Absrac. In hs paper we consder general -norm n he defnon of fuzzy normed lnear space whch s nroduced by he auhors n an earler paper. I s proved ha f -norm s chosen oher han mn hen decomposon heorem of a fuzzy norm no a famly of crsp norms may no hold. We sudy some basc resuls on fne dmensonal fuzzy normed lnear spaces n general -norm seng AMS Classfcaon: 54A40, 03E72 Keywords: Correspondng Auhor: Fuzzy norm, -norm, Fuzzy normed lnear space. Tarapada Bag (arapadavb@gmal.com) 1. Inroducon Theory of fuzzy ses was nroduced by Zadeh [19] n Afer he poneerng work of Zadeh, here has been a grea effor o oban fuzzy analogues of classcal heores. Among oher felds, a progressve developmens are made n he feld of fuzzy merc spaces and fuzzy normed lnear spaces [3, 4, 6, 7, 8, 9, 11]. The noon of nuonsc fuzzy se has been nroduced by Aanassov [1] as a generalzed fuzzy se. J.H.Park [14], who frs nroduced he dea of nuonsc fuzzy merc space and suded some basc properes. On he oher hand, Saada & Park [15] have an mporan conrbuon on he nuonsc fuzzy opologcal spaces. They have also nroduced he noon of nuonsc fuzzy normed lnear space and suded some basc properes n such spaces. There have been a good amoun of work done n nuonsc fuzzy se such as T.K. Mandal & S.K.Samana [12, 13], N. Thllagovndan e al.[17]. Recenly Vjayabalaj e al.[18] nroduced a concep of nuonsc fuzzy n-normed lnear space and developed some resuls. T.K.Samana e al. [16] consdered a fuzzy normed lnear space whch was nroduced by Bag & Samana [2, 5] and defned an nuonsc fuzzy normed lnear space n general seng ( akng and as -norm and -co-norm respecvely ). They manly suded dfferen resuls on fne dmensonal nuonsc fuzzy normed lnear space. Bu her

2 resuls depend on he decomposon heorem of he nuonsc fuzzy norm no a famly of pars of crsp norms for whch hey have aken he addonal condons on -norm and -conorm as a a = a and a a = a a [0, 1] whch resuled =mn and a a=max. So effecvely he generaly of he -norm and -conorm are los. On he oher hand, because of he relaon M(x, ) + N(x, ) 1, some of he condons nvolvng he funcons M(x, ) and N(x, ) n he defnon consdered by T.K.Samana e al.[16] led o such a suaon ha n some defnons and resuls relaed o convergence and Cauchyness of a sequence saemens nvolvng one of he funcons M and N follows from he oher. To avod hese rvaly, n hs paper, we have modfed he defnon of nuonsc fuzzy normed lnear space nroduced by R. Saada e al. [15] and sudy fne dmensonal nuonsc fuzzy normed lnear space. In our defnon boh he condons vz. (1) a a = a, a a = a and (2) M(x, ) + N(x, ) 1 are waved. In our presen approach we have avoded he decomposon echnque whch s very much dependen on he resrced -norm vz. mn and -conorm vz. max. The organzaon of he paper s as n he followng: Secon 1 comprses some prelmnary resuls. In Secon 2, we nroduce a defnon of nuonsc fuzzy normed lnear space. Some basc resuls on compleeness and compacness are esablshed n fne dmensonal nuonsc fuzzy normed lnear spaces n Secon Prelmnares Defnon 2.1 ([10]). A bnary operaon : [0, 1] [0, 1] [0, 1] s a -norm f sasfes he followng condons: (1) s assocave and commuave; (2) a 1 = a a [0, 1]; (3) a b c d whenever a c and b d for each a, b, c, d [0, 1]. If s connuous hen s called connuous -norm. Followng are examples of some -norms ha are frequenly used as fuzzy nersecons defned for all a, b [0, 1]. () Sandard nersecon: a b = mn(a, b). () Algebrac produc: a b = ab. () Bounded dfference: a b = max(0, a + b 1). (v) Drasc nersecon: a b = The relaons among hese -norms are a for b = 1 b for a = 1 0 for oherwse. a b(drasc) max(0, a + b 1) ab mn(a, b). Defnon 2.2 ([10]). A bnary operaon : [0, 1] [0, 1] [0, 1] s a -co-norm f sasfes he followng condons: (1) s assocave and commuave; (2) a 0 = a a [0, 1]; (3) a b c d whenever a c and b d for each a, b, c, d [0, 1]. 246

3 If s connuous hen s called connuous -co-norm. examples of -co-norms. () Sandard unon: a b = max(a, b). () Algebrac sum: a b = a + b ab. () Bounded sum : a b = mn(1, a + b). (v) Drasc unon: a for b = 0 a b = b for a = 0 1 for oherwse. Followng are some Relaons among hese -co-norms are a b (Drasc) mn(1, a + b) a + b ab max(a, b) Defnon 2.3 ([5]). Le U be a lnear space over he feld F (C or R). A fuzzy subse N of U R (R- se of real numbers) s called a fuzzy norm on U f (N1) R wh 0, N(x, ) = 0; (N2) ( R, > 0, N(x, ) = 1) ff x = 0; (N3) R, > 0, N(cx, ) = N(x, ) f c 0; (N4) s, R; x, u U; N(x + u, s + ) N(x, s) N(u, ); (N5) N(x,.) s a non-decreasng funcon of R and lm N(x, ) = 1. The par (U, N) wll be referred o as a fuzzy normed lnear space. In [2], parcular -norm mn s aken for. Defnon 2.4 ([15]). The 5-uple (V, µ, ν,, ) s sad o be an nuonsc fuzzy normed lnear space f V s a vecor space, s connuous -norm, s a connuous -conorm and µ, ν are fuzzy ses on V (0, ) sasfyng he followng condons for every x, y V and s, > 0; (a) µ(x, ) + ν(x, ) 1; (b) µ(x, ) > 0; (c) µ(x, ) = 1 f and only f x = 0; (d) µ(cx, ) = µ(x, ) f c 0; (e) µ(x + u, s + ) µ(x, s) µ(u, ); (f) µ : (0, ) [0, 1] s connuous; (g) lm µ(x, ) = 1 and lm µ(x, ) = 0; 0 (h) ν(x, ) < 1; () ν(x, ) = 0 ff x = 0; (j) ν(cx, ) = ν(x, ) f c 0; (k) ν(x + u, s + ) ν(x, s) ν(u, ); (l) ν : (0, ) [0, 1] s connuous; (m) lm ν(x, ) = 0 and lm ν(x, ) = 1. 0 In hs case (µ, ν) s called an nuonsc fuzzy norm. T.K.Samana e al. [16] consder he above defnon by omng he condons (f) and (l) as n he followng. 247

4 Defnon 2.5 ([16]). Le be a connuous -norm, be a connuous -conorm and V be a lnear space over he feld F(R/C). An nuonsc fuzzy norm ( IFN ) on V s an objec of he form A = {((x, ), N(x, ), M(x, )) : (x, ) V R + } where N,M are fuzzy ses on V R +, N denoes he degree of membershp and M denoes he degree of non-membershp () N(x, ) + M(x, ) 1 (x, ) V R + ; () N(x, ) > 0; () N(x, ) = 1 f and only f x = 0; (v) N(cx, ) = N(x, ) f c 0, c F; (v) s, R + ; x, u V ; N(x + u, s + ) N(x, s) N(u, ); (v) N(x,.) s a non-decreasng funcon of R + and lm N(x, ) = 1; (v) M(x, ) > 0; (v) ( R, > 0, M(x, ) = 0) ff x = 0; (x) M(cx, ) = M(x, ) f c 0, c F; (x) s, R + ; x, u V ; M(x + u, s + ) M(x, s) M(u, ); (x) M(x,.) s a non-ncreasng funcon of R + and lm M(x, ) = 0. Then we say (V, A) s an nuonsc fuzzy normed lnear space. Defnon 2.6 ([16]). A sequence {x n } n an IFNLS (V, A) s sad o converge o x V f gven r > 0, > 0, 0 < r < 1 here exss a posve neger n 0 such ha N(x n x, ) > 1 r and M(x n x, ) < r n n 0. Theorem 2.7 ([16]). In an IFNLS (V, A), a sequence {x n } converges o x ff lm N(x n x, ) = 1 and lm M(x n x, ) = 0. n n Theorem 2.8 ([16]). If a sequence {x n } n an IFNLS (V, A), s convergen, s lm s unque. Defnon 2.9 ([16]). A sequence {x n } n an IFNLS (V, A) s sad o be a Cauchy sequence f lm N(x n+p x n, ) = 1 and lm M(x n+p x n, ) = 0 unformly n n on p = 1, 2,..., > 0. Defnon 2.10 ([16]). Le (V, A) be an IFNLS. A subse P of V s sad o be closed f for any sequence {x n } n P converges o x P. Defnon 2.11 ([16]). Le (V, A) be an IFNLS and P V. Then he closure of P denoed by P, s defned by P = {x V : a sequence {x n } n P convergng o x}. Defnon 2.12 ([16] ). Le (V, A) be an IFNLS. A subse P of V s sad o be compac f any sequence {x n } n P has a subsequence whch converges o some elemen n P. 3. Inuonsc fuzzy normed lnear spaces Followng s our modfed defnon of nuonsc fuzzy normed lnear space. 248

5 Defnon 3.1. Le be a -norm, be a -conorm and V be a lnear space over he feld F(R or C). An nuonsc fuzzy norm ( IFN ) on V s an objec of he form A = {((x, ), N(x, ), M(x, )) : (x, ) V R} where N,M are fuzzy ses on V R, N denoes he degree of membershp and M denoes he degree of non-membershp (x, ) V R sasfyng he followng condons: (IFN1) R wh 0, N(x, ) = 0; (IFN2) ( R, > 0, N(x, ) = 1) ff x = 0; (IFN3) R, > 0, N(cx, ) = N(x, (IFN4) s, R; x, u U; N(x + u, s + ) N(x, s) N(u, ); (IFN5) lm N(x, ) = 1. (IFN6) R wh 0, M(x, ) = 1; (IFN7) ( R, > 0, M(x, ) = 0) ff x = 0; ) f c 0; (IFN8) R, > 0, M(cx, ) = M(x, ) f c 0; (IFN9) s, R; x, u V ; M(x + u, s + ) M(x, s) M(u, ); (IFN10) lm M(x, ) = 0. Then we say (V, A) s an nuonsc fuzzy normed lnear space. Remark 3.2. From (IFN2) and (IFN4), follows ha N(x,.) s a non-decreasng funcon of R. From (IFN7) and (IFN9), follows ha M(x,.) s a non-ncreasng funcon of R. Example 3.3. Le (V, ) be a normed lnear space. Defne wo fuzzy subses N and M :V R [0, 1] by N(x, ) = M(x, ) = { + x for > x 0 x { x x + for > x 1 for x Take a b = ab and a b = mn{1, a + b}. Then (V, A) s an nuonsc fuzzy normed lnear space. Proof. All he condons excep (IFN4) and (IFN9) are easly verfed. Frs we verfy (IFN4);.e., N(x + y, s + ) N(x, s) N(y, ). Le x, y V and s, R. Suppose s, > 0 (Snce n oher cases (IFN4) s obvous). We have, s+ s++ x+y s+ = s++ x + y s s+ (s+ x )(+ y ) s++ x + y s s+ x +s y + x y s (s+ x )(+ y ) = s2 y +(s+) x y + 2 x A 0. where A = (s + + x + y )(s + x + s y + x y ). So N(x + u, s + ) N(x, s) N(u, ). Nex we verfy M(x + u, s + ) M(x, s) M(u, ). We only consder he case when s > x and > y (snce n oher cases (IFN9) 249

6 s obvous). We have, M(x, s) M(y, ) M(x + y, s + ) = x x +s + y y +s x+y x+y +s+ = [{ x ( + y ) + y (s + x )}{ x + y + s + } x + y (s + x )( + y )]/A where A = ( x + s)( + y ){ x + y + s + }. = [(s + ) x ( + y ) + (s + ) y (s + x ) + x x + y ( + y ) + x + y y (s + x ) x + y (s + x )( + y )]/A = [(s + ) x ( + y ) + (s + ) y (s + x ) + x x + y ( + y ) + x + y (s + x ){ y y }]/A. = [(s+) x (+ y )+(s+) y (s+ x )+ x+y ( x + x y s x )]/A. = [(s + ) x ( + y ) + (s + ) y (s + x ) + x y x + y s x + y ]/A. [(s + ) x ( + y ) + (s + ) y (s + x ) + x y x + y s x s y ]/A..e. M(x, s) M(y, ) M(x + y, s + ) 0..e. M(x, s) M(y, ) M(x + y, s + ). Noe 3.4. In he conex of modfed Defnon 3.1, we consder he same Defnon 2.6, Defnon 2.9, Defnon 2.10, Defnon 2.11, Defnon 2.12 and s easy o verfy ha he Theorem 2.7 and Theorem 2.8 are vald n respec of modfed defnon. 4. Fne dmensonal nuonsc fuzzy normed lnear spaces In hs secon we sudy compleeness and compacness properes of fne dmensonal nuonsc fuzzy normed lnear spaces. Frsly consder he followng Lemma whch plays he key role n sudyng properes of fne dmensonal nuonsc fuzzy normed lnear spaces. Lemma 4.1. Le (V, A) be an nuonsc fuzzy normed lnear space wh he underlyng -norm connuous a (1, 1) and he underlyng -conorm connuous a (0, 0) and {x 1, x 2,..., x n } be a lnearly ndependen se of vecors n V. Then c 1, c 2 > 0 and δ 1, δ 2 (0, 1) such ha for any se of scalars {α 1, α 2,..., α n }; N(α 1 x 1 + α 2 x α n x n, c 1 M(α 1 x 1 + α 2 x α n x n, c 2 n n α j ) < 1 δ 1. (4.1.1a) α j ) > δ 2. (4.1.1b) Proof. Le s = α 1 + α α n. If s = 0 hen α j = 0 j = 1, 2,..., n and he relaon (4.1.1a) holds for any c > 0 and δ (0, 1). Nex we suppose ha s > 0. Then (4.1.1a) s equvalen o N(β 1 x 1 +β 2 x β n x n, c 1 ) < 1 δ 1 (4.1.2a) for some c 1 > 0 and δ 1 (0, 1), and for all scalars β s wh β j = 1. If possble suppose ha (4.1.2a) does no hold. Thus for each c > 0 and δ (0, 1), a se of scalars {β 1, β 2,..., β n } wh β j = 1 for whch N(β 1 x 1 + β 2 x β n x n, c) 1 δ. 250

7 Then for c = δ = 1 m, m = 1, 2,..., a se of scalars {β(m) 1, β (m) 2,..., β n (m) } wh 1 j = 1 such ha N(y m, m ) 1 1 m where y m = β (m) 1 x 1 + β (m) 2 x β (m) n x n. Snce j = 1, we have 0 j 1 for j = 1, 2,..., n. So for each fxed j he sequence {β (m) j } s bounded and hence {β (m) 1 } has a convergen subsequence. Le β 1 denoe he lm of ha subsequence and le {y 1,m } denoe he correspondng subsequence of {y m }. By he same argumen {y 1,m } has a subsequence {y 2,m } for whch he correspondng subsequence of scalars {β (m) 2 } converges o β 2 ( say ). Connung n hs way, afer n seps we oban a subsequence {y n,m } where y n,m = γ (m) j x j wh γ (m) j = 1 and γ (m) j β j as m. Le y = β 1 x 1 + β 2 x β n x n. Now we show ha lm N(y n,m y, ) = 1 > 0. We have N(y n,m y, ) = N( (γ (m) j β j }x j, ) So, N(x 1, n γ (m) )... N(x n, ). 1 β 1 n γ n (m) β n lm N(y n,m y, ) lm N(x 1, )... lm n γ (m) 1 β 1 N(x n, n γ n (m) β n ). lm N(y n,m y, ) ( by he connuy of -norm a (1, 1)). lm N(y n,m y, ) = 1 > 0 (4.1.3a). Now for k > 0, choose m such ha 1 m < k. 1 We have N(y n,m, k) = N(y n,m + 0, m + k 1 m ) N(y 1 n,m, m ) N(0, k 1 m ) (1 1 m ) N(0, k 1 m )..e. N(y n,m, k) (1 1 m ) N(0, k 1 m ) = (1 1 m ) 1 = 1 1 m )..e. lm N(y n,m, k) 1..e. lm N(y n,m, k) = 1. (4.1.4a) Now N(y, 2k) = N(y y n,m + y n,m, k + k) N(y y n,m, k) N(y n,m, k) N(y, 2k) lm N(y y n,m, k) lm N(y n,m, k) ( by he connuy of -norm a (1, 1)). N(y, 2k) 1 1 by (4.1.3a)&(4.1.4a) N(y, 2k) = 1 1 = 1. Snce k > 0 s arbrary, by (IFN2) follows ha y = 0. Agan snce j = 1 and {x 1, x 2,..., x n } are lnearly ndependen se of vecors, so y = β 1 x 1 + β 2 x β n x n 0. Thus we arrve a a conradcon. Now we prove he relaon (4.1.1b). If s = 0 hen α j = 0 j = 1, 2,..., n and he relaon (4.1.1b) holds for any c > 0 251

8 and δ (0, 1). Nex we suppose ha s > 0. Then (4.1.1b) s equvalen o M(β 1 x 1 +β 2 x β n x n, c 2 ) > δ 2 for some c 2 > 0 and δ 2 (0, 1), and for all scalars β j s wh β j = 1. (4.1.2b) If possble suppose ha (4.1.2b) does no hold. Thus for each c > 0 and δ (0, 1), a se of scalars {β 1, β 2,..., β n } wh β j = 1 for whch M(β 1 x 1 + β 2 x β n x n, c) δ. Then for c = δ = 1 m, m = 1, 2,..., a se of scalars {γ(m) 1, γ (m) 2,..., γ n (m) } wh γ (m) 1 j = 1 such ha M(z m, m ) 1 m where z m = γ (m) 1 x 1 + γ (m) 2 x γ (m) n x n. Snce γ (m) j = 1, we have 0 γ (m) j 1 for j = 1, 2,..., n. Then by same argumen as above, we oban a subsequence {z n,m } where z n,m = η (m) j x j wh η (m) j = 1 and η (m) j η j as m. Thus η j = 1 Le z = η 1 x 1 + η 2 x η n x n. Then we have lm M(z n,m z, ) = 0 > 0. (4.1.3b) Now for k > 0, choose m such ha 1 m < k. 1 We have M(z n,m, k) = M(z n,m + 0, m + k 1 m ) M(z 1 n,m, m ) M(0, k 1 m ) 1 m M(0, k 1 m )..e. M(z n,m, k) 1 m M(0, k 1 m ) = 1 m 0 = 1 m..e. lm M(z n,m, k) 0..e. lm M(z n,m, k) = 0. (4.1.4b) Now M(z, 2k) = M(z z n,m + z n,m, k + k) M(z z n,m, k) M(z n,m, k) M(z, 2k) lm M(z z n,m, k) lm M(z n,m, k) ( by he connuy of -conorm a (0, 0)). M(z, 2k) 0 0 by (4.1.3b)&(4.1.4b) M(z, 2k) = 0 0 = 0. Snce k > 0 s arbrary, by (IFN7) follows ha z = 0. Agan snce η (m) j = 1 and {x 1, x 2,..., x n } are lnearly ndependen se of vecors, so z = η 1 x 1 + η 2 x η n x n 0. Thus we arrve a a conradcon. Ths complees he lemma. 252

9 Theorem 4.2. Every fne dmensonal nuonsc fuzzy normed lnear space (V, A) wh he connuy of he underlyng -norm a (1, 1) and -co-norm a (0, 0) s complee. Proof. Le (V, A) be an nuonsc fuzzy normed lnear space and dmv =k (say ). Le {e 1, e 2,..., e k } be a bass for V and {x n } be a Cauchy sequence n V. Le x n = β (n) 1 e 1 + β (n) 2 e β (n) k e k where β (n) 1, β (n) 2,..., β (n) k are suable scalars. So lm N(x m x n, ) = 1 > 0 (4.2.1a) m,n and lm M(x m x n, ) = 0 > 0 (4.2.1b) m,n Now from Lemma 4.1, follows ha c 1, c 2 > 0 and δ 1, δ 2 (0, 1) such ha N( (β (m) β (n) )e, c 1 β (n) ) < 1 δ 1. (4.2.2a) M( (β (m) )e, c 2 ) > δ 2 (4.2.2b) Agan for 1 > δ 1 > 0, from (4.2.1a), follows ha a posve neger n 0 (δ 1, ) such ha, N( (β (m) β (n) )e, ) > 1 δ 1 m, n n 0 (δ 1, ). (4.2.3a) Now from (4.2.2a) and (4.2.3a), we have, N( (β (m) )e, ) > 1 δ 1 > N( β (n) ) m, n n 0 (δ 1, ) c 1 (β (m) )e, c 1 < m, n n 0 (δ 1, ) ( snce N(x,.) s nondecreasng n ). < c 1 m, n n 0 (δ 1, ) Snce > 0 s arbrary, from above we have, lm m,n β(m) < c 1 m, n n 0 (δ 1, ) and = 1, 2,..., k. = 0 for = 1, 2,..., k. {β (n) } s a Cauchy sequence of scalars for each = 1, 2,..., k. So each sequence {β (n) } converges. Le lm n β(n) = β for = 1, 2,..., k. and x = β e. Clearly x V Now > 0, N(x n x, ) = N( β (n) e k β (n) β e, ) = N( (β (n) β )e, )..e. N(x n x, ) N(e 1, ) N(e 2, )... 1 β 1 k β (n) 2 β 2... N(e k, ). (4.2.4a) k β (n) k β k 253

10 When n hen k β (n) β ( snce β(n) β ) for = 1, 2,..., k and > 0. From (4.2.4a) we ge, usng he connuy of -norm a (1, 1), lm N(x n x, ) > 0 n lm N(x n x, ) = 1 > 0. n Now from (4.2.2b) and (4.2.3b), we have, M( (β (m) β (n) )e, ) < δ 2 < M( m 0 (δ, ) c 2 (β (m) β (n) )e, c 2 (4.2.5a) β (n) ) m, n < m, n n 0 (δ 2, ) ( snce M(x,.) s non-ncreasng n ) < c 2 m, n n 0 (δ 2, ) < c 2 m, n n 0 (δ 2, ) and = 1, 2,..., k. Snce > 0 s arbrary, from above we have, lm m,n β(m) = 0 for = 1, 2,..., k. {β (n) } s a Cauchy sequence of scalars for each = 1, 2,..., k. So each sequence {β (n) } converges. Le lm n β(n) = β for = 1, 2,..., k. and x = β e. Clearly x V. Now > 0, M(x n x, ) = M( β (n) e k β (n) β e, ) = M( (β (n) β )e, )..e. M(x n x, ) M(e 1, ) M(e 2, )... 1 β 1 k β (n) 2 β 2... M(e k, ). (4.2.4b) k β (n) k β k When n hen k β (n) β ( snce β(n) β ) for = 1, 2,..., k and > 0. From (4.2.4b) we ge, usng he connuy of -co norm a (0, 0), lm M(x n x, ) > 0 n lm M(x n x, ) = 0 > 0. n (4.2.5b) From (4.2.5a) and (4.2.5b), we have x n x as n. Hence (V, A) s complee. Defnon 4.3. Le (V, A) be an nuonsc fuzzy normed lnear space and F V. F s sad o be bounded f for each r, 0 < r < 1, 1, 2 > 0 such ha N(x, 1 ) > 1 r and M(x, 2 ) < r x F. Theorem 4.4. In a fne dmensonal nuonsc fuzzy normed lnear space (V, A) n whch he underlyng -norm s connuous a (1, 1) and -co-norm s connuous a (0, 0), a subse F s compac ff s closed and bounded. Proof. Frs we suppose ha F s compac. We have o show ha F s closed and bounded. 254

11 Le x F. Then a sequence {x n } n F such ha lm x n = x. n Snce F s compac, a subsequence {x nk } of {x n } converges o a pon n F. Agan {x n } x so {x nk } x and hence x F. So F s closed. If possble suppose ha F s no bounded. Then r 0 wh 0 < r 0 < 1 such ha for each posve neger n, x n F such ha N(x n, n) 1 r 0 or M(x n, n) r 0. So here exss a subsequence of {x n } ( whou loss of generaly we assume {x n } o be ha subsequence ) for whch a leas one of he relaons N(x nk, n k ) 1 r 0 n N (4.4.1a) M(x nk, n k ) r 0 n N (4.4.1b) holds. Frs we assume ha (4.4.1a) holds. Now for > 0, 1 r 0 N(x nk, n k ) = N(x nk x + x, n k + ) where > 0 1 r 0 N(x nk x, ) N(x, n k ) 1 r 0 lm N(x n k x, ) lm N(x, n k ) k k 1 r = 1 ( usng he connuy of -norm a (1, 1)) r 0 0 whch s a conradcon. In case (4.4.1b) holds, by consderng he funcon M(x, ), proceedng as above, we arrve a a conradcon. Hence F s bounded. Conversely suppose ha F s closed and bounded and we have o show ha F s compac. Le dm V=n and {e 1,, e 2,..., e n } be a bass for V. Choose a sequence {x k } n F and suppose x k = β (k) 1 e 1 + β (k),..., β(k) n are scalars. Now from Lemma 4.1, c 1, c 2 > 0 and δ 1, δ 2 (0, 1) such ha β (k) 1, β(k) 2 N( β (k) e, c 1 and M( β (k) e, c 2 2 e β n (k) e n where β (k) ) < 1 δ 1 (4.4.2a) β (k) ) > δ 2 (4.4.2b) Agan snce F s bounded, for δ 1 (0, 1), 1 > 0 such ha N(x, 1 ) > 1 δ 1 and 2 > 0 such ham(x, 2 ) < δ 1 x F. So N( β (k) e, 1 ) > 1 δ 1 (4.4.3a) and M( β (k) e, 2 ) < δ 1 (4.4.3b) From (4.4.2a) and (4.4.3a) we ge, N( β (k) e, c 1 β (k) ) < 1 δ 1 < N( β (k) e, 1 ) N( β (k) e, c 1 β (k) ) < N( 255 β (k) e, 1 )

12 c 1 n β (k) < 1 ( snce N(x,.) s non-decreasng ) β (k) 1 c1 for k = 1, 2,... and = 1, 2,..., n. So each sequence {β (k) } ( = 1, 2,..., n) s bounded. By repeaed applcaons of Bolzano-Weersrass heorem, follows ha each of he sequences {β (k) } has a convergen subsequence say {β k l }, = 1, 2,..., n. Le x kl = β (k l) 1 e 1 + β (k l) 2 e β (k l) n e n and {β (k l) 1 }, {β (k l) 2 },..., {β (k l) n } are all convergen. Le β = lm l β (k l), = 1, 2,..., n. and x = β 1 e 1 + β 2 e β n e n. Now for > 0 we have, N(x kl x, ) = N( n (βk l N(e 1, β )e, ) n β k l lm N(x kl x, ) (β k l l of -norm a (1, 1)) lm N(x kl x, ) = 1. l Now for > 0 we ge, M(x kl x, ) = M( n (βk l M(e 1, β1 )... N(e n, ) 1 n β k l n β n β as l ) ( usng he connuy n β k l β )e, ) β1 )... M(e n, 1 n β k l lm M(x kl x, ) (β k l l l and usng he connuy of -co-norm a (0, 0)) lm M(x kl x, ) = 0. l n β n ) β as l ) (snce β k l (4.4.4a) β as (4.4.4b) From (4.4.4a ) and (4.4.4b), follows ha x kl x. Thus x F (snce F s closed). Hence F s compac. Ths complees he proof. 5. Concluson In hs paper, he defnon of nuonsc fuzzy normed lnear space (IFNLS) nroduced by T.K.Samana e al. s generalzed. In he new defnon of IFNLS, he underlyng -norm and -conorm are consdered n general seng n he sense ha only connuy of -norm and -conorm a (1, 1) and (0, 0) respecvely are used. We are able o esablsh some basc heorems n fne dmensonal IFNLS n hs seng and our approach s fundamenally dfferen because we have no used he decomposon heorem of nuonsc fuzzy norm whose valdy requre a srngen resrcon ha -norm s mn and -conorm s max. Also we have only use he mplc nuonsc properes among N(x, ) and M(x, ) funcons and skp he explc resrcon N(x, )+M(x, ) 1 for whch n several defnons nvolvng convergence, boundeness boh he funcons N(x, ) and M(x, ) have equal role. We hnk ha here s a scope of furher work n hs seng. Acknowledgemens. The auhors are graeful o he referees for her valuable suggesons n rewrng he paper n he presen form. The auhors are also hankful o he Edor-n-Chef of he journal ( AFMI ) for valuable commens whch help us o revse he paper. The presen work s parally suppored by Specal Asssance 256

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