Abstract
AbstractIn this paper, we define -fuzzy boundedness for linear operators and we prove that every finite dimensional -fuzzy normed space is complete.
Fuzzy Inf. Eng. (2011) 4: 385-391 DOI 10.1007/s12543-011-0093-6 ORIGINAL ARTICLE Some Results on the L-fuzzy Topological Isomorphism Reza Saadati Received: 30 January 2011/ Revised: 25 August 2011/ Accepted: 15 November 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we defineL-fuzzy boundedness for linear operators and we prove that every finite dimensionalL-fuzzy normed space is complete. Keywords Bounded linear operators · Finite dimensional normed spaces·L-fuzzy normed spaces 1. Introduction In this section, we generalize the idea of intuitionistic fuzzy normed spaces and we define the notion of L-fuzzy normed spaces. Then some of the basic facts are re- viewed on L-fuzzy normed spaces. Definition 1.1 Let L = (L,≤ ) be a complete lattice, and U a non-empty set called universe. AnL-fuzzy setA on U is defined as a mappingA : U −→ L. For each u in U, A(u) represents the degree (in L) to which u satisfiesA. We define 0 = inf L and 1 = sup L. L L Definition 1.2 A triangular norm (t-norm) onL is a mappingT : L → L satisfying the following conditions: (i) (∀x ∈ L)(T (x, 1 ) = x); (boundary condition) (ii) (∀(x, y) ∈ L )(T (x, y) = T (y, x)); (commutativity) (iii) (∀(x, y, z) ∈ L )(T (x,T (y, z)) = T (T (x, y), z)); (associativity) (iv) (∀(x, x , y, y ) ∈ L )(x ≤ x and y ≤ y ⇒T (x, y) ≤ T (x , y )). (mono- L L L tonicity) R. Saadati () Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran. email: rsaadati@eml.cc 386 R. Saadati (2011) A t-norm can also be defined recursively as an (n+ 1)-ary operation (n ∈ N\{0}) byT = T and n n−1 T (x ,··· , x ) = T (T (x ,··· , x ), x ) (1) (n+1) (1) (n) (n+1) for n ≥ 2 and x ∈ L. (i) Definition 1.3 A negation on L is any decreasing mapping N : L → L satisfying N(0 ) = 1 and N(1 ) = 0 .If N(N(x)) = x, for all x ∈ L, then N is called an L L L L involutive negation. Definition 1.4 [1] The 3-tuple (V,P,T ) is said to be anL-fuzzy normed space if V is a vector space,T is a continuous t-norm onL andP is anL-fuzzy set on V× ]0,+∞[ satisfying the following conditions for every x, y in V and t, sin ]0,+∞[: (a) P(x, t) > 0 ; L L (b) P(x, t) = 1 if and only if x = 0; (c) P(αx, t) = P(x, ) for eachα 0; |α| (d) T (P(x, t),P(y, s)) ≤ P(x+ y, t+ s); (e) P(x,·): ]0,∞[→ L is continuous; (f) lim P(x, t) = 0 and lim P(x, t) = 1 . t→0 L t→∞ L In this case, P is called anL-fuzzy norm. Definition 1.5 A sequence {x } in an L-fuzzy normed space (V,P,T ) is called a n n∈N Cauchy sequence if for eachε ∈ L\{0 } and t > 0, there exists n ∈ N such that L 0 P(x − x , t) > N(ε) n m L for each n, m ≥ n ; where N is an involutive negation. The sequence {x } is said 0 n n∈N to be convergent to x ∈ V in the L-fuzzy normed space (V,P,T ) and denoted by x −→ xif P(x − x, t) → 1 whenever n → +∞ for every t > 0.An L-fuzzy normed n n L space is said to be complete if and only if every Cauchy sequence is convergent. Lemma 1.1 [2] Let P be an L-fuzzy norm on V. Then, (i) P(x, t) is nondecreasing with respect to t, for all x in V; (ii) P(x− y, t) = P(y− x, t), for all x, y in V and t ∈ ]0,+∞[. Definition 1.6 Let (V,P,T ) be an L-fuzzy normed space. A subset A of V is said to be LF-bounded if there exist t > 0 and r ∈ L\{0 , 1 } such that P(x, t) > N(r) for L L L each x ∈ A. Next, we assume that, for everyμ ∈ L\{0 , 1 }, there isλ ∈ L\{0 , 1 } such that L L L L n−1 T (N(λ),··· ,N(λ)) > N(μ), (1) L Fuzzy Inf. Eng. (2011) 4: 385-391 387 where N is an involutive negation on L. Lemma 1.2 [3] Let (V,P,T ) be an L-fuzzy normed space and define E : V −→ λ,P R ∪{0} by E (x) = inf{t > 0: P(x, t) > N(λ)} λ,P L for eachλ ∈ L\{0 , 1 } and x ∈ V. Then we get L L (i) E (αx) = |α|E (x), for every x ∈ V and α ∈ R; λ,P λ,P (ii) If T satisfies (1) for anyμ ∈ L\{0 , 1 }, there existsλ ∈ L\{0 , 1 } such that L L L L E (x +···+ x ) ≤ E (x )+···+ E (x ) μ,P 1 n λ,P 1 λ,P n for any x, y ∈ V; (iii) The sequence {x } is convergent with respect to an L-fuzzy norm P if and n n∈N only if E (x − x) → 0. Also the sequence {x } is Cauchy with respect to λ,P n n n∈N anL-fuzzy norm P if and only if it is Cauchy with E . λ,P Lemma 1.3 [3] A subset A of R is LF-bounded in (R,P,T ) if and only if it is bounded in R. Lemma 1.4 [3] A sequence {β } is convergent in the L-fuzzy normed space n n∈N (R,P,T ) if and only if it is convergent in (R,|·|). Corollary 1.1 If the real sequence {β } is LF-bounded, then it has at least one n n∈N limit point. Lemma 1.5 [3] A subset A of R is LF-bounded in (R,P,T ) if and only if it is bounded in R. Lemma 1.6 [3] A sequence{β } is convergent in theL-fuzzy normed space (R,P,T ) if and only if it is convergent in (R,|·|). Corollary 1.2 If the real sequence {β } is LF-bounded, then it has at least one limit point. Definition 1.7 Let T and T be two continuous t-norms. Then T dominates T , denoted by T T , if for all x , x , y , y ∈ L, L 1 2 1 2 T [T (x , x ),T (y , y )] ≤ T [T (x , y ),T (x , y )]. 1 2 1 2 L 1 1 2 2 Definition 1.8 The 3-tuple (R ,Φ,T ) is called anL-Fuzzy Euclidean normed space if T is a t-norm and Φ(x, t) is anL-fuzzy Euclidean norm defined by Φ(x, t) = P(x , t), j=1 n−1 where a = T (a ,··· , a ), T T,x = (x ,··· , x ),t > 0, and P is an j 1 n L 1 n j=1 L-fuzzy norm. 388 R. Saadati (2011) Lemma 1.7 [3, 4] Suppose that the hypotheses of Definition 1.8 are satisfied, then (R ,Φ,T ) is an L-fuzzy normed space. Corollary 1.3 TheL-fuzzy Euclidean normed space (R ,Φ,T ) is complete. Proof Let{x } be a Cauchy sequence in theL-fuzzy Euclidean normed space (R ,Φ, T ). Since E (x − x )= inf{t > 0: Φ(x − x , t) > N(λ)} λ,Φ n m n m L = inf{t > 0: P(x − x , t) > N(λ)} m, j n, j L j=1 ≥ inf{t > 0: P(x − x , t) > N(λ)} m, j n, j L = E (x − x ) = |x − x |E (1), λ,P m, j n, j m, j n, j λ,P the sequence {x } in which j = 1,··· , n is Cauchy sequence in R and convergent m, j to x ∈ R then by Lemma 1.6 the sequence {x } is convergent in L-fuzzy normed j m, j space (R,P,T ). We prove that {x } convergent to x = (x ,··· , x ). m 1 n n−1 limΦ(x − x, t) = lim P(x − x , t) = T (1 ,··· , 1 ) = 1 . m m, j j L L L m m j=1 2. Main Results Theorem 2.1 [5, 6] Let {x ,··· , x } be a linearly independent set of vectors in vec- 1 n tor space V and (V,P,T ) be an L-fuzzy normed space. Then there is c 0 and an L-fuzzy norm space (R,P ,T ) such that for every choice of the n real scalars α ,··· ,α , we have 1 n P(α x +···+α x , t) ≤ P (c |α |, t). (2) 1 1 n n L 0 j j=1 Definition 2.1 Let (V,P,T ) and (V,P ,T ) be fuzzy normed space. Then twoL-fuzzy norms P and P are said to be equivalent whenever x −→ xin (V,P,T ) if and only if x −→ xin (V,P ,T ). Theorem 2.2 On a finite dimensional vector space V, every two L-fuzzy norms P and P are equivalent. Proof Let dim V = n and{v ,··· , v } be a basis for V. Then every x ∈ V has a unique 1 n representation x = α v . Let x −→ x in (V,P,T ), but for each m ∈ N, x has a j j m m j=1 unique representation, i.e., x = α v +···+α v . m 1,m 1 n,m n By Theorem 2.1 there is a c 0 and an L-fuzzy norm P such that (2) holds. So P(x − x, t) ≤ P (c |α −α |, t) ≤ P (c|α −α |, t). m L 0 j,m j L 0 j,m j j=1 Fuzzy Inf. Eng. (2011) 4: 385-391 389 Now if m −→ ∞, thenP(x − x, t) −→ 1 for every t > 0 and hence|α −α |−→ 0 m L j,m j in R. On the other hand, by Lemma 1.2 (ii) for any μ ∈ L\{0 , 1 }, there exists L L λ ∈ L\{0 , 1 } such that L L E (x − x) ≤ |α −α |E (v ). μ,P m j,m j λ,P j j=1 Since|α −α |−→ 0, then we have x −→ x in (V,P ,T ). With the same argument, j,m j m x −→ x in (V,P ,T ) imply x −→ x in (V,P,T ). m m Definition 2.2 A linear operatorΛ :(V,P,T ) −→ (V ,P ,T ) is said to be L-fuzzy bounded if there exists a constant h ∈ R−{0} such that for every x ∈ V and for every t > 0 P (Λx, t) ≥ P(hx, t). (3) Note that, by Lemma 1.2 and last definition we have E (Λx)= inf{t > 0: P (Λx, t) > N(λ)} λ,P L ≤ inf{t > 0: P(x, t/|h|) > N(λ)} = |h| inf{t > 0: P(x, t) > N(λ)} L L =|h|E (x). λ,P Theorem 2.3 Every linear operatorΛ :(V,P,T ) −→ (V ,P ,T ) isL-fuzzy bounded if and only if it is continuous. Proof By (3) every L-fuzzy bounded linear operator is continuous. Conversely, let the linear operator Λ be continuous but is not L-fuzzy bounded. Then, for each n in N there is an x in V such that E (Λx ) ≥ nE (p ). If we let y = , then n λ,P n λ,P n n nE (x ) λ,P n it is easy to see y → 0, butΛy do not tend to 0. n n Definition 2.3 A linear operator Λ :(V,P,T ) −→ (V ,P ,T ) is an L-fuzzy topo- −1 logical isomorphism ifΛ is one-to-one and onto, and bothΛ andΛ are continuous. L-Fuzzy normed spaces (V,P,T ) and (V ,P ,T ) for which such a Λ exists are L- fuzzy topologically isomorphic. Lemma 2.1 A linear operator Λ :(V,P,T ) −→ (V ,P ,T ) is L-fuzzy topological isomorphism if Λ is onto and there exists constants a, b 0 such that P(ax, t) ≤ P (Λx, t) ≤ P(bx, t). Proof By hypothesis Λ is L-fuzzy bounded and by last theorem is continuous and since Λx = 0 implies 1 = P (Λx, t) ≤ P(x, t/|b|) and consequently x = 0, then L L −1 Λ is one-to-one. Thus Λ exists and, since P (Λx, t) ≤ P(bx, t) is equivalent to −1 −1 −1 P (y, t) ≤ P(bΛ y, t) = P(Λ y, t/|b|)or P ( y, t) ≤ P(T y, t) where y =Λx, L L −1 we see Λ is L-fuzzy bounded and by last theorem is continuous. Hence Λ is an L-fuzzy topological isomorphism. Corollary 2.1 L-Fuzzy topologically isomorphism preserves completeness. Proof By Lemma 2.1, the proof is trivial. 390 R. Saadati (2011) Theorem 2.4 Every linear operatorΛ :(V,P,T ) −→ (V ,P ,T ) where dim V < ∞ but other, not necessarily finite dimensional, is continuous. Proof If we define P (x, t) = T (P(x, t),P (Λx, t)), (4) where T T . Then (V,P ,T)isan L-fuzzy normed space because (a),(b),(c),(f) and (e) are immediate from Definition 1.4, for triangle inequality (d), T (P (x, t),P (z, s)) = T [T (P(x, t),P (Λx, t)),T (P(z, s),P (Λz, s))] ≤ T [T (P(x, t),P(z, s))T (P (Λx, t),P (Λz, s))] ≤ T (P(x+ z, t+ s),P (Λ(x+ z), t+ s)) = P (x+ z, t+ s). P P Now, let x −→ x. Then by Theorem 2.3 x −→ x, but since by (4), P (Λx, t) ≥ n n L P (x, t), then Λx −→ Λx. HenceΛ is continuous. Corollary 2.2 Every linear isomorphism between finite dimensionalL-fuzzy normed spaces is topological isomorphism. Proof By Theorem 2.4, the linear isomorphism between finite dimensionalL-fuzzy normed spaces is continuous and hence is topological isomorphism. Corollary 2.3 Every finite dimensional L-fuzzy normed space (V,P,T ) is complete. Proof By Corollary 2.2, (V,P,T )isL-fuzzy topologically isomorphism to (R ,Φ,T ). Since (R ,Φ,T ) is complete and L-fuzzy topological isomorphism preserves com- pleteness, (V,P,T ) is complete. 3. Conclusion In this paper, we consideredL-fuzzy normed spaces and reviewed some fundamental problems atL-fuzzy functional analysis. Acknowledgements The author would like to thank the referee, Dr. Hadi Naseri and editorial office for giving useful suggestions on the improvement of this paper. References 1. Hosseini S B, O’Regan D, Saadati R (2007) Some results on intuitionistic fuzzy spaces. Iran J. Fuzzy Syst. 4: 53-64 2. Saadati R (2009) A note on “Some results on the IF-normed spaces”. Chaos Solitons Fractals 41: 206–213 3. Deschrijver G, O’Regan D, Saadati R, Vaezpour S M (2009) L-fuzzy Euclidean normed spaces and compactness. Chaos Solitons Fractals 42: 40-45 4. Saadati R, Park J H (2006) Intuitionistic fuzzy Euclidean normed spaces. Commun. Math. Anal. 1: 85-90 5. Saadati R, Cho Y J (2011) Cheng-Mordeson L-fuzzy normed spaces and application in stability of functional equation. Acta Mathematica Academiae Paedagogicae Ny´ıregyhaziensis ´ 27: 127-143 Fuzzy Inf. Eng. (2011) 4: 385-391 391 6. Agarwal R P, Cho Y J, Saadati R (2011) On random topological structures. Abstract and Applied Analysis Volume 2011: 1-41 7. Saadati R, Park J H (2006) On the intuitionistic fuzzy topological spaces. Chaos, Solitons and Fractals 27: 331-344 8. Saadati R, Vaezpour S M, Cho Y J (2009) Quicksort algorithm: Application of a fixed point theorem in intuitionistic fuzzy quasi-metric spaces at a domain of words. J. Comput. Appl. Math. 228: 219-225
Fuzzy Information and Engineering – Taylor & Francis
Published: Dec 1, 2011
Keywords: Bounded linear operators; Finite dimensional normed spaces; -fuzzy normed spaces
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