Fuzzy inner product space and its properties 1

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1 International Journal of Fuzzy Mathematis and Systems IJFMS). ISSN Volume 5, Number 1 015), pp Researh India Publiations Fuzzy inner produt spae and its properties 1 S. Mukherjee and T. Bag Department of Mathematis, Visva Bharati, Santiniketan-73135, W. Bengal, India tarapadavb@gmail.om Abstrat Definition of fuzzy inner produt on a omplex linear spae is introdued. A deomposition theorem from a fuzzy inner produt into a family of risp inner produts is established. It is shown that fuzzy inner produt indues a fuzzy norm Bag & Samanta) type. AMS subjet lassifiation: 54A40, 03E7. Keywords: Fuzzy inner produt spae, deomposition theorem. 1. Introdution In 1997, C. Alsina et al. [1] first introdued the idea of real probabilisti inner produt spae. Following their onept, R. Biswas [], A. M. El-Abyed & H. M. Hamouly [3] were the first who gave a meaningful definition of fuzzy inner produt spaes. On the other hand Kohli & Kumar [4], Mazumdar & Samanta [5], Hasankhani, Nazari & Saheli [6], Goudarzi & Vaezpour [7], S. Vijayabalaji [8], Mukherjee & Bag [9] studied different properties of fuzzy inner produt spaes and fixed point theorems. Following the definition of fuzzy real inner produt spae introdued by M. Goudarzi & Vaezpour, in [9], we redefine the definition of fuzzy real inner produt spae in a sense to establish a deomposition theorem whih helps us to develop more results of funtional analysis in fuzzy setting. In this paper, we onsider omplex linear spae and introdue a definition of fuzzy inner produt omplex) spae. We establish a deomposition theorem from a fuzzy 1 The present work is partially supported by Speial Assistane Programme SAP) of UGC, New Delhi, India [Grant No. F. 510/8/DRS/004SAP-I)].

2 S. Mukherjee and T. Bag inner produt into a family of risp inner produts. Finally it is shown that a fuzzy norm Bag & Samanta type) is indued from fuzzy inner produt. The organization of the paper is as follows: Setion, provides some preliminary results whih are used in this paper. In Setion 3, definition of fuzzy inner produt is given. Deomposition theorem from a fuzzy inner produt into a family of risp inner produt is established in Setion 4.. Some preliminary results In this setion some definitions and preliminary results are given whih are used in this paper. Definition.1. Bag & Samanta [10]) Let U be a linear spae over a field F field of real/omplex numbers). A fuzzy subset N of U R R is the set of real numbers) is alled a fuzzy norm on U if x,u U and F, following onditions are satisfied: N1) t R with t 0, Nx,t)= 0; N) t R, t > 0, Nx,t)= 1 ) iff x = 0; t N3) t R, t > 0, Nx,t)= Nx, ) if = 0; N4) s, t R, x,u U; Nx + u, s+ t) min{nx,s),nu,t)} N5) Nx,.) is a non-dereasing funtion of R and lim Nx, t) = 1. The pair t U, N) will be referred to as a fuzzy normed linear spae. Theorem.. [10] Let U, N) be a fuzzy normed linear spae. Assume further that, N6) t > 0, Nx,t)>0 implies x = 0. Define x = {t > 0 : Nx, t) }, 0, 1). Then { : 0, 1)} is an asending family of norms on U and they are alled -norms on U orresponding to the fuzzy norm N on U. Theorem.3. [10] Let U, N) be a fuzzy normed linear spae satisfying N6). Assume further that, N7) for x = 0, Nx,.) is a ontinuous funtion of R. Let x = {t > 0 : Nx, t) }, 0, 1) and N : U R [0, 1] be a funtion defined by { N { 0, 1) : x t} if x, t) = 0, 0) x, t) = 0 if x, t) = 0, 0). Then

3 Fuzzy inner produt spae and its properties 3 i) { : 0, 1)} is an asending family of norms on U. ii) N is a fuzzy norm on U. iii) N = N. Definition.4. Goudarzi & Vaezpour [7]) A fuzzy inner produt spae FIP-spae) is a triplet X, F, ), where X is a real vetor spae, is a ontinuous t-norm, F is a fuzzy set on X R and the following onditions hold for every x, y, z X and s, t, r R. FI-1) Fx, x,0) = 0 and Fx, x, t) > 0, for eah t>0; FI-) Fx, x, t) = Ht)for some t R if and only if x = 0; FI-3) Fx, y, t) = Fy, x, t); FI-4) For any real number, Fx,y,t)= F x, y, t ) if >0 Ht) if = 0 t 1 Fx, y, ) if <0 Where Ht) = { 1 if t>0 0 if t 0. FI-5) sup s+r=t F x, z, s) Fy, z, r)) = Fx + y, z, t); FI-6) Fx, y,.) : R [0, 1] is ontinuous on R {0}; FI-7) lim Fx, y, t) = 1. t + Definition.5. [9] Let X be a linear spae over R the set of real numbers). Let F be a fuzzy subset of X X R. Then F is alled fuzzy real inner produt on X if x, y, z X and t R, FI-1) Fx,x,t)= 0 t 0 FI-) Fx,x,t)= 1 t>0) iff x = 0 FI-3) Fx,y,t)= Fy,x,t)

4 4 S. Mukherjee and T. Bag FI-4) Fx, y, t) = F x, y, t ) if >0 Ht) if = 0 t 1 Fx, y, ) if <0 FI-5) Fx + y, z, t + s) Fx,z,t) Fy,z,s) FI-6) lim Fx,y,t)= 1. The pair X, F) is said to be a fuzzy real inner produt t + spae. 3. Fuzzy inner produt spae In this setion definition of fuzzy inner produt spae on a omplex linear spae have introdued and study some results. Definition 3.1. Let V be a linear spae over F R or C). Define µ : V V F [0, 1] suh that x, y, z V, t F, the following onditions hold: CFI-1) µx, x, t) = 0 t having Rl t < 0 and Im t = 0. CFI-) µx, x, t) = 1 t having Rl t > 0 and Im t = 0) iff x = θ. CFI-3) µx, y, t) = µy, x, t). CFI-4) For any salar k having Im k = 0 and t = 0, 1 µ x, y, t ) if k R k µkx,y,t)= Ht) if k = 0 µ x, y, t ) otherwise k Where H : F [0, 1] defined by { 1 if t R + Ht) = 0 otherwise. CFI-5) For t, s R + µx + y, z, t + s) µx, z, t) µy, z, s). CFI-6) lim Rl t µx, y, t) = 1. Then µ is said to be a fuzzy inner produt and V, µ) is a fuzzy inner produt spae.

5 Fuzzy inner produt spae and its properties 5 Remark 3.. If V is a linear spae over R, then V, µ) is a real fuzzy inner produt spae Mukherjee & Bag). Example 3.3. Let V,, ) be an inner produt spae. Define µ : V V F [0, 1] by, For = 0, µx, y, t) = Ht).For = 0, 1 ifrl t > Rl x, y and Im t = Im x, y 1 µx,y,t)= if Rl t = Rl x, y or Im t = Im x, y 0 ifrl t < Rl x, y and Im t = Im x, y. Then µ is a fuzzy inner produt on V. Proof. Note that for = 1, 1 ifrl t > Rl x, y and Im t = Im x, y 1 µx,y,t)= µx, y, t) = if Rl t = Rl x, y or Im t = Im x, y 0 ifrl t < Rl x, y and Im t = Im x, y. CFI-1) Let Rl t < 0 and Im t = 0. Sine x, x 0, therefore Im x, x = 0 Rl t < Rl x, x and Im t = Im x, x µx, x, t) = 0. CFI-) µx, x, t) = 1 t having Rl t > 0 and Im t = 0 Rl t > Rl x, x and Im t = Im x, x t having Rl t > 0 and Im t = 0 Rl x, x = 0 x = θ. Conversely let x = θ. Therefore x, x = 0 t having Rl t > 0 and Im t = 0 Rl t > Rl x, x and Im t = Im x, x = 0. Thus µx, x, t) = 1. CFI-3) Suppose µx, y, t) = 1 Rl t > Rl x, y and Im t = Im x, y Rl t >Rl x, y and Im t = Im x, y Rl t >Rl y, x and Im t = Im y, x µy, x, t) = 1. Similarly we have the result for µx, y, t) = 1 and µx, y, t) = 0. Therefore µx, y, t) = µy, x, t) in any ase. CFI-4) For = 0, by definition µx,y,t)= Ht). Let R, then µx,y,t)= 1 Rl t > Rl x, y and Im t = Im x, y Rl t <Rl x, y and Im t = Im x, y µ x, y, t ) = 0.

6 6 S. Mukherjee and T. Bag Therefore µx,y,t)= 1 = 1 0 = 1 µ x, y, t ). Now µx,y,t)= 1 Rl t = Rl x, y or Im t = Im x, y Rl t = Rl x, y or Im t = Im x, y µ x, y, t ) = 1. Therefore µx,y,t)= 1 = 1 1 = 1 µ x, y, t Rl t < Rl x, y and Im t = Im x, y Rl t >Rl x, y and Im t = Im x, y µ x, y, t ) = 1. Therefore µx, y, t) = 0 = 1 1 = 1 µ 1 µ x, y, t ). Now let >0. Therefore µx,y,t)= 1 Rl t > Rl x, y and Im t = Im x, y ). Again µx,y,t)= 0 x, y, t ). Thus µx, y, t) = Rl t >Rl x, y and Im t = Im x, y µ x, y, t ) = 1 Similarly µx,y,t)= 1 µ x, y, t ) = 1 and µx,y,t)= 0 µ x, y, t ) = 0. Thus µx,y,t)= µ x, y, t ). CFI-5) a) If µx, z, t) µy, z, s) = 0, then there is nothing to prove. If µx, z, t) µy, z, s) = 1, three ases may arise. Case-i) µx, z, t) = µy, z, s) = 1. Case-ii) µx, z, t) = 1,µy,z,s)= 1. Case-iii) µx, z, t) = 1, µy,z,s)= 1. Case-i) µx, z, t) = µy, z, s) = 1 Therefore t = Rl x, z and s = Rl y, z

7 Fuzzy inner produt spae and its properties 7 t + s = Rl x, z + Rl y, z = Rl x + y, z. Then µx + y, z, t + s) = 1. Thus µx + y, z, t + s) µx, z, t) µy, z, s) in this ase. Case-ii) µx, z, t) = 1,µy,z,s)= 1. Therefore t = Rl x, z and s>rl y, z t + s>rl x, z + Rl y, z = Rl x + y, z µx + y, z, t + s) = 1 µx, z, t) µy, z, s). Similarly we an prove Case-iii). Nowifµx, z, t) µy, z, s) = 1 then µx, z, t) = µy, z, s) = 1 t>rl x, z and s>rl y, z t + s>rl x, z + Rl y, z = Rl x + y, z µx + y, z, t + s) = 1 µx, z, t) µy, z, s). CFI-6) Clearly lim Rl t µx, y,t) = 1. Theorem 3.4. Let V, µ) be a fuzzy inner produt spae. Further assume that for x, y V CFI-7) µx, y, st) µx, x, s ) µy, y, t ), s, t R and x, y X. Define a funtion N : X R [0, 1] by { µ x, x, t Nx, t) = ) if t>0 0 otherwise Then N is a fuzzy norm on X. We all this norm as indued norm of µ. Proof. N1) t 0, Nx, t) = 0 by definition). N) Nx, t) = 1, t>0 µx, x, t ) = 1, t>0 x = 0. N3) If >0, then for t 0, Nx,t)= 0 = N x, t ) and for t>0, Nx, t) = µ x, x, t ) = µ If <0, then for t 0, Nx,t)= 0 = N for t>0, x, Nx, t) = µ x, x, t ) = µ ) t Thus Nx, t) = N x,. x, x, ) t and x, x, t ) = N x, t ) = N x, N4) Nx + y, s + t) = µx + y, x + y, s + t) ) ) t ) t.

8 8 S. Mukherjee and T. Bag µx, x, s ) µx, y, st) µy, x, ts) µy, y, t ) = µx, x, s ) µx, y, st) µy, y, t ). By CFI-7) we have Nx + y, s + t) µx, x, s ) µy, y, t ) = Nx, s) Ny, t). N5) This follows from lim Rl t + µx, y, t) = 1 and Nx, t) = 0 for t Deomposition theorem In this setion deomposition theorem from a fuzzy inner produt to a family of risp inner produt has been proved. Theorem 4.1. Let V, µ) be a fuzzy inner produt spae. Further assume that for x, y V CFI-8) µx, x, t) = 0 Im t = 0 and { t>0 : µx, x, t) > 0} x = θ}. Also assume that CFI-9) µix, y, Rl t) = µ x, y, Im t i ), µix, y, Im t) = µ x, y, Rl t i Define for 0, 1), x, y = {t>0 : µx, y, t) }+ {t<0 : Fx,y,t) 1 }if t R x, y = { Rl t > 0) + Rl t < 0) + i Im t > 0) + i Im t < 0) : µx, y, Rl t), µx, y, Im t), µx, y, t) } otherwise. Then {, : 0, 1)} is a family of inner produt in V. We all these inner produts as -inner produts orresponding to the fuzzy inner produt µ. Proof. I) We have to show that for 0, 1), x + y, z = x, z + y, z. Let x, y = Rl 1 x, y + Rl x, y + iim 1 x, y + iim x, y where Rl 1 x, y = {Rl t > 0 : µx, y, Rl t) }, Rl x, y = {Rl t < 0 : µx, y, Rl t) } Im 1 x, y = {Im t > 0 : µx, y, Im t) } Im x, y = {Im t < 0 : µx, y, Im t) }. First we shall show that Rl 1 x + y, z = Rl 1 x, z + Rl. For ɛ>0 µx + y, z, Rl 1 x, z + Rl + ɛ) µx, z, Rl 1 x, z + ɛ ) µy, z, Rl + ɛ ) =. Therefore {t R : µx + y, z, t) } Rl 1 x, z + Rl + ɛ. Sine ɛ is arbitrary, Rl 1 x + y, z Rl 1 x, z + Rl 4.1.1). Again if [ A = 1 1 µ x, z, Rl 1 x, z ɛ )) 1 µ y, z, Rl ɛ ))] ).

9 Fuzzy inner produt spae and its properties 9 [ = 1 µ x, z, x, z ɛ ) µ y, z, Rl ɛ ) 1 µ x y, z, Rl 1 x, z + Rl ɛ)] = µx + y, z, Rl 1 x, z + Rl ɛ) 4.1.). Now µ x, z, Rl 1 x, z ɛ ) <, µ y, z, Rl ɛ ) < 1 µ x, z, Rl 1 x, z ɛ ) > 1, 1 µ y, z, Rl ɛ ) > 1 [ 1 µ x, z, Rl 1 x, z ɛ )] [ 1 µ y, z, Rl ɛ ) ] > 1 [ A = 1 1 µ x, z, Rl 1 x, z ɛ )) 1 µ y, z, Rl ɛ ))] < 1 1 ) =. From 4.1.) we have µx + y, z, Rl 1 x, z + Rl ɛ) A< x + y, z Rl 1 x, z + Rl ɛ. Sine ɛ is arbitrary, we have x + y, z Rl 1 x, z + Rl 4.1.3). From 4.1.1) and 4.1.3) we have x + y, z = Rl 1 x, z + Rl. Similarly we an prove that Rl x + y, z = Rl x, z + Rl y, z, Im 1 x + y, z = Im 1 x, z + Im, Im x + y, z = Im x, z + Im y, z. Thus x + y, z = x, z + y, z. II) x, x = {t>0 : µx, x, t) }+ {t<0 : µx, x, t) 1 }. Note that µx, x, t) = 0 t C R {0})by CFI-1 & CFI-8), x, x 0 0, 1)}. Now let for 0, 1), x, x = 0 {t>0 : µx, x, t) } =0 t>0 µx, x, t) >0. x = 0 by CFI-8). Conversely assume that x = 0. Then µx, x, t) = Ht). Therefore x, x = {t>0 : µx, x, t) }+ {t <0 : µx, x, t) 1 }, 0, 1) = {t>0 : t>0}+ {t<0 : t<0} =0, 0, 1). III) x, y = { Rl t > 0) + Rl t < 0) i Im t > 0) i Im t < 0) : µx, y, Rl t), µx, y, Im t), µx, y, t) } ={ Rl t >0)+ Rl t <0) i Im t >0) i Im t <0) : µy, x, Rl t), µy, x, Im t), µy, x, t) } ={ Rl t >0) + Rl t <0) + i Im t <0) + i Im t >0) : µy, x, Rl t), µy, x, Im t), µy, x, t) } = y, x.

10 10 S. Mukherjee and T. Bag IV) a) Let >0. Then x, y ={ Rl t > 0) + Rl t < 0) + i Im t > 0) + i Im t < 0) : µx,y,rlt),µx,y,imt),µx,y,t) } ={ Rl t > 0) + Rl t < 0) + i Im t > 0) + i Im t < 0) : µx, y, Rl t ), µx, y, Im t ), µx, y, t ) } = { Rl s > 0) + Rl s < 0) + i Im s > 0) + i Im s < 0) : µx, y, Rl s), µx, y, Im s), µx, y, s) } where s = t = x, y. b) Let = 0. Then x, y ={ Rl t > 0) + Rl t < 0) + i Im t > 0) + i Im t < 0) : µx,y,rlt),µx,y,imt),µx,y,imt) } = {Rl t > 0 : Ht) } = {Rl t > 0 : Rl t > 0} =0 = x, y. ) Let <0and = mfor some m>0. Then mx + mx, y = mx, y + mx, y 0,y = mx, y + mx, y 4.1.4) Now by FI-4), µ0, y,t)= Ht)and thus by definition of x, y, it follows that 0,y = 0. Hene from 4.1.4) we have 0 = mx, y + mx, y mx, y = mx, y mx, y = m x, y x, y = x, y. d) Let = 1 + i. Then x, y = 1 x + i x, y = 1 x, y + i x, y = 1 x, y + ix, y 4.1.5) Now ix, y = { Rl t + Rl t + i Im t + i Im t : µix, y, Rl t), µix, y, Im t) } ={ Rl t + Rl t + i Imt + i Imt : µx, y, Im t i ), µx, y, Rlt i ) }[by CFI-9)] 4.1.6). Let t = s, thus Rl s = Im t, Im s = Rl t. i From 4.1.6) we get ix, y ={ Im s > 0) + Im s < 0) + i Rl s > 0) + i Rl s < 0) : µx, y, Im s), µx, y, Rl s) } ={ Im s < 0) Im s > 0)+i Rl s > 0)+i Rl s < 0) : µx, y, Im s), µx, y, Rl s) } = i{ Rl s > 0) + Rl s < 0) + i Im s > 0) + i Im s < 0) : µx, y, Rl s), µx, y, Im s) } = i x, y.

11 Fuzzy inner produt spae and its properties 11 Therefore 4.1.5) gives x, y = 1 x, y + i x, y = 1 + i ) x, y = x, y. Thus in any ase x, y = x, y. Theorem 4.. Let {, : 0, 1)} be a family of inner produt on a linear spae V over F. Now define a funtion µ : V V F [0, 1] for x, y V suh that for = 0, µ x,y,t)= Ht)for = 0 µ x,y,t) { 0, 1) : Rl x, y 1 Rl t and Im x, y = Im t} if Rl t > 0 1 = 0 ifrl t = 0 { 0, 1) : Rl x, y Rl t and Im x, y = Im t} if Rl t < 0. Then µ is a fuzzy inner produt spae. Proof. CFI-1) Let Rl t < 0 and Im t = 0. Sine Rl x, x 0 and Im x, x = 0 Therefore µ x,x,t)= { 0, 1) : Rl x, x Rl t and Im x, y = Im t} { 0, 1) : 0, 1)} =0. CFI-) Let x = 0. For Rl t > 0 and Im t = 0, then µ x, x, t) = { 0, 1) : Rl x, x 1 Rl t and Im x, x = Im t} 1 = { 0, 1) : 0 t} = { 0, 1) : 0, 1)} =1. Conversely assume that µ x,x,t)= 1 Rl t > 0 and Im t = 0. { 0, 1) : Rl x, x 1 Rl t and Im x, x = Im t}=1. 1 Choose any ɛ 0, 1), then for any Rl t > 0, Imt= 0, t ɛ, 1] suh that Rl x, x Rl t and hene Rl x, x Rl t. Sine Rl t > 0, Imt= 0 is arbitrary, this implies Rl x, x = 0 and hene x = 0. CFI-3) Let Rl t > 0. Then Rl t >0. So µ x,y,t)= { 0, 1) : Rl x, y 1 Rl t and Im x, y 1 = Imt} = { 0, 1) : Rl y, x 1 t andim y, x 1 = Im t} = µ y, x, t). Similar results for Rl t = 0 and Rl t < 0 also hold. CFI-4) Let >0. If Rl t > 0 then Rl t > 0. So µ x, y, t) = { 0, 1) : Rl x, y 1 Rl t and Im x, y = Im t} { 1 = 0, 1) : Rl x, y 1 Rl t and Im x, y } 1 = Imt

12 1 S. Mukherjee and T. Bag = µ x, y, t ). Similarly if Rl t < 0 then µ x, y, t) = µ x, y, t ). If = 0 then by definition µ x, y, t) = Ht). Let <0. If Rl t < 0 then Rl t > 0. So µ x, y, t) = { 0, 1) : Rl x, y Rl t and Im x, y = Im t} { = 0, 1) : Rl x, y Rl t and Im x, y } = Imt Let = 1 β. { Then µ x, y, t) = 1 β 0, 1) : Rl x, y 1 β Rl t and Im x, y } 1 β = Imt { = 1 β 0, 1) : Rl x, y 1 β Rl t and Im x, y } 1 β = Imt = 1 µ x, y, t ). Similarly if Rl t < 0 then µ x, y, t) = 1 µ x, y, t CFI-5) Let t>0, s>0 then t + s>0. Let µ x, z, t) µ y, z,s)>. Therefore µ x, z,t)>and µ y, z,s)> x, z t and y, z s x + y, z = x, z + y, z t + s µ x + y, z, t + s) µ x + y, z, t + s) µ x,z,t) µ y,z,s)>. CFI-6) Clearly lim Rl t + µ x,y,t)= 1. Therefore µ is a fuzzy real inner produt on X. ). 5. Conlusion Conept of probabilisti real inner produt spae, fuzzy real inner produt spae are introdued by different authors and study some properties. On the other hand, although the fuzzy inner produt on omplex linear spae has been defined by some authors but further development is not satisfatory. In this paper, we try to introdue a definition of fuzzy inner produt on omplex linear spae in a sense to establish a deomposition theorem from a fuzzy inner produt into a family of risp inner produts. We think that this theorem will be helpful to fuzzify more results of funtional analysis. The present work is partially supported by Speial Assistane Programme SAP) of UGC, New Delhi, India [Grant No. F. 510/4/DRS/009 SAP-I)].

13 Fuzzy inner produt spae and its properties 13 Referenes [1] C. Alsina, B. Shweizer, C. Sempi, A. Sklar, On the definition of a probabilisti inner produt spae, Rendionti di Matematia, Serie VII, Vol-17, Roma1997), [] R. Biswas, Fuzzy inner produt spaes and fuzzy norm funtion, Information Sienes, ), [3] A. M. El-Abyad, H. M. El-Hamouly, Fuzzy inner produt spaes, Fuzzy Sets and Systems, 44) 1991) [4] J. K. Kohli, R.Kumar, On fuzzy inner produt spaes and fuzzy o-inner produt spaes, Fuzzy Sets and Systems 53) 1993) 7 3. [5] Pinaki Mazumdar, S. K. Samanta, On fuzzy inner produt spaes, The Journal of Fuzzy Mathematis, 16) 008) [6] A. Hasankhani, A. Nazari, M. Saheli, Some properties of fuzzy Hilbert spaes and norm of operators, Iranian Journal of Fuzzy Systems, 73) 010) [7] M. Goudarzi, S. M. Vaezpour, On the definition of fuzzy Hilbert spaes and its appliation, The Journal of Nonlinear Siene and Appliations 1) 009) [8] S.Vijayabalaji, Fuzzy strong n-inner produt spae, International Journal ofapplied Mathematis, 1) 010) [9] S. Mukherjee, T. Bag, Fuzzy real inner produt spae and its properties, Ann. Fuzzy Math. Inform. 6) 013) [10] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaes, Journal of Fuzzy Mathematis, 113) 003)

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