Engineering and Technology Journal Vol. 37, Part B, No. 01, 2019 DOI: http://dx.doi.org/10.30684/etj.37.1B.5 Jehad R. Kider Department of Mathematics and Computer Applications, School of Applied Sciences, University of Technology. jehadkider@gmail.com Noor A. Kadhum Department of Mathematics and Computer Applications, School of Applied Sciences, University of Technology. Properties of Fuzzy Closed Linear Operator Abstract- In this paper we recall the definition of fuzzy norm of a fuzzy bounded linear operator and the fuzzy convergence of sequence of fuzzy bounded linear operators in order to prove the uniform fuzzy bounded theorem and fuzzy open mapping theorem. The definition of fuzzy closed linear operators on fuzzy normed spaces is introduced in order to prove the fuzzy closed graph theorem. Keywords- Fuzzy bounded linear operator, fuzzy norm of a fuzzy bounded operator, Fuzzy open mapping, Fuzzy closed graph, Fuzzy closed operator. Received on: 12/03/2018 Accepted on: 13/12/2018 Published online: 25/04/2019 How to cite this article: J.R. Kider and N.A. Kadhum, “Properties of Fuzzy Closed Linear Operator,” Engineering and Technology Journal, Vol. 37, Part B, No. 1, pp. 25-31, 2019. 1. Introduction Zadeh in 1965[1] was the first author who find the theory of fuzzy set. When Katsaras in 1984 [2] studying the notion of fuzzy topological vector spaces he was the first researcher who find the notion of fuzzy norm on a linear vector space. A fuzzy metric space was found by Kaleva and Seikkala in 1984 [3]. The type of fuzzy norm on a linear space was found by Felbin in 1992 [4] where the corresponding fuzzy metric is of Kaleva and Seikkala type. Another type of fuzzy metric space was found by Kramosil and Michalek in [5]. The type of fuzzy norm on a linear space was found by Cheng and Mordeson in 1994 [6] so that the corresponding fuzzy metric is of Kramosil and Michalek type. A finite dimensional fuzzy normed linear spaces was studied by Bag and Samanta in 2003 [7]. Some results on fuzzy complete fuzzy normed spaces was studied by Saadati and Vaezpour in 2005 [8]. Fuzzy bounded linear operators on a fuzzy normed space was studied by Bag and Samanta in 2005 [9]. The fixed point theorems on fuzzy normed linear spaces of Cheng and Mordeson type was proved by Bag and Samanta in 2006, 2007 [10], [11]. The fuzzy normed linear space and its fuzzy topological structure of Cheng and Mordeson type was studied by Sadeqi and Kia in 2009 [12]. Properties of fuzzy continuous mapping on a fuzzy normed linear spaces of Cheng and Mordeson type was studied by Nadaban in 2015 [13]. The definition of the fuzzy norm of a fuzzy bounded linear operator was introduced by Kider and Kadhum in 2017 [14]. Fuzzy functional analysis is developed by the concepts of fuzzy norm and a large number of Copyright © 2019 by UOT, IRAQ researches by different authors have been published for reference please see [ 16, 17, 18, 19,20, 21, 22, 23].The structure of this paper is as follows: In section two we recall the definition of fuzzy normed space duo to Cheng and Mordeson [6] and we recall some basic definitions and properties of this space that we will need it later in this paper. In section three the definition of three types of fuzzy convergence sequence of operators is recalled in order to prove the uniform fuzzy bounded theorem. Also the fuzzy open mapping theorem is proved and the fuzzy closed graph theorem is proved after defining the fuzzy closed linear operator. 2. Properties of Fuzzy normed space In this section we recall basic properties of fuzzy normed space Definition 2.1: [1] Suppose that U is any set, a fuzzy set 𝐴̃ in U is equipped with a membership function, μ𝐴̃ (u): U→ ̃ is represented by 𝐴̃ ={(u,μ ̃ (𝑢)): [0,1]. Then 𝐴 𝐴 uU, 0 ≤ μ𝐴̃ (𝑢) ≤ 1}. Definition 2.2: [25] Let ⨂ ∶ [0, 1] × [0, 1 ] → [0, 1] be a binary operation then ⨂ is called a continuous t -norm (or triangular norm) if for all α , β , γ , δ ∈ [0, 1] it has the following properties (1)α⨂β = β⨂α, (2)α⨂1 = α, (3)(α ⨂ β)⨂γ = α ⨂(β⨂γ) (4)If α ≤ β and γ ≤ δ then α⨂γ ≤ β⨂δ Remark 2.3: [22] (1)If α > 𝛽 then there is 𝛾 such that α⨂γ ≥ β 52 Vol. 37, Part B. No. 01, 2019 Engineering and Technology Journal (2)There is 𝛿 such that δ⨂δ ≥ σ where α , β , γ , δ, σ ∈ [0,1] Definition 2.4: [8] The triple (V, L, ⨂ ) is said to be a fuzzy normed space if V is a vector space over the field 𝔽, ⨂ is a t-norm and L: V × [0 , ∞ ) →[0,1] is a fuzzy set has the following properties for all a, b∈ V and α , β > 0. 1-L(a , α) > 0 2-L( a , α) = 1 ⇔ a = 0 α 3-L(ca , α) = L (a, | | ) for all c ≠ 0 ∈ 𝔽 c 4-L(a , α)⨂L( b , β ) ≤ L (a + b , α + β) 5-L(a , . ): [0, ∞) → [0,1] is continuous function of 𝛼. 6- lim L(a, α) = 1 α→∞ Remark 2.5: [24] Assume that (V, L, ⨂) is a fuzzy normed space and let a ∈ V , t > 0 , 0 < 𝑞 < 1. If L(a, t) > (1 − q) then there is s with 0 < s < 𝑡 such that L(a, s) > (1 − q). Definition 2.6: [6] Suppose that (V, L, ⨂) is a fuzzy normed space. Put FB(a, p, t) = {b ∈ V ∶ L (a − b, t) > (1 − p)} FB[a, p, t] = {b ∈ V ∶ L (a − b, t) ≥ (1 − p)} Then FB(a, p, t) and FB[a, p, t] is called open and closed fuzzy ball with the center a ∈Vand radius p, with p > 0. Lemma 2.7: [8] Suppose that (V, L, ⨂) is a fuzzy normed space then L(x − y, t) = L(y − x, t) for all x, y ∈ V and t>0 Definition 2.8: [6] Assume that (V, L, ⨂) is a fuzzy normed space. W ⊆V is called fuzzy bounded if we can find t > 0 and 0 < 𝑞 < 1 such that L(w, t) > (1 − q) for each w ∈ W. Definition 2.9: [8] A sequence (vn ) in a fuzzy normed space (V, L, ⨂) is called converges to v ∈ V if for each q > 0 and t > 0 we can find N with L[v n − v, t] > (1 − q) for all n ≥ N. Or in other word lim vn = n→∞ v or simply represented by vn →v, v is known the limit of (vn ) or lim L[vn − v, t] = 1. n→∞ Definition 2.10: [8] A sequence (v n ) in a fuzzy normed space (V, L, ⨂) is said to be a Cauchy sequence if for all 0 < 𝑞 < 1 , 𝑡 > 0 there is a number N with L[v m − vn ,t] > (1 − q) for all m , n ≥ N. Definition 2.11: [4] Suppose that (V, L, ⨂) is a fuzzy normed space and let W be a subset of V. Then the closure of W is ̅ or CL(W) and which is W ̅= written by W ⋂{W⊆B: B is closed in V}. Lemma 2.12: [14] Assume that (V, L, ⨂) is a fuzzy normed space and ̅ if suppose that W is a subset of V. Then w ∈ W and only if there is a sequence (wn) in W with (wn ) converges to w. Definition 2.13: [14] Suppose that (V, L, ⨂) is a fuzzy normed space and ̅= W ⊆ V. Then W is called dense in V when W V. Theorem 2.14: [14] Suppose that (V, L, ⨂) is a fuzzy normed space and assume that W is a subset of V. Then W is dense in V if and only if for every u ∈ V there is w ∈ W such that L[u − w , t] > (1 − ε) for some 0 < 𝜀 < 1 and t > 0. Definition 2.15: [10] A fuzzy normed space (V, L, ⨂) is said to be complete if every Cauchy sequence in V converges to a point in V. Definition 2.16: [8] Suppose that (V, LV , ⨂) and (W, LW , ⨀) are two fuzzy normed spaces .The operator S: V → W is said to be fuzzy continuous at 𝐯𝟎 ∈ V if for all t > 0 and for all 0 < 𝛼 < 1 there is s[depends on t , α and v 0 ] and there is β [depends on t, α and v 0 ] with, LV [v − v0 ,s] > (1 − p) we have LW [S(v) − S(v 0 ), t] > (1 − α) for all v ∈V. Definition 2.17: [14] Let (V, LV , ⨂) and (U, LU ,⨀) be two fuzzy normed spaces. The operator T:D(T)→ U is said to be fuzzy bounded if there exists r, 0 < r < 1 such that LU (Tv , t) ≥ (1 − r) ⨂LV (v,t) for each v ∈ D(T) ⊆ V and t > 0 where D(T) is the domain of T. Theorem 2.18: [14] Suppose that (V, LV , ⨂) and (U, LU ,⨀) are two fuzzy normed spaces. The operator S: D(S) →U is fuzzy bounded if and only if S(A) is fuzzy bounded for every fuzzy bounded subset A of D(S). Put FB(V,U) ={S:V→U, S is a fuzzy bounded operator} when (V, LV , ⨂) and (U, LU , ⨀) are two fuzzy normed spaces [14]. Lemma 2.19: [14] Let (V, LV , ⨂) be a fuzzy normed space. If A and B are fuzzy bounded subset of V then A+B and 𝛼A are fuzzy bounded for any 𝛼 ≠ 0 ∈ 𝔽. Theorem 2.20: [14] Suppose that (V, LV , ⨂) and (U, LU ,⨀) are two fuzzy normed spaces. Define L(T, t) = 𝐢𝐧𝐟𝐱∈𝐃(𝐓)𝐋𝐘(𝐓𝐱,𝐭) for all T ∈ FB(V, U) and t > 0 then (FB(V, U), L,∗) is fuzzy normed space. Theorem 2.21: [14] Suppose that (V, LV , ⨂) and (U, LU ,⨀) are two fuzzy normed spaces with S: D(S) →U is a linear 62 Vol. 37, Part B. No. 01, 2019 Engineering and Technology Journal operator where D(S) ⊆ V. Then S is fuzzy bounded if and only if S is fuzzy continuous. Corollary 2.22: [14] Let (V, LV , ⨂) and (U, LU , ⨀) be two fuzzy normed spaces. Assume that T: D(T) →U is a linear operator where D(T) ⊆V. Then T is a fuzzy continuous if T is a fuzzy continuous at x ∈ D(T). Theorem 2.23: [14] Let (V, LV , ⨂) and (U, LU , ⨀) be two fuzzy normed spaces. If U is fuzzy complete then FB(V, U) is fuzzy complete. Definition 2.24: [14] A linear functional f from a fuzzy normed space (V, LV , ⨂) into the fuzzy normed space (F, LF , ⨀) is said to be fuzzy bounded if there exists r, 0 < 𝑟 < 1 such that LF [f(x), t] ≥ (1 − r)⨂LX [x, t] for all x ∈ D(f) and t > 0. Furthermore, the fuzzy norm of f is L(f, t) = inf LF (f (x),t) and LF (f(x), t) ≥ L(f, t)⨂LX (x, t). Definition 2.25: [14] Suppose that (V, LV , ⨂) is a fuzzy normed space. Then the vector space FB(V,𝔽 ) ={f:V→ 𝔽, f is fuzzy bounded linear functional } with a fuzzy norm defined by 𝐋 (𝐟, 𝐭) = 𝐢𝐧𝐟 𝐋𝐅 (𝐟(𝐱), 𝐭) form a fuzzy normed space which is called the fuzzy dual space of V. 3. Fuzzy Convergence of Sequence of Operators and Functional Definition 3.1: [24] A sequence (vn ) in a fuzzy normed space (V, LV , ⨂) is said to weakly fuzzy convergent if we can find v ∈ V with every h ∈ FB(V,ℝ) lim h(v n) = h(v ). This is written vn →w v the n→∞ element v is said to be the weak limit of (v n) and (vn ) is said to be fuzzy converges weakly to v. Theorem 3.2: [24] Suppose that (vn ) is in the fuzzy normed space(V, LV , ⨂). 1. If vn →v then vn →wv. w 2.If vn → v and dimension of V is finite then vn →v. Definition3.3: [24] Suppose that (V, LV , ⨂) and (U, LU , ⨀) are two fuzzy normed spaces. A sequence (Tn ) operators Tn ∈ FB(V, U) is said to be 1. Uniform operator fuzzy convergent if there is T: V → U with L[Tn –T, t] → 1 for any t > 0 and n≥N. 2. Strong operator fuzzy convergent if there is T: V → U with LU [Tnv– Tv,t] → 1 for every t > 0, v ∈ V and n ≥ N. 3. Weak operator fuzzy convergent if there is T: V → U with LR [f (Tnv) − f (Tv ), t] for every t > 0, f ∈ FB(U, R) and n≥N. Respectively, T is called uniform strong and weak operator limit of (Tn ) Definition3.5: [24] Let (V, LV , ⨂) be a fuzzy normed space. A sequence (hn ) of functional hn ∈ FB(V,ℝ) is called 1) Strong fuzzy converges if there is h ∈ FB(V, ℝ) with L[hn − h, t] → 1 for all t > 0 this written hn → h 2) Weak fuzzy converges if there is h ∈FB(V,R) with hn (v) → h(v) for every v ∈ V written by lim hn (v) = h(v). n→∞ Definition 3.6: A subset E of a fuzzy normed space (V, L, ⨂) is said to be 1- Rare (or no where dense ) in V if its closure Ē has no interior points 2-Meager (or of the first category) in V if E is the union of countable many sets each of which is rare in V 3-Nonmeager (or of the second category) in V if E is not meager in V Theorem [Fuzzy Baire’s theorem] 3.7: Let (V, LV , ⨂) be a complete fuzzy normed space such that V ≠ ∅ , then it is non meager in itself. Proof: We must prove that when V ≠ ∅ is complete and V=∪∞ k=1 Vk ( Vk closed) then there is one Vk contains open subset which is not equal to ∅. Let V be meager in itself that is V=∪∞ k=1 Mk ….(1) such that every Mk is rare in V, we will find a Cauchy sequence (pk ) whose limit is p is in no Mk this contradict the equation (1) by ̅̅̅̅1 does not assumption M1 is rare in V so that M contains a nonempty open set but V does. It ̅̅̅̅c = V − M ̅̅̅̅1 ≠ ∅ and follows ̅̅̅̅ M1 ≠ V. Thus M 1 c ̅̅̅̅ open. Now choose a point q1 in M1 and an open fuzzy ball about it, say ̅̅̅̅c . By assumption M2 is B1 = FB(q1 , ε1 , t) ⊂ M 1 ̅̅̅̅2 does not contains a nonempty rare in V so that M ε open set hence FB (q1 , 1 , t) ⊈ ̅̅̅̅ M2 this implies 2 ̅̅̅̅c ∩ FB (q1 , ε 1 , t) is not empty and open, let that M 2 2 ̅̅̅̅2 ∩ FB(q1 , ε1 , t) B2 = FB(q2 , ε2 , t) ⊆ M by induction we obtain a sequence of open fuzzy balls Bk = FB(qk , εk , t), such that ε Bk ∩ Mk = ∅ and Bk+1 ⊂ FB (qk , k , t) ⊂ Bk . 2 Let εk < 2−k and (1 − εk ) > 1 − 2−k the sequence (qm ) of the centers is a Cauchy and converges say qm ⟶ q ∈ V because V is complete as we assume. Now for all m and n with n > 𝑚 with Bn ⊂ FB (qm , LV (qm − q, t) εm 2 , t) so that 62 Vol. 37, Part B. No. 01, 2019 Engineering and Technology Journal t t ≥ LV ( qm − qn , ) ⨂LV (qn − q, ) 2 2 ε t ≥ (1 − m) ⨂LV (qn − q, ) = (1 − 2 𝜀𝑚 2 ) ⨂1 = (1 − 𝜀𝑚 2 2 ) as n → ∞ c , so Hence q ∈ BM for every m since Bm ⊂ ̅̅̅̅̅ Mm ∞ q ∉ Mm for any m, so that q ∉∪m=1 Mm = V. This contradicts q ∈ V. We need the following theorem in the next results Uniform fuzzy Bounded Theorem 3.8: Let (Tn ) be sequence in FB(V, U) where (V, LV , ⨂) is a complete fuzzy normed space and (U, LU ,⨀) is a fuzzy normed space such that (LU [Tn v, t]) is fuzzy bounded for every v ∈ V with LU [Tnv, t] ≥ (1 − cv ) ….(2) n=1, 2,… where 0 < cv < 1. Then (L[Tn, t]) is fuzzy bounded so there is 0 < 𝑐 < 1 with L[Tn , t] ≥ (1 − c)….(3) n ∈ ℕ. Proof: For m ∈ (0, 1) let Vm={ v∈V: LU [Tnv,t] ≥ (1 − m) } for all v ∈ ̅ Vm there is a sequence (vj ) in Vm converges to v. This means that for every fixed 𝑛 we have LU [Tnv j ,t] ≥ (1 − m) and obtain LU [Tnv,t] ≥ (1 − m) since Tn is fuzzy continuous. Hence v ∈ Vm and Vm is closed now by (2) each v ∈ V belongs to some Vm. Hence V= ⋃∞ 𝑚=1 Vm . Since V is complete at this point Fuzzy Baire’s theorem implies that some Vm contains an open fuzzy ball say, B0 = FB(v0 , r, t) ⊂ Vm0…(4) Let v ∈ V be nonzero, let z = v0 + v with r that LV [z − v 0 , t] > LV [z − v0 ,t] > (1 − ) so 2 (1 − 𝑟) which implies 𝑧 ∈ B0. Now by (4) and from definition of Vm0 we have t LU [Tnz, ] ≥ (1 − m0 ) for all n. Also 2 t LU [Tn v 0, ] ≥ (1 − m0 ) since v0 ∈ B0 . 2 Now for all v ∈ V LU [Tn v, t] = LU [Tn(z − v 0 ), t] t t ≥ LU [Tn z, ] ⨀LU [Tnv 0 , ] 2 2 ≥ (1 − m0 )⨀(1 − m0 ). Hence for all n, L[Tn , t] = inf LU [Tnv,t] ≥ (1 − c) where for some (1 − m0 )⨀(1 − m0 ) ≥ (1 − c) for some 0 < 𝑐 < 1. The following lemma is needed later Lemma 3.9: Let Tn ∈ FB(V, U) where (V, LV , ⨂) is a complete fuzzy normed space and (U, LU ,⨀) is a fuzzy normed space if (Tn ) is a strong operator fuzzy convergent with limit T then T ∈ FB(V, U). Proof Suppose that T is linear since Tn v → Tv for every v ∈ V and Tn is linear. The sequence (Tn v) is fuzzy bounded for any v ∈ V. Now since V is complete so (L[Tn ,t]) is fuzzy bounded by uniform fuzzy bounded Theorem 3.8, say L[Tn ,t] ≥ (1 − c ) for all n and for some 0 < 𝑐 < 1. From this it follows that LU [Tnv, t] ≥ L[T, t] ⨂LV [v,t]. This implies LU [Tv, t] ≥ (1 − c)⨂LV [v, t] hence T ∈ FB(V, U). Definition 3.10: Let (V, LV , ⨂) be a fuzzy normed space then a subset D of V is called a fuzzy total set if ̅̅̅̅̅̅̅̅ spanD =V. Theorem 3.11: A sequence (Tn ) ∈ FB(V, U) where (V, LV , ⨂) and (U, LU , ⨀) are complete fuzzy normed spaces is strong operator fuzzy convergent if and only if (i) (L[Tn ,t]) is fuzzy bounded sequence. (ii) (Tn d) is a Cauchy sequence in U for any d ∈ E where E is a fuzzy total subset of V. Proof: Since Tn v → Tv for every v ∈ V then (𝑖) is satisfied from Theorem 3.8 since V is complete, also (ii) is satisfied. Conversely suppose that (i) and (ii) holds let t L [Tn , ] ≥ (1 − c) for all n and t >0 3 0 < 𝑐 < 1. We consider any v ∈ V and prove that (Tn v) converges strongly in U. Let 0 < 𝜀 < 1 and t > 0 be given since spanE is dense in V we t have y ∈ spanE with LV [v − y , ] > (1 − 𝜀) for 3 some 0 < 𝜀 < 1 the sequence (Tn y) is Cauchy by (ii) hence for given 0 < 𝑟 < 1 and t > 0 there is a t number N such that LU [Tn y –Tm y, ] > (1 − r). 3 Now for n, m ≥N we get LU [Tn v –Tm v, t] ≥ t t LU [Tnv –Tny, ] ⨀LU [Tn y – Tm y, ] 3 3 t ⨀LU [Tmy –Tmv, ] 3 t t t ≥ L [Tn , ] ⨂LV [v − y, ] ⨀LU [Tn y –Tm y, ] 3 3 3 t t ⨀L [Tm , ] ⨂LV [v − y, ] 3 3 > (1 − c)⨂(1 − ε)⨀(1 − r)⨀(1 − c)⨂(1 − ε). Now we can find 0 < (1 − 𝑝) < 1 so that (1 − c)⨂(1 − ε)⨀(1 − r)⨀(1 − c)⨂(1 − ε) > (1 − p) .Therefore LU [Tnv − Tm v,t] > (1 − 𝑝) for every n, m ≥N. Hence (Tnv ) is Cauchy in U but U is complete (Tn v) convergence in U. Since v ∈ V was arbitrary this proves strong operator fuzzy convergence of (Tn ). 62 Engineering and Technology Journal Definition 3.12: Suppose that (V, LV , ⨂) and (U, LU ,⨀) are two fuzzy normed spaces. The operator T: D(T) → U where D(T) ⊆ V is said to be a fuzzy open if for any open set E in D(T) T(E) is open set in U. The following lemma is the key of proving the next main results Lemma 3.13: Suppose that (V, LV , ⨂)and (U, LU , ⨀) are two complete fuzzy normed spaces and let T: V → U be a fuzzy bounded linear surjective operator. Then T(B0 ) the image of the open fuzzy ball B0 = FB(0,1, t) ⊂ V contains an open fuzzy ball a bout 0 ∈ U. Proof: First step we will prove that the closure of the image of the open fuzzy ball B1 = 1 FB (0, , t) contains an open fuzzy ball B ∗. For a 2 subset E of V we know that αE = {v ∈ V: v = 𝛼z, z ∈ E} where 𝛼 ≠ 0 ∈ 𝐹 and E + w ={v ∈ V: v = z + w, z ∈ E}. We consider the fuzzy open ball 1 Bk = FB (0, k , t). Hence V= ⋃∞ k=1 Bk since T is 2 surjective and linear U= T(V) = T(⋃∞ k=1 Bk ) ∞ ∞ ̅̅̅̅̅̅̅̅ =⋃ k=1 T(Bk )= ⋃ k=1 T(Bk ) since U is complete it is nonmeager in itself by Theorem 3.7. Hence we concluded that ̅̅̅̅̅̅̅̅ T(B1 ) must contain some open ̅̅̅̅̅̅̅̅ (B1 ). It fuzzy ball, say B ∗ = FB(y 0 ,ε, t) ⊂ T follows that B ∗ −y0 = FB(0, ε, t) ⊂ ̅̅̅̅̅̅̅̅ T(B1 ) − y 0 … (5). In the second step we will show that ̅̅̅̅̅̅̅̅ T(B n ) contains an open fuzzy ball Dn about 0 ∈ U where Bn = FB(0, 2−n , t) ⊂V. To prove ̅̅̅̅̅̅̅̅ T(B1 ) − ̅̅̅̅̅̅̅̅ (B0 ) let y ∈ ̅̅̅̅̅̅̅̅ y0 ⊂ T T(B1 ) − y0 . Then y + y0 ∈ ̅̅̅̅̅̅̅̅ T(B1 ) we know that y 0 ∈ ̅̅̅̅̅̅̅̅ T(B1 ). Then by lemma 2.12 there are un = Twn ∈ T(B1 ) such that un → y + y0 , vn = Tzn ∈ T(B1 ) such that vn → y0 since wn ,zn ∈ B2 it follows that t t LV [wn − zn ,t] ≥ LV [wn, ] ⨂LV [zn, ] 2 2 1 1 ≥ (1 − ) ⨂ (1 − ) 2 2 so that wn − zn ∈ B0 . From T(wn − zn ) = Twn − Tzn = un − vn → y ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ we say that y ∈ ̅̅̅̅̅̅̅̅ T(B0 ). So T(B 1 ) − y 0 ⊂ T(B0 ). Thus from (5) we get B ∗ − y 0 = FB(0, ε, t) ⊂ ̅̅̅̅̅̅̅̅ T(B0 ) from this we obtain ε ̅̅̅̅̅̅̅̅ (Bn )…(6) Vn = FB (0, n , t) ⊂ T 2 −n where Bn = FB(0, 2 , t) ⊂ V with B0 ⊃ B1 ⊃ ̅̅̅̅̅̅̅̅ (Bn ) ⊃ ⋯ T(B1 ) ⊃ ⋯ ⊃ T B2 ⊃ ⋯ so ̅̅̅̅̅̅̅̅ T(B0 ) ⊃ ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ that is T(Bn ) ⊂ T(B0 ). In the final step we prove that T(B0 ) contains an open fuzzy ball a bout 0 ∈ ε U that is we must prove that V1 = FB (0, , t) ⊂ 2 T(B0 ) let y ∈ V1 from (6) with n = 1 we Vol. 37, Part B. No. 01, 2019 T(B1 ) there must be have V1 ⊂ ̅̅̅̅̅̅̅̅ T(B1 ) hence y ∈ ̅̅̅̅̅̅̅̅ ε v ∈ T(B1 ) such that LU [y − v, t] > (1 − ). 4 Now v ∈ T(B1 ) implies v = T(x1 ) for some x1 ∈ ϵ B1 . Hence LU [(y − Tx1 ), t] > (1 − ) from this 4 and (6) with n = 2 we see that y − Tx1 ∈ V2 ⊂ ̅̅̅̅̅̅̅̅ (B2 ) . As before there is x2 ∈ B2 such that T ϵ LU [(y − Tx1 ) − Tx2 , t] > (1 − ) hence 8 y − Tx1 − Tx2 ∈ V3 ⊂ ̅̅̅̅̅̅̅̅ T(B3 ) and so on. In the nth step we can find xn ∈ Bn such that ε LU [y − ∑nk=1 Txk , t] > (1 − n+1) (n = 2 1,2,3, … ) …(7) Let zn = ∑ni=1 xi since xk ∈ Bk we have 1 LV [xk , t] > (1 − k). For m>n 2 LV [zn − zm ,t] > 1 1 1 (1 − m+1 ) ⨂ (1 − m+2 ) ⨂ … ⨂ (1 − n ) 2 2 2 1 1 1 Put (1 − m+1) ⨂ (1 − m+2) ⨂ … ⨂ (1 − n ) > 2 2 2 (1 − r) for some 0 < 𝑟 < 1 hence LV [zn − zm , t] > (1 − r) for all n, m > N for some positive number N. Thus (zn ) is Cauchy so zn → x because V is complete. Also x ∈ B0 since x = ∑∞ k=1 xk and B0 ⊃ B1 ⊃ B2 ⊃ ⋯ and T is fuzzy continuous, Tzn → Tx and (7( show that T(x)= y. Hence y ∈ T(B0 ). The following result is the main result Theorem [fuzzy open mapping theorem] 3.14: Suppose that (V, LV , ⨂) and (U, LU ,⨀) are two complete fuzzy normed spaces and let S: V → U be a surjective fuzzy bounded linear operator then S is a fuzzy open mapping. Hence if S is bijective then S −1 is fuzzy continuous hence fuzzy bounded. Proof: We will prove that if E⊂V is open then S(E) is open in U that is we will show that for every u= S(v) ∈ S(E) the set S(E) contains an open fuzzy ball about u= S(v) ∈ S(E). Let u= S(v) ∈ S(E ) but E is open so it contains an open fuzzy ball with center v. Hence E−v contain an open fuzzy ball with center 0 let the radius of the fuzzy ball be r, 0 < 𝑟 < 1. Since FB(0, 1, t) ⊃ FB(0, r, t) for each 0 < 𝑟 < 1. Hence E−v contains the open fuzzy ball FB(0, 1, 𝑡). Now Lemma 3.13 implies that S(E − v) = S(E) − S(v ) contains an open fuzzy ball a bout 0 and so dose S (E) − S(v). Thus S(E) contains an open fuzzy ball about u= S(v ). Since u ∈ S(E) was arbitrary S(E) is open in U. Now if S −1 :U → V exists it is continuous since S is fuzzy open. Because S −1 is linear then by Theorem 2.21 it is fuzzy bounded. 62 Vol. 37, Part B. No. 01, 2019 Engineering and Technology Journal Definition 3.15: Suppose that (V, LV , ⨂ ) and (U, LU , ⨂) are two fuzzy normed spaces and let S: D(S) →U be a linear operator with D(S) is a subset of V then S is said to be a fuzzy closed linear operator if its graph G(S) = {(v, u): v ∈ D(S), u = S(v)} is closed in the fuzzy normed space (V × U, L, ⨂) where L[(v, u), t] = LV [v, t]⨂LU [u, t] Theorem [fuzzy closed graph theorem] 3.16: Suppose that (V, LV , ⨂) and (U, LU ,⨂) are complete fuzzy normed spaces and let S: D(S) →U be a closed linear operator with D(S) is a subset of V. Then S is fuzzy bounded if D(S) is closed in V. Proof: First (V × U, L,∗) is complete fuzzy normed space [24]. By assumption G(S) is closed in V × U and D(S) is closed in V. Hence G(S) and D(S) are complete [24]. Now define P: G(S) → D(S) by P [(v, S(v))] =v then P is linear and P is fuzzy bounded since L[P (v,S (v)),t] = LV [v, t] ≥ LV [v, t]⨂LU [Sv,t] = L[(v, S(v)), t] and P is bijective so P −1 exists where P −1 :D(S) → G(S) defined by P −1 [v] = (v,S (v )) since G(S) and D(S) are complete we can apply Theorem 3.14 and see P −1 is fuzzy bounded, say L[(v,S(v))] ≥ (1 − r) ⨂LV [v,t] for some 0 < 𝑟 < 1 and for all v ∈ D(S). Hence S is fuzzy bounded because LU [Sv, t] ≥ LU [Sv, t]⨂LV [v,t] = L[(v, S(v )), t] ≥ (1 − r)⨂LV [v,t] for all v ∈ D(S). We need the following lemma in the next result. Lemma 3.17: Suppose that (V, LV , ⨂ ) and (U, LU , ⨂) are two fuzzy normed spaces and let S: D(S) →U be a linear operator with D(S) is a subset of V then S is fuzzy closed if and only if when vn → v where vn ∈ D(S) and Svn → u then v ∈ D(S) and S(v ) = u where u ∈ U. Proof: G(S) is closed if and only if z = (v, u) ∈ ̅̅̅̅̅̅ G(S) implies z ∈ G(S), Now from Theorem 3.16 we see that z ∈ ̅̅̅̅̅̅ G(S) if and only if there are zn = (vn, S(v n )) ∈ G(S) such that zn → z hence vn → v, Svn → u and z = (v,u) ∈ G(S) if and only if v ∈ D(S) and u = S (v). Theorem 3.18: Suppose that (V, LV , ⨂ ) and (U, LU , ⨂) are two fuzzy normed spaces and let S: D(S) →U be fuzzy bounded linear operator with D(S) is a subset of V then: 1-If D(S) is closed in V then S is closed 2-If S is closed and U is complete then D(S) is closed in V. Proof: 1-Suppose that (vn ) ∈ D(S) and v n → v such ̅̅̅̅̅̅ (S) = D(S) since D(S) is that Sv n → Sv then v ∈ D closed and Sv n → Sv since S is fuzzy continuous. Hence S is fuzzy closed by Lemma 3.17. ̅̅̅̅̅̅ (S) there is (vn ) in D(S) with vn → v 2-Let v ∈ D since S is fuzzy bounded LU [Sv n − Svm, t] = LU [S(v n − v m ),t] ≥ L[S, t]⨂LV [vn − vm ,t]. 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