FUZZY n-inner PRODUCT SPACE

Transcription

1 Bull. Korean Mah. Soc. 43 (2007), No. 3, pp FUZZY n-inner PRODUCT SPACE Srinivaan Vijayabalaji and Naean Thillaigovindan Reprined from he Bullein of he Korean Mahemaical Sociey Vol. 43, No. 3, Augu 2007 c 2007 The Korean Mahemaical Sociey

2 Bull. Korean Mah. Soc. 43 (2007), No. 3, pp FUZZY n-inner PRODUCT SPACE Srinivaan Vijayabalaji and Naean Thillaigovindan Abrac. The purpoe of hi paper i o inroduce he noion of fuzzy n-inner produc pace. Acending family of quai α -n-norm correponding o fuzzy quai n-norm i inroduced and we provide ome reul on i. 1. Inroducion An inereing heory of 2-inner produc pace and n-inner produc pace ha been effecively conruced by C. R. Diminnie, S. Gähler and A. Whie in [6, 7]. I wa furher inveigaed and developed by A. Miiak in [20, 21]. Recen reul abou n-inner produc pace can be viewed in [4, 5]. In [11, 12, 13, 15, 19] we can udy abou he origin and developmen of n-normed linear pace. Differen auhor inroduced he definiion of fuzzy inner produc pace in [1, 16, 17] and fuzzy normed linear pace in [2, 3, 8, 9, 10, 14, 18, 23]. Recenly in [22] we have inroduced he noion of fuzzy n-normed linear pace. In hi preen paper we inroduce he noion of fuzzy n-inner produc pace a a furher generalizaion of n-inner produc pace found in [4]. We furher generalize our fuzzy n-normed linear pace [22] o fuzzy quai n-normed linear pace provide ome reul on i. 2. Preliminarie Before proceeding furher, in hi ecion le u recall ome familiar concep which will be needed in he equel. Definiion 2.1 ([4]). Le n be a naural number greaer han 1 and X be a real linear pace of dimenion greaer han or equal o n and le (,,..., ) be a real valued funcion on X X }{{} = X n+1 aifying he following n+1 condiion: (1) (i) (x, x x 2,..., x n ) 0, (ii) (x, x x 2,..., x n ) = 0 if any only if x, x 2,..., x n are linearly dependen, Received June 25, Mahemaic Subjec Claificaion. 46S40, 03E72. Key word and phrae. n-inner produc, fuzzy n-inner produc, fuzzy quai n-norm, quai α-n-norm. 447 c 2007 The Korean Mahemaical Sociey

3 448 SRINIVASAN VIJAYABALAJI AND NATESAN THILLAIGOVINDAN (2) (x, y x 2,..., x n ) = (y, x x 2,..., x n ), (3) (x, y x 2,..., x n ) i invarian under any permuaion of x 2,..., x n, (4) (x, x x 2,..., x n ) = (x 2, x 2 x, x 3,..., x n ), (5) (ax, x x 2,..., x n ) = a(x, x x 2,..., x n ) for every a R(real), (6) (x + x, y x 2,..., x n ) = (x, y x 2,..., x n ) + (x, y x 2,..., x n ). Then (,,..., ) i called an n-inner produc on X and (X, (,,..., )) i called an n-inner produc pace. Definiion 2.2 ([4]). Le n N (naural number) and X be a real linear pace of dimenion greaer han or equal o n. A real valued funcion,..., on X X = X }{{} n aifying he following four properie: n (1) x 1, x 2,..., x n = 0 if any only if x 1, x 2,..., x n are linearly dependen, (2) x 1, x 2,..., x n i invarian under any permuaion, (3) x 1, x 2,..., ax n = a x 1, x 2,..., x n, for any a R (real), (4) x 1, x 2,, x n 1, y + z x 1, x 2,..., x n 1, y + x 1, x 2,..., x n 1, z, i called an n-norm on X and he pair (X,,..., ) i called an n-normed linear pace. Remark 2.3. In he above definiion if we replace (3) by, (3) x 1, x 2,..., ax n = a p x 1, x 2,..., x n, for any a R(real) and 0 p < 1, hen (X,,..., ) i called a quai n-normed linear pace. Remark 2.4 ([4]). If an n-inner produc pace (X, (,,..., )) i given hen x 1, x 2,..., x n = (x 1, x 1 x 2,..., x n ) define an n-norm on X. Furher he following exenion of Cauchy-Buniakowki inequaliy i alo rue (x, y x 2,..., x n ) (x, x x 2,..., x n ) (y, y x 2,..., x n ). Definiion 2.5 ([22]). Le X be a linear pace over a real field F. A fuzzy ube N of X n R (R-e of real number) i called a fuzzy n-norm on X if and only if: (N1) For all R wih 0, N(x 1, x 2,..., x n, ) = 0. (N2) For all R wih > 0, N(x 1, x 2,..., x n, ) = 1 if and only if x 1, x 2,..., x n are linearly dependen. (N3) N(x 1, x 2,..., x n, ) i invarian under any permuaion of x 1, x 2,..., x n. (N4) For all R wih > 0, N(x 1, x 2,..., cx n, ) = N(x 1, x 2,..., x n, c ), if c 0, c F (field). (N5) For all, R, N(x 1, x 2,..., x n +x n, +) min{n(x 1, x 2,..., x n, ), N(x 1, x 2,..., x n, )}. (N6) N(x 1, x 2,..., x n, ) i a non-decreaing funcion of R and lim N(x 1, x 2,..., x n, ) = 1. Then (X, N) i called a fuzzy n-normed linear pace or in hor f-n- NLS.

4 FUZZY n-inner PRODUCT SPACE 449 Remark 2.6. In he above Definiion 2.5, if we replace (N4) by, (N4) For all R wih > 0 N(x 1, x 2,..., cx n, ) = N(x 1, x 2,..., x n, c ), if c 0, p c F (field),0 p < 1. Then (X, N) i called a fuzzy quai n-normed linear pace or in hor f-q-n-nls. Theorem 2.7 ([22]). Le (X, N) be a f-n-nls. Aume he condiion ha (N )N(x 1, x 2,..., x n, ) > 0 for all > 0 implie x 1, x 2,..., x n are linearly dependen. Define x 1, x 2,..., x n α =inf { : N(x 1, x 2,..., x n, ) α},α (0, 1). Then {,,..., α : α (0, 1)},i an acending family of n-norm on X. We call hee n-norm a α-n-norm on X correponding o he fuzzy n-norm on X. 3. Fuzzy n-inner produc pace In hi ecion we inroduce he aifacory noion of fuzzy n-inner produc pace a a generalizaion of Definiion 2.1 a follow. Definiion 3.1. Le X be a linear pace over a field F. A fuzzy ube J : X n+1 R (R-e of real number) i called a fuzzy n-inner produc on X if and only if: (1) For all R wih 0, J(x, x x 2,..., x n, ) = 0. (2) For all R wih > 0, J(x, x x 2,..., x n, ) = 1 if and only if x, x 2,..., x n are linearly dependen. (3) For all > 0, J(x, y x 2,..., x n, ) = J(y, x x 2,..., x n, ). (4) J(x, y x 2,..., x n, ) i invarian under any permuaion of x 2,..., x n. (5) For all > 0, J(x, x x 2,..., x n, ) = J(x 2, x 2 x, x 3,..., x n, ). (6) For all > 0, J(ax, bx x 2,..., x n, ) = J(x, x x 2,..., x n, ab (real). (7) For all, R, J(x + x, y x 2,..., x n, + ) min{j(x, y x 2,..., x n, ), J(x, y x 2,..., x n, )}. (8) For all, R wih > 0, > 0, J(x, y x 2,..., x n, ) min{j(x, x x 2,..., x n, ), J(y, y x 2,..., x n, )}. (9) J(x, y x 2,..., x n, ) i a non-decreaing funcion of R and ), a, b R lim J(x, y x 2,..., x n, ) = 1. Then (X, J) i called a fuzzy n-inner produc pace or in hor f-n-ips. Example 3.2. Le (X, (,,..., )) be an n-inner produc pace. Define + (x,y x 2,...,x n ), when > 0, R, J(x, y x 2,..., x n, ) = (x, y x 2,..., x n ) X n+1 0, when 0.

5 450 SRINIVASAN VIJAYABALAJI AND NATESAN THILLAIGOVINDAN Then (X, J) i a f-n-ips. Proof. The nine condiion for f-n-ips are verified below: (1) For all R wih 0 we have by our definiion, J(x, x x 2,..., x n, ) = 0. (2) For all R wih > 0 we have J(x, x x 2,..., x n, ) = 1 + (x, x x 2,..., x n ) = 1 (x, x x 2,..., x n ) = 0 (x, x x 2,..., x n ) = 0 x, x 2,..., x n are linearly dependen. (3) For all > 0, J(x, y x 2,..., x n, ) = + (x, y x 2,..., x n ) = + (y, x x 2,..., x n ) = J(y, x x 2,..., x n, ). (4) A (x, x x 2,..., x n ) i invarian under any permuaion of x 2,..., x n, we have J(x, y x 2,..., x n, ) i invarian under any permuaion. (5) For all > 0, (6) For all > 0, J(x, x x 2,..., x n, ) = + (x, x x 2,..., x n ) = + (x 2, x 2 x, x 3,..., x n ) = J(x 2, x 2 x, x 3,..., x n, ). J(x, x x 2,..., x n, ab ) = ab ab + (x, x x 2,..., x n ) = + ab (x 2, x 2 x, x 3,..., x n ) = + (ax, bx x 2,..., x n ) = J(ax, bx x 2,..., x n, ). (7) If (a) + < 0 (b) = = 0 (c) + > 0; > 0, < 0; < 0, > 0, hen he above relaion i obviou.if (d) > 0, > 0, + > 0. Then

6 FUZZY n-inner PRODUCT SPACE 451 wihou lo of generaliy aume ha J(x, y x 2,..., x n, ) J(x, y x 2,..., x n, ) + (x, y x 2,..., x n ) + (x, y x 2,..., x n ) + (x, y x 2,..., x n ) 1 + (x, y x 2,..., x n ) (x, y x 2,..., x n ) (x, y x 2,..., x n ) + (x, y x 2,..., x n ) 1 + (x, y x 2,..., x n ) (x, y x 2,..., x n ) (x, y x 2,..., x n ) (x, y x 2,..., x n ) + (x, y x 2,..., x n ) (x, y x 2,..., x n ) + (x, y x 2,..., x n ) (1 + ) (x, y x 2,..., x n ) (x + x, y x 2,..., x n ) ( + ) (x, y x 2,..., x n ) (x + x, y x 2,..., x n ) (x, y x 2,..., x n ) 1 + (x, y x 2,..., x n ) + (x, y x 2,..., x n ) + (x, y x 2,..., x n ) (x + x, y x 2,..., x n ) (x + x, y x 2,..., x n ) (x + x, y x 2,..., x n ) (x + x, y x 2,..., x n ) min{j(x, y x 2,..., x n, ), J(x, y x 2,..., x n, )} J(x + x, y x 2,..., x n, + ). (8) Wihou lo of generaliy aume ha J(x, x x 2,..., x n, ) J(y, y x 2,..., x n, ), for all, R wih > 0, > 0. + (x, x x 2,..., x n ) + (y, y x 2,..., x n ) + (x, x x 2,..., x n ) + (y, y x 2,..., x n )

7 452 SRINIVASAN VIJAYABALAJI AND NATESAN THILLAIGOVINDAN 1 + (x, x x 2,..., x n ) (x, x x 2,..., x n ) (x, x x 2,..., x n ) 1 + (y, y x 2,..., x n ) (y, y x 2,..., x n ) (y, y x 2,..., x n ) (x, x x 2,..., x n ) (x, x x 2,..., x n ) (x, x x 2,..., x n ) (y, y x 2,..., x n ). By Remark 2.4, (x, x x 2,..., x n ) 2 (x, y x 2,..., x n ) 2 (x, x x 2,..., x n ) 2 2 (x, y x 2,..., x n ) 2 (x, x x 2,..., x n ) 2 2 (x, y x 2,..., x n ) 2. Taking quare roo on boh ide, (x, x x 2,..., x n ) 1 + (x, x x 2,..., x n ) + (x, x x 2,..., x n ) + (x, x x 2,..., x n ) (x, y x 2,..., x n ) 1 + (x, y x 2,..., x n ) + (x, y x2,..., x n ) + (x, y x2,..., x n ) min{j(x, x x 2,..., x n, ), J(y, y x 2,..., x n, )} J(x, y x 2,..., x n, ). (9) For all 1, 2 R if 1 < 2 0 hen, by our definiion, J(x, y x 2,..., x n, 1 ) = J(x, y x 2,..., x n, 2 ) = 0. Suppoe 2 > 1 > 0 hen, (x, y x 2,..., x n ) (x, y x 2,..., x n ) (x, y x 2,..., x n ) ( 2 1 ) = ( 2 + (x, y x 2,..., x n ) )( 1 + (x, y x 2,..., x n ) ) 0, for all (x, y x 2,..., x n ) X n+1

8 FUZZY n-inner PRODUCT SPACE (x, y x 2,..., x n ) (x, y x 2,..., x n ) J(x, y x 2,..., x n, 2 ) J(x, y x 2,..., x n, 1 ). Thu J(x, y x 2,..., x n, ) i a non-decreaing funcion. Alo, lim J(x, y x 2,..., x n, ) = lim + (x, y x 2,..., x n ) = lim (1 + 1 (x, y x 2,..., x n ) ) = 1. Thu (X, J) i a f-n-ips. 4. Quai α-n-normed linear pace A a conequence of Definiion 2.5 we inroduce an inereing noion of acending family of quai α-n-norm correponding o he fuzzy quai n-norm in he following Theorem. Theorem 4.1. Le (X, N) be a f-q-n-nls. Aume he condiion ha (N7) N(x 1, x 2,..., x n, ) > 0 for all > 0 implie x 1, x 2,..., x n are linearly dependen. Define x 1, x 2,..., x n α = inf { : N(x 1, x 2,..., x n, ) α}, α (0, 1). Then {,,..., α : α (0, 1)},i an acending family of quai n-norm on X. We call hee quai n-norm a quai α-n-norm on X correponding o he fuzzy quai n-norm on X. Proof. (1) x 1, x 2,..., x n α = 0 inf { : N(x 1, x 2,..., x n, ) α} = 0, For all R, > 0, N(x 1, x 2,..., x n, ) α > 0, α (0, 1) By (N7) x 1, x 2,..., x n are linearly dependen. Converely aume ha x 1, x 2,..., x n are linearly dependen. By (N2), N(x 1, x 2,..., x n, ) = 1 for all > 0. For all α (0, 1), inf { : N(x 1, x 2,..., x n, ) α} = 0. x 1, x 2,..., x n α = 0. (2) A N(x 1, x 2,..., x n, ) i invarian under any permuaion i follow ha x 1, x 2,..., x n α i invarian under any permuaion. (3) For all c F, 0 p < 1, hen, x 1, x 2,..., cx n α = inf { : N(x 1, x 2,..., cx n, ) α} { } = inf : N(x 1, x 2,..., x n, c p ) α. Le = c hen, x p 1, x 2,..., cx n α =inf{ c p : N(x 1, x 2,..., x n, ) α} = c p inf{ : N(x 1, x 2,..., x n, ) α} = c p x 1, x 2,..., x n α.

9 454 SRINIVASAN VIJAYABALAJI AND NATESAN THILLAIGOVINDAN (4) x 1, x 2,..., x n α + x 1, x 2,..., x n α = inf { : N(x 1, x 2,..., x n, ) α} { } + inf : N(x 1, x 2,..., x n, ) α { } = inf + : N(x 1, x 2,..., x n, ) α, N(x 1, x 2,..., x n, ) α { } inf + : N(x 1, x 2,..., x n + x n, + ) α { } inf r : N(x 1, x 2,..., x n + x n, r) α, r = + = x 1, x 2,..., x n + x n α. Therefore, x 1, x 2,..., x n +x n α x 1, x 2,..., x n α + x 1, x 2,..., x n α. Thu {,,..., α : α (0, 1)} i a quai α-n-norm on X. Le 0 < α 1 < α 2. Then x 1, x 2,..., x n α1 = inf { : N(x 1, x 2,..., x n, ) α 1 } x 1, x 2,..., x n α2 = inf { : N(x 1, x 2,..., x n, ) α 2 }. A α 1 < α 2 { : N(x 1, x 2,..., x n, ) α 2 } { : N(x 1, x 2,..., x n, ) α 1 } inf { : N(x 1, x 2,..., x n, ) α 2 } inf { : N(x 1, x 2,..., x n, ) α 1 } x 1, x 2,..., x n α2 x 1, x 2,..., x n α1. Hence {,,..., α : α (0, 1)}, i an acending family of quai α-n-norm on X correponding o he fuzzy quai n-norm on X. Theorem 4.2. Le {,,..., α : α (0, 1)} be an acending family of quai n-norm correponding o (X, N). Now we define a funcion N : X n R [0, 1] by, up{α (0, 1) : x 1, x 2,..., x n α }, N (x 1, x 2,..., x n, ) = when x 1, x 2,..., x n are linearly independen, 0. 0, oherwie. Then (X, N ) i a f-q-n-nls. Proof. Le u verify he ix condiion of he f-q-n-nls a follow: (N1) For all R wih < 0 we have N (x 1, x 2,..., x n, ) = 0, for all (x 1, x 2,..., x n ) X n, a {α : x 1, x 2,..., x n α } = φ when < 0. For = 0 and x 1, x 2,..., x n are linearly independen, {α : x 1, x 2,..., x n α } = φ N (x 1, x 2,..., x n, ) = 0. When x 1, x 2,..., x n are linearly dependen and = 0 hen from he definiion, N (x 1, x 2,..., x n, ) = 0. Thu for all R wih 0, N (x 1, x 2,...,

10 FUZZY n-inner PRODUCT SPACE 455 x n, ) = 0. (N2) Le N (x 1, x 2,..., x n, ) = 1. Tha i for > 0, N (x 1, x 2,..., x n, ) = 1. Chooe any ɛ (0, 1). Then for > 0, here exi α (ɛ, 1] uch ha x 1, x 2,..., x n α and hence x 1, x 2,..., x n ɛ. Since > 0 i arbirary, hi implie ha x 1, x 2,..., x n ɛ = 0 x 1, x 2,..., x n are linearly dependen. Converely, if x 1, x 2,..., x n are linearly dependen, hen for > 0, N (x 1, x 2,..., x n, ) = up{α : x 1, x 2,..., x n α } = up{α : α (0, 1)} = 1. Thu for all > 0, N (x 1, x 2,..., x n, ) = 1 if and only if x 1, x 2,..., x n are linearly dependen. (N3) A x 1, x 2,..., x n α i invarian under any permuaion of x 1, x 2,..., x n we have N (x 1, x 2,..., x n, ) i invarian under any permuaion of x 1, x 2,..., x n. (N4) For all R wih > 0, c F, 0 p < 1, N (x 1, x 2,..., cx n, ) = up{α : x 1, x 2,..., cx n α } = up{α : x 1, x 2,..., x n α c } = N (x p 1, x 2,..., x n, c ). (N5) We have p o how ha for all, R, N (x 1, x 2,..., x n + x n, + ) { } min N (x 1, x 2,..., x n, ), N (x 1, x 2,..., x n, ). If (a) + < 0 (b) = = 0 (c) + > 0; > 0, < 0; < 0, > 0, hen in hee cae he relaion i obviou. If (d) > 0, > 0, le p = N (x 1, x 2,..., x n, ), q = N (x 1, x 2,..., x n, ) and p q. If p = 0 and q = 0 hen obviouly (N5) hold. Le 0 < r < p q. Then here exi α > r uch ha x 1, x 2,..., x n α and here exi β > r uch ha x 1, x 2,..., x n α. Le γ = min{α, β} > r. Thu x 1, x 2,..., x n γ x 1, x 2,..., x n α and x 1, x 2,..., x n γ x 1, x 2,..., x n α. Now x 1, x 2,..., x n + x n γ x 1, x 2,..., x n α + x 1, x 2,..., x n α +. Therefore, N (x 1, x 2,..., x n + x n, + ) γ > r. Since 0 < r < γ i arbirary, N (x 1, x 2,..., x n + x n, + ) p = min{n (x 1, x 2,..., x n, ), N (x 1, x 2,..., x n, )}. Similarly if p q, hen alo he relaion hold. Thu, N (x 1, x 2,..., x n, + ) min{n (x 1, x 2,..., x n, ), N (x 1, x 2,..., x n, )} (N6) Le (x 1, x 2,..., x n ) X n and α (0, 1). Now > x 1, x 2,..., x n α N (x 1, x 2,..., x n, ) = up{β : x 1, x 2,..., x n β } α. So, lim N (x 1, x 2,..., x n, ) = 1. If 1 < 2 0 hen N (x 1, x 2,..., x n, 1 ) = N (x 1, x 2,..., x n, 2 ) = 0 for all (x 1, x 2,..., x n ) X n. If 2 > 1 > 0 hen {α : x 1, x 2,..., x n α 1 } {α : x 1, x 2,..., x n α 2 } up{α : x 1, x 2,..., x n α 1 } up{α : x 1, x 2,..., x n α 2 } N (x 1, x 2,..., x n, 1 ) N (x 1, x 2,..., x n, 2 ). Thu N (x 1, x 2,..., x n, ) i a non-decreaing funcion of R. Hence (X, N ) i a f-q-n-nls. Remark 4.3. Aume furher ha for x 1, x 2,..., x n are linearly independen, (N8) N(x 1, x 2,..., x n, ) i a coninuou funcion of R (R-e of real number) and ricly increaing in he ube { : 0 < N(x 1, x 2,..., x n, ) < 1} of R.

11 456 SRINIVASAN VIJAYABALAJI AND NATESAN THILLAIGOVINDAN Theorem 4.4. Le (X, N) be a f-q-n-nls aifying he condiion (N7) and (N8) and {,,..., α : α (0, 1)} be an acending family of quai n-norm correponding o (X, N). Then for (y 1, y 2,..., y n ) X n wih y 1, y 2,..., y n are linearly independen, N(y 1, y 2,..., y n, y 1, y 2,..., y n α ) α, α (0, 1). Proof. Le y 1, y 2,..., y n α = T, hen T > 0. Alo here exi a equence { n } n=1 uch ha N(y 1, y 2,..., y n, n ) α and lim n = T. So, lim N(y 1, y 2, n n..., y n, n ) α By (N8), N(y 1, y 2,..., y n, lim n) α. N(y 1, y 2,..., y n, n y 1, y 2,..., y n α ) α, for all α (0, 1). Theorem 4.5. Le (X, N) be a f-q-n-nls aifying he condiion (N7) and (N8) and {,,..., α : α (0, 1)} be an acending family of quai n-norm correponding o (X, N). Then for (y 1, y 2,..., y n ) X n wih y 1, y 2,..., y n are linearly independen, α (0, 1) and (> 0) R, y 1, y 2,..., y n α = if and only if N(y 1, y 2,..., y n, ) = α. Proof. Le α (0, 1), y 1, y 2,..., y n are linearly independen and = y 1, y 2,..., y n α = inf{ : N(y 1, y 2,..., y n, ) α}. Since N(y 1, y 2,..., y n, ) i coninuou (by (N8)), we have by Theorem 4.4 (4.1) N(y 1, y 2,..., y n, ) α. Alo, N(y 1, y 2,..., y n, ) N(y 1, y 2,..., y n, ) if N(y 1, y 2,..., y n, ) α. If poible, le N(y 1, y 2,..., y n, ) > α, hen again by (N8), here exi < uch ha N(y 1, y 2,..., y n, ) > α which i impoible, ince = inf{ : N(y 1, y 2,..., y n, ) α}. Thu (4.2) N(y 1, y 2,..., y n, ) α By (4.1) and (4.2) we ge N(y 1, y 2,..., y n, ) = α. Thu (4.3) = y 1, y 2,..., y n α N(y 1, y 2,..., y n, ) = α. Nex if N(y 1, y 2,..., y n, ) = α, hen from he definiion (4.4) y 1, y 2,..., y n α = inf{ : J(w, z y 2,..., y n, ) α} =. (Since N(x 1, x 2,..., x n, ) i ricly increaing in { : 0 < N(x 1, x 2,..., x n, ) < 1}.) From (4.3) and (4.4) we have, for y 1, y 2,..., y n are linearly independen, α (0, 1) and (> 0) R, y 1, y 2,..., y n α = if and only if N(y 1, y 2,..., y n, ) = α. Theorem 4.6. Le (X, N) be a f-q-n-nls aifying he condiion (N7) and (N8). Le x 1, x 2,..., x n α =inf { : N(x 1, x 2,..., x n, ) α}, α (0, 1) and N : X n R [0, 1] i defined by, up{α (0, 1) : x 1, x 2,..., x n α }, N (x 1, x 2,..., x n, ) = when x 1, x 2,..., x n are linearly independen, 0. 0, oherwie.

12 FUZZY n-inner PRODUCT SPACE 457 Then (a) {,,..., α : α (0, 1)} i an acending family of quai α-n-norm correponding o (X, N). (b) (X, N ) i a f-q-n-nls. (c) N = N. Proof. (a) and (b) follow from Theorem 4.1 and Theorem 4.2. (c) To prove hi we conider he following cae. Le (y 1, y 2,..., y n, 0 ) X n R and N(y 1, y 2,..., y n, ) = α 0. Cae(i) If y 1, y 2,..., y n are linearly dependen and 0 0, hen N(y 1, y 2,..., y n, 0 ) = N (y 1, y 2,..., y n, 0 ) = 0. Cae(ii) If y 1, y 2,..., y n are linearly dependen and 0 > 0, hen N(y 1, y 2,..., y n, 0 ) = N (y 1, y 2,..., y n, 0 ) = 1. Cae(iii) If y 1, y 2,..., y n are linearly independen and 0 0, hen N(y 1, y 2,..., y n, 0 ) = N (y 1, y 2,..., y n, 0 ) = 0. Cae(iv) Suppoe y 1, y 2,..., y n are linearly independen and 0 (> 0) R uch ha N(y 1, y 2,..., y n, 0 ) = 0. For α (0, 1), y 1, y 2,..., y n α = inf{ : N(y 1, y 2,..., y n, 0 ) α} By Theorem 4.4, N(y 1, y 2,..., y n, y 1, y 2,..., y n α ) α, for all α (0, 1). Since, N(y 1, y 2,..., y n, 0 ) = 0 < α i follow ha 0 < y 1, y 2,..., y n α, for all α > 0. So, N (y 1, y 2,..., y n, 0 ) = up{α : y 1, y 2,..., y n α 0 } = up{φ} = 0. Therefore N(y 1, y 2,..., y n, 0 ) = N (y 1, y 2,..., y n, 0 ). Cae(v) If y 1, y 2,..., y n are linearly independen and 0 (> 0) R, uch ha 0 < N(y 1, y 2,..., y n, 0 ) < 1. Le N(y 1, y 2,..., y n, 0 ) = α 0. Then 0 < α 0 < 1. Now N (x 1, x 2,..., x n, ) = up{α : x 1, x 2,..., x n α }, when x 1, x 2,..., x n are linearly independen, (4.5) 0 and (4.6) x 1, x 2,..., x n α = inf{ : N(x 1, x 2,..., x n, ) α}, α (0, 1). Since N(y 1, y 2,..., y n, 0 ) = α 0, we have from (4.6), (4.7) y 1, y 2,..., y n α0 0. Uing (4.7) we ge from (4.5), (4.8) N (y 1, y 2,..., y n, 0 ) α 0 N (y 1, y 2,..., y n, 0 ) N(y 1, y 2,..., y n, 0 ). Now from Theorem 4.5, N(y 1, y 2,..., y n, 0 ) = α 0 y 1, y 2,..., y n α0 = 0. Now for 1 > α > α 0, le y 1, y 2,..., y n α =, hen 0. Then by Theorem 4.5, N(y 1, y 2,..., y n, ) = α. So, N(y 1, y 2,..., y n, ) = α > α 0 = N(y 1, y 2,..., y n, 0 ). Since N(y 1, y 2,..., y n, ) i ricly increaing and N(y 1, y 2,..., y n, ) > N(y 1, y 2,..., y n, 0 ), i follow ha > 0. So for 1 > α > α 0, y 1, y 2,..., y n α = 0. Hence (4.9) N (y 1, y 2,..., y n, 0 ) α 0 = N(y 1, y 2,..., y n, 0 ). By (4.8) and (4.9) we have, N (y 1, y 2,..., y n, 0 ) = N(y 1, y 2,..., y n, 0 ). Cae (vi) If y 1, y 2,..., y n are linearly independen and 0 (> 0) R, uch ha N(y 1, y 2,..., y n, 0 ) = 1. Then by (4.5) and (4.6) i follow ha, y 1, y 2,..., y n α 0 N (y 1, y 2,..., y n, 0 ) = 1. Thu N(y 1, y 2,..., y n, 0 ) = N (y 1, y 2,

13 458 SRINIVASAN VIJAYABALAJI AND NATESAN THILLAIGOVINDAN..., y n, 0 ) = 1. Hence N(x 1, x 2,..., x n, ) = N (x 1, x 2,..., x n, ) for all (x 1, x 2,..., x n, ) X n R. Reference [1] A. M. El-Abyad and H. M. El-Hamouly, Fuzzy inner produc pace, Fuzzy Se and Syem 44 (1991), no. 2, [2] T. Bag and S. K. Samana, Finie dimenional fuzzy normed linear pace, J. Fuzzy Mah. 11 (2003), no. 3, [3] S. C. Cheng and J. N. Mordeon, Fuzzy linear operaor and fuzzy normed linear pace, Bull. Calcua Mah. Soc. 86 (1994), no. 5, [4] Y. J. Cho, M. Maić, and J. Pečarić, Inequaliie of Hlawka ype in n-inner produc pace, Commun. Korean Mah. Soc. 17 (2002), no. 4, [5] Y. J. Cho, P. C. S. Lin, S. S. Kim, and A. Miiak, Theory of 2-inner produc pace, Nova Science Publiher, Inc., Huningon, NY, [6] C. Diminnie, S. Gähler, and A. Whie, 2-inner produc pace, Demonraio Mah. 6 (1973), [7], 2-inner produc pace. II, Demonraio Mah. 10 (1977), no. 1, [8] C. Felbin, Finie-dimenional fuzzy normed linear pace, Fuzzy Se and Syem 48 (1992), no. 2, [9], The compleion of a fuzzy normed linear pace, J. Mah. Anal. Appl. 174 (1993), no. 2, [10], Finie dimenional fuzzy normed linear pace. II, J. Anal. 7 (1999), [11] S. Gähler, Lineare 2-normiere Räume, Mah. Nachr. 28 (1964), [12], Uneruchungen über verallgemeinere m-meriche Raume, I, II, III, Mah. Nachr. 40 (1969), [13] H. Gunawan and M. Mahadi, On n-normed pace, In. J. Mah. Mah. Sci. 27 (2001), no. 10, [14] A. K. Kaara, Fuzzy opological vecor pace. II, Fuzzy Se and Syem 12 (1984), no. 2, [15] S. S. Kim and Y. J. Cho, Sric convexiy in linear n-normed pace, Demonraio Mah. 29 (1996), no. 4, [16] J. K. Kohli and R. Kumar, On fuzzy inner produc pace and fuzzy co-inner produc pace, Fuzzy Se and Syem 53 (1993), no. 2, [17], Linear mapping, fuzzy linear pace, fuzzy inner produc pace and fuzzy co-inner produc pace, Bull. Calcua Mah. Soc. 87 (1995), no. 3, [18] S. V. Krihna and K. K. M. Sarma, Separaion of fuzzy normed linear pace, Fuzzy Se and Syem 63 (1994), no. 2, [19] R. Malčeki, Srong n-convex n-normed pace, Ma. Bilen No. 21 (1997), [20] A. Miiak, n-inner produc pace, Mah. Nachr. 140 (1989), [21], Orhogonaliy and orhonormaliy in n-inner produc pace, Mah. Nachr. 143 (1989), [22] Al. Narayanan and S. Vijayabalaji, Fuzzy n-normed linear pace, In. J. Mah. Mah. Sci (2005), no. 24, [23] G. S. Rhie, B. M. Choi, and D. S. Kim, On he compleene of fuzzy normed linear pace, Mah. Japon. 45 (1997), no. 1,

14 FUZZY n-inner PRODUCT SPACE 459 Srinivaan Vijayabalaji Deparmen of Mahemaic Annamalai Univeriy Annamalainagar , Tamilnadu, India addre: balaji Naean Thillaigovindan Deparmen of Mahemaic Annamalai Univeriy Annamalainagar , Tamilnadu, India addre: hillai