L_FNS1c.dvi CUBO A Mathematical Journal Vol.12, No¯ 03, (71–81). October 2010 L -Random and Fuzzy Normed Spaces and Classical Theory DONAL O’REGAN Department of Mathematics, National University of Ireland, Galway, Ireland email: donal.oregan@nuigalway.ie AND REZA SAADATI Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran email: rsaadati@eml.cc ABSTRACT In this paper we study L -random and L -fuzzy normed spaces and prove open mapping and closed graph theorems for these spaces. RESUMEN En este artículo estudiamos espacios normados L -random and L -fuzzy. Probamos el teo- rema de la aplicación abierta y el teorema del gráfico cerrado. Key words and phrases: L -random normed space, L -fuzzy normed space, completeness, quotient space, open mapping and closed graph. Math. Subj. Class.: PLEASE INFORM. 72 Donal O’Regan & Reza Saadati CUBO 12, 3 (2010) 1 Introduction and Preliminaries In this paper we study L -random and L -fuzzy normed spaces and study completeness for these spaces. Further we prove open mapping and closed graph theorems in this setting. The ideas here are motivated from the functional analysis literature. The plan in sections 1-3 is to present in detail the L -random normed space setting. In section 4 we see from the definition how easily the theory extends to the L -fuzzy normed space situation. Let L = (L,≥L ) be a complete lattice, i.e. a partially ordered set in which every nonempty subset admits a supremum and infimum, and 0L = inf L, 1L = sup L. The space of lattice random distribution functions, denoted by ∆+ L , is defined as the set of all mappings F : R∪ {−∞,+∞} → L such that F is continuous and non-decreasing on R, F(0) = 0L , F(+∞) = 1L . Now D+ L ⊆ ∆ + L is defined as D+ L = {F ∈ ∆+ L : l−F(+∞) = 1L }, where l − f (x) denotes the left limit of the function f at the point x. The space ∆+ L is partially ordered by the usual point-wise ordering of functions, i.e., F ≥ G if and only if F(t) ≥L G(t) for all t in R. The maximal element for ∆+ L in this order is the distribution function given by ε0(t) =    0L , if t ≤ 0, 1L , if t > 0. Define the mapping T∧ from L 2 to L by: T∧(x, y) =      x, if y ≥L x, y, if x ≥L y. Recall (see [4], [5]) that if {xn} is a given sequence in L, (T∧) n i=1 xi is defined recurrently by (T∧) 1 i=1 xi = x1 and (T∧) n i=1 xi = T∧((T∧) n−1 i=1 xi , xn) for n ≥ 2. A negation on L is any decreasing mapping N : L → L satisfying N (0L ) = 1L and N (1L ) = 0L . If N (N (x)) = x, for all x ∈ L, then N is called an involutive negation. In the following L is endowed with a (fixed) negation N . Definition 1.1. A lattice random normed space (briefly, L -random normed space) is a triple (X , P , T ), where X is a vector space, T is a t–norm on the lattice L and P is a mapping from X × [0,∞) into D+ L such that the following conditions hold: (LRN1) P (x, t) = ε0(t) for all t > 0 if and only if x = 0; (LRN2) P (αx, t) = P ( x, t |α| ) for all x in X , α 6= 0 and t ≥ 0; (LRN3) P (x + y, t + s) ≥L T (P (x, t), P ( y, s)) for all x, y ∈ X and t, s ≥ 0. CUBO 12, 3 (2010) L -Random and Fuzzy Normed Spaces ... 73 We note from (LPN2) that P (−x, t) = P (x, t) (x ∈ X , t ≥ 0). Example 1.2. Let L = [0, 1] × [0, 1] and operation ≤L defined by: L = {(a1, a2) : (a1, a2) ∈ [0, 1] × [0, 1] and a1 + a2 ≤ 1}, (a1, a2) ≤L (b1, b2) ⇐⇒ a1 ≤ b1, a2 ≥ b2, ∀a = (a1, a2), b = (b1, b2) ∈ L. Then (L,≤L ) is a complete lattice (see [2]). In this complete lattice, we denote its units by 0L = (0, 1) and 1L = (1, 0). Let (X ,‖ · ‖) be a normed space. Let T (a, b) = (min{a1, b1}, max{a2, b2}) for all a = (a1, a2), b = (b1, b2 ) ∈ [0, 1] × [0, 1] and µ be a mapping defined by P (x, t) = ( t t +‖x‖ , ‖x‖ t +‖x‖ ) , ∀t ∈ R+. Then (X , P , T ) is a L -random normed space. Definition 1.3. Let (X , P , T ) be a L -random normed space. (1) A sequence {xn} in X is said to be convergent to x in X if, for every t > 0 and ε ∈ L\{0L }, there exists a positive integer N such that P (xn − x, t) >L N (ε) whenever n ≥ N. (2) A sequence {xn} in X is called Cauchy sequence if, for every t > 0 and ε ∈ L\{0L }, there exists a positive integer N such that P (xn − xm, t) >L N (ε) whenever n ≥ m ≥ N. (3) A L -random normed space (X , P , T ) is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X . Theorem 1.4. If (X , P , T ) is a L -random normed space and {xn} is a sequence such that xn → x, then limn→∞ P (xn, t) = P (x, t). Proof. The proof is the same as in [9]. Let (X , P , T ) be a L -random normed space. For t > 0 we define the open ball B(x, r, t) with center x and radius r ∈ L \ {0L , 1L } as B(x, r, t) = { y ∈ X : P (x − y, t) >L N (r)}. Henceforth we assume that T is a continuous t–norm on the lattice L such that for every µ ∈ L \ {0L , 1L }, there is a λ ∈ L \ {0L , 1L } such that T n−1(N (λ), ..., N (λ)) >L N (µ). Lemma 1.5. Let (X , P , T ) be a L -random normed space. Let N be a continuous negator on L . Define Eλ,P : V → R + ∪ {0} by Eλ,P (x) = inf{t > 0 : P (x, t) >L N (λ)} for each λ ∈ L \ {0L , 1L } and x ∈ V . Then we have the following properties. 74 Donal O’Regan & Reza Saadati CUBO 12, 3 (2010) (i) For any µ ∈ L \ {0L , 1L } there exists λ ∈ L \ {0L , 1L } such that Eµ,P (x + y) ≤ Eλ,P (x) + Eλ,P ( y) for any x, y ∈ V . (ii) The sequence (xn)n∈N is convergent w.r.t. a L -random norm P if and only if Eλ,P (xn − x) → 0. Also the sequence (xn)n∈N is Cauchy w.r.t. a L -random norm P if and only if it is Cauchy w.r.t. Eλ,P . Proof. For (i), by the continuity of the t-norm T and the negator N , for every µ ∈ L \{0L , 1L } we can find a λ ∈ L \ {0L , 1L } such that T (N (λ), N (λ)) ≥L N (µ). By Definition 1.1 we have P (x + y, Eλ,P (x) + Eλ,P ( y) + 2δ) ≥L T (P (x, Eλ,M (x) +δ), P ( y, Eλ,P ( y) +δ)) ≥L T (N (λ), N (λ)) ≥L N (µ), for every δ > 0, which implies that Eµ,P (x + y) ≤ Eλ,P (x) + Eλ,P ( y) + 2δ. Since δ > 0 was arbitrary, we have Eµ,P (x + y) ≤ Eλ,P (x) + Eλ,P ( y). For (ii), we have P (xn − x,η) >L N (λ) ⇐⇒ Eλ,P (xn − x) < η for every η > 0. 2 Quotient Spaces Definition 2.1. Let (V , P , T ) be a L -random normed space, W a linear manifold in V and let Q : V −→ V /W be the natural map, Q x = x + W . For t > 0, we define: P̄ (x + W , t) = sup{P (x + y, t) : y ∈ W }. CUBO 12, 3 (2010) L -Random and Fuzzy Normed Spaces ... 75 Theorem 2.2. Let W be a closed subspace of a L -random normed space (V , P , T ). If x ∈ V and ǫ > 0, then there is an x′ in V such that x′ + W = x + W , Eλ,P (x ′) < E ¯λ,P (x + W ) +ǫ. Proof. By the properties of sup, there always exists y ∈ W such that Eλ,P (x + y) < E ¯λ,P (x + W ) +ǫ. Now it is enough to put x ′ = x + y. Theorem 2.3. Let W be a closed subspace of a L -random normed space (V , P , T ) and P̄ be given in the above definition. Then: (1) P̄ is a L -random normed space, on V /W . (2) P̄ (Q x, t) ≥L P (x, t). (3) If (V , P , T ) is a complete L -random normed space, then so is (V /W , P̄ , T ). Proof. It is clear that P̄ (x + W , t) >L 0L . Let P̄ (x + W , t) = 1L . By definition there is a sequence {xn} in W such that P (x+ xn, t) −→ 1L . Thus, x+ xn −→ 0 or equivalently xn −→ (−x) and since W is closed, x ∈ W and x + W = W , the zero element of V /W . Then we have P̄ ((x + W ) + ( y + W ), t) = P̄ ((x + y) + W , t) ≥L P ((x + m) + ( y + n), t) ≥L T (P (x + m, t1), P ( y + n, t2 )) for m, n ∈ W , x, y ∈ V and t1 + t2 = t. Now if we take the sup, then we have P̄ ((x + W ) + ( y + W ), t) ≥L T (P̄ (x + W , t1), P̄ ( y + W , t2 )). Therefore P̄ is a L -random norm on V /W . (2) By Definition 2.1, we have P̄ (Q x, t) = P̄ (x + W , t) = sup{P (x + y, t) : y ∈ W } ≥L P (x, t). Note that, by Lemma 1.5, Eλ,P̄ (Q x) = inf{t > 0 : P̄ (Q x, t) >L N (λ)} ≤ inf{t > 0 : P (x, t) >L N (λ)} = Eλ,P (x). (3) Let {xn + W } be a Cauchy sequence in V /W . Then there exists n0 ∈ N such that for every n ≥ n0, Eλ,P̄ ((xn + W ) − (xn+1 + W )) ≤ 2 −n. Let y1 = 0. Choose y2 ∈ W such that Eλ,P (x1 − (x2 − y2), t) ≤ Eλ,P̄ ((x1 − x2) + W ) + 1/2. However E ¯λ,P ((x1 − x2) + W ) ≤ 1/2 and so Eλ,P (x1 − (x2 − y2)) ≤ 1/2 2. 76 Donal O’Regan & Reza Saadati CUBO 12, 3 (2010) Now suppose yn−1 has been chosen, so choose yn ∈ W such that Eλ,P ((xn−1 + yn−1) − (xn + yn)) ≤ Eλ,P̄ ((xn−1 − xn) + W ) + 2 −n+1. Hence we have Eλ,P ((xn−1 + yn−1) − (xn + yn)) ≤ 2 −n+2. However for every positive integer m > n and by Lemma 1.5 for λ ∈ L there exists γ ∈ L, such that Eλ,P ((xm + ym) − (xn + yn)) ≤ Eγ,P ((xn+1 + yn+1) − (xn + yn)) + ···+ Eγ,P ((xm + ym) − (xm−1 + ym−1)) ≤ m ∑ i=n 2−i. By Lemma 1.5, {xn + yn} is a Cauchy sequence in V . Since V is complete, there is an x0 in V such that xn + yn −→ x0 in V . On the other hand, xn + W = Q(xn + yn) −→ Q(x0) = x0 + W . Therefore, every Cauchy sequence {xn + W } is convergent in V /W and so V /W is complete. Thus (V /W , P̄ , T ) is a complete L -random normed space. Theorem 2.4. Let W be a closed subspace of a L -random normed space (V , P , T ). If two of the spaces V , W and V /W are complete, then so is the third one. Proof. If V is a complete L -random normed space, then so are V /W and W . Hence all that needs to be checked is that V is complete whenever both W and V /W are complete. Suppose that W and V /W are complete L -random normed spaces and let {xn} be a Cauchy sequence in V . Since Eλ,P̄ ((xn − xm) + W ) ≤ Eλ,P (xn − xm) for each m, n ∈ N, the sequence {xn + W } is Cauchy in V /W and so converges to y + W for some y ∈ W . Thus there is a n0 ∈ N such that for every n ≥ n0, we have Eλ,P̄ ((xn − y) + W ) < 2 −n. Now by the last theorem there exist a sequence { yn} in V such that yn +W = (xn − y) +W , Eλ,P ( yn) < Eλ,P̄ ((xn − y) +W ) + 2 −n. Thus we have limn Eλ,P ( yn) ≤ 0 by Lemma 1.5, P ( yn, t) → 1L for every t > 0, i.e. limn yn = 0. Therefore, {xn − yn − y} is a Cauchy sequence in W and thus is convergent to a point z ∈ W . This implies that {xn} converges to z + y and hence V is complete. 3 Open Mapping and Closed Graph Theorems Definition 3.1. A linear operator T : (V , P , T ) −→ (V ′, P ′, T ′) is said to be L -random bounded if there exist constants h ∈ R+ such that for every x ∈ V and for every t > 0, P ′(T x, t) ≥L P (x, t/h). (3.1) CUBO 12, 3 (2010) L -Random and Fuzzy Normed Spaces ... 77 Note that, by (3.1) we have Eλ,P ′ (T x) = inf{t > 0 : P ′(T x, t) >L N (λ)} ≤ inf{t > 0 : P (x, t/h) >L N (λ)} = = h inf{t > 0 : P (x, t) >L N (λ)} = hEλ,P (x). Theorem 3.2. Every linear operator T : (V , P , T ) −→ (V ′, P ′, T ′) is L -random bounded if and only if it is continuous. Proof. By (3.1) every L -random bounded linear operator is continuous. Now, we prove the converse. Let the linear operator T be continuous but not L -random bounded. Then, for each n in N there is a xn in V such that Eλ,P ′ (T xn) ≥ nEλ,P ( pn). If we let yn = xn nEλ,P (xn ) then it is easy to see yn → 0 but T yn do not tend to 0. Theorem 3.3. (Open mapping theorem) If T is a L -random bounded linear operator from a complete L -random normed space (V , P , T ) onto a complete L -random normed space (V ′, P ′, T ) then T is an open mapping. Proof. The theorem will be proved in several steps. Step1: Let E be a neighborhood of the 0 in V . We show 0 ∈ ( T(E) )o . Let W be a balanced neighborhood of 0 such that W + W ⊂ E. Since T(V ) = V ′ and W is absorbing, it follows that V ′ = ∪n T(nW ), so by Theorem 3.17 in [6], there exists a n0 ∈ N such that T(n0W ) has nonempty interior. Therefore, 0 ∈ ( T(W ) )o − ( T(W ) )o . On the other hand, ( T(W ) )o − ( T(W ) )o ⊂ T(W ) − T(W ) = T(W ) + T(W ) ⊂ T(E). Thus the set T(E) includes the neighborhood ( T(W ) )o − ( T(W ) )o of 0. Step 2: We show 0 ∈ (T(E))o. Since 0 ∈ E and E is an open set, there exists 0L n. On the other hand, 0 ∈ T(B(0,ǫn, t ′ n )), where t ′ n = 1 2n t0, so by step 1, there exist 0L 0 such that B(0,σn, tn ) ⊂ T(B(0,ǫn, t ′ n )). Since the set {B(0, r, 1/n)} is a countable local base at zero and t′n −→ 0 as n −→ ∞, so tn and σn can be chosen such that tn −→ 0 and σn −→ 0L as n → ∞. Now we show B(0,σ1, t1) ⊂ (T(E)) o. Suppose y0 ∈ B(0,σ1, t1 ). Then y0 ∈ T(B(0,ǫ1, t ′ 1)) and so for 0L 0 the ball B( y0,σ2, t2) intersects T(B(0,ǫ1, t ′ 1)). Therefore there exists x1 ∈ B(0,ǫ1, t ′ 1) such that T x1 ∈ B( y0,σ2, t2), i.e. P ′( y0 − T x1, t2 ) >L N (σ2) or equivalently y0 − T x1 ∈ B(0,σ2, t2) ⊂ T(B(0,ǫ1, t ′ 1)). By the similar argument there exist x2 in B(0,ǫ2, t ′ 2) such that P ′( y0 − (T x1 + T x2), t3) = P ′(( y0 − T x1) − T x2, t3) >L N (σ3). 78 Donal O’Regan & Reza Saadati CUBO 12, 3 (2010) If this process is continued, it leads to a sequence {xn} such that xn ∈ B(0,ǫn, t ′ n ), P ′ ( y0 − ∑n−1 j=1 T x j, tn ) >L N (σn). Now if n, m ∈ N and m > n, then P ( n ∑ j=1 x j − m ∑ j=n+1 x j , t ) = µ ( m ∑ j=n+1 x j , t ) ≥L T m−n(P (xn+1, tn+1), P (xm, tm )) where tn+1 + tn+2 + ··· + tm = t. Put t ′ 0 = min {tn+1, tn+2,··· , tm }. Since t ′ n −→ 0, there exists n0 ∈ N such that 0 < t ′ n ≤ t ′ 0 for n > n0. Therefore, for m > n we have T m−n(P (xn+1, t ′ 0 ), P (xm, t ′ 0 )) ≥L T m−n(P (xn+1, t ′ n+1), P (xm, t ′ m )) ≥L T m−n(N (ǫn+1), N (ǫm)). Hence, lim n−→∞ P ( m ∑ j=n+1 x j , t ) ≥L lim n−→∞ T m−n(N (ǫn+1), N (ǫm)) = 1L . That is, P ( ∑m j=n+1 x j , t ) −→ 1L , for all t > 0. Thus the sequence { ∑n j=1 x j } is a Cauchy se- quence and consequently the series { ∑ ∞ j=1 x j } converges to some point x0 ∈ V , because V is a complete space. By fixing t > 0, there exists n0 ∈ N such that t > tn for n > n0, because tn −→ 0. Thus P ′ ( y0 − T ( n−1 ∑ j=1 x j ) , t ) ≥L P ′ ( y0 − T ( n−1 ∑ j=1 x j ) , tn ) ≥L N (σn) and thus P ′ ( y0 − T ( ∑ n−1 j=1 x j ) , t ) −→ 1L . Therefore, y0 = lim n T ( n−1 ∑ j=1 x j ) = T ( lim n n−1 ∑ j=1 x j ) = T x0. But, by Proposition 1 of [7], P (x0, t0 ) = lim n P ( n ∑ j=1 x j , t0 ) ≥L T n(lim n (P (x1, t ′ 1), P (xn, t ′ n )) ≥L lim n T n−1(N (ǫ1), ..., N (ǫn)) >L N (α) Hence x0 ∈ B(0,α, t0). Step 3: Let G be an open subset of V and x ∈ G. Then we have T(G) = T x + T(−x + G) ⊃ T x + (T(−x + G))o. Hence T(G) is open, because it includes a neighborhood of each of its point. CUBO 12, 3 (2010) L -Random and Fuzzy Normed Spaces ... 79 Corollary 3.4. Every one-to-one L -random bounded linear operator from a complete L - random normed space onto a complete L -random normed space has a L -random bounded inverse. Definition 3.5. Let T and T ′ be two continuous t-norms. Then T ′ dominates T , denoted by T ′ ≫L T , if for all x1, x2, y1, y2 ∈ L , T [T ′(x1, x2), T ′( y1, y2)] ≤L T ′[T (x1, y1), T (x2, y2)]. Theorem 3.6. (Closed graph theorem) Let T be a linear operator from the complete L -random normed space (V , P , T ) into the complete L -random normed space (V ′, P ′, T ). Suppose for every sequence {xn} in V such that xn −→ x and T xn −→ y for some elements x ∈ V and y ∈ V ′ it follows that T x = y. Then T is L -random bounded. Proof. For any t > 0, x ∈ V and y ∈ V ′, define Φ((x, y), t) = T ′(P (x, t), P ′( y, t)), where T ′ ≫L T . First we show that (V × V ′,Φ, T ) is a complete L -random normed space. The properties of (LRN1) and (LRN2) are immediate from the definition. For the triangle inequality (LRN3), suppose that x, z ∈ V , y, u ∈ V ′ and t, s > 0, then T (Φ((x, y), t),Φ((z, u), s)) = T [T ′(P (x, t), P ′( y, t)), T ′(P (z, s), P ′(u, s))] ≤L T ′[T (P (x, t), P (z, s)), T (P ′( y, t), P ′(u, s))] ≤L T ′(P (x + z, t + s), P ′( y + u, t + s)) = Φ((x + z, y + u), t + s). Now if {(xn, yn)} is a Cauchy sequence in V ×V ′, then for every ǫ ∈ L \ {0L } and t > 0 there exists n0 ∈ N such that Φ((xn, yn) − (xm, ym ), t) >L N (ǫ) for m, n > n0. Thus for m, n > n0, T ′(P (xn − xm, t), P ′( yn − ym, t)) = Φ((xn − xm, yn − ym), t) = Φ((xn, yn) − (xm, ym), t) >L N (ǫ). Therefore {xn} and { yn} are Cauchy sequences in V and V ′, respectively, and there exist x ∈ V and y ∈ V ′ such that xn −→ x and yn −→ y and consequently (xn, yn) −→ (x, y). Hence (V × V ′,Φ, T ) is a complete L -random normed space. The remainder of the proof is the same as the classical case. 4 L -fuzzy normed space We conclude the paper with the setting of L -fuzzy normed spaces. Consider the L -fuzzy normed space (X , F , T ) in which F is a L -fuzzy set on X× ]0,+∞[ satisfying the following 80 Donal O’Regan & Reza Saadati CUBO 12, 3 (2010) conditions for every x, y in X and t, s in (0,+∞): (a) 0L