J. Appl. Math. & Computing Vol. 17(2005), No. 1 - 2, pp. 475 - 484
SOME RESULTS ON FUZZY BANACH SPACES R. SAADATI AND S. M. VAEZPOUR∗
Abstract. The main aim of this paper is to consider the fuzzy norm, define the fuzzy Banach spaces, its quotients and prove some theoremes and in particular Open mapping and Closed graph theoremes on these spaces. AMS Mathematics Subject Classification : 46S40. Key words and phrases : Fuzzy space, fuzzy normed space, fuzzy Banach space.
1. Introduction The theory of fuzzy sets was introduced by L. Zadeh [8] in 1965. Many mathematicians considered the fuzzy metric spaces in different view ( see [1], [3], [4] and [5]). First we recall the definition of continuous t-norm, fuzzy metric spaces and Cauchy sequences introduced by George and Veermani [3]. Definition 1. A binary operation ∗ : [0, 1] × [0, 1] −→ [0, 1] is said to be a continuous t-norm if ([0, 1], ∗) is a topological monoid with unit 1 such that a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d (a, b, c, d ∈ [0, 1]). It is clear that if we define a∗b = ab or a∗b = min(a, b) then ∗ is a continuous t-norm. Definition 2. The 3-tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞) satisfying the following conditions for all x, y, z ∈ X and t, s > 0, (i) M (x, y, 0) > 0, Received October 21, 2003. Revised April 11, 2004. ∗ Corresponding author. c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.
475
476
R. Saadati and S. M. Vaezpour
(ii) (iii) (iv) (v)
M (x, y, t) = 1 for all t > 0 if and only if x = y, M (x, y, t) = M (y, x, t), M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s) for all t, s > 0, M (x, y, .) : (0, ∞) −→ [0, 1] is continuous.
Definition 3. A sequence {xn }n in a fuzzy metric space (X, M, ∗) is a Cauchy sequence if and only if for each 0 < < 1 and t > 0 there exist n0 ∈ N such that for all n, m ≥ n0 we have, M (xn , xm , t) > 1 − . A fuzzy metric space is said be complete if and only if every Cauchy sequence is convergent. At last we state the following lemma which will be used later. Lemma 1. If A is a subset of [0, 1] and supA = β then for ∈ [0, β) there exists α0 in A such that ∗ β ≤ α0 . Proof. Since ∗ β < β, so it is enough to choose α0 ∈ [ ∗ β, β] ∩ A.
2. Fuzzy Banach spaces In this section at first we define the fuzzy normed spaces and give several examples of these spaces, and then we define the fuzzy Banach spaces. Definition 4. The 3-tuple (X, N, ∗) is said to be a fuzzy normed space if X is a vector space, ∗ is a continuous t-norm and N is a fuzzy set on X × (0, ∞) satisfying the following conditions for every x, y ∈ X and t, s > 0: (i) N (x, t) > 0, (ii) N (x, t) = 1 iff x = 0, (iii) N (αx, t) = N (x, t/|α|), for all α 6= 0, (iv) N (x, t) ∗ N (y, s) ≤ N (x + y, t + s), (v) N (x, .) : (0, ∞) −→ [0, 1] is continuous, (vi) limt→∞ N (x, t) = 1. Lemma 2. Let N be a fuzzy norm. Then (i) N (x, t) is nondecreasing with respect to t for each x ∈ X. (ii) N (x − y, t) = N (y − x, t).
Some results on fuzzy Banach spaces
477
Proof. Let t < s. Then k = s − t > 0 and we have N (x, t) = N (x, t) ∗ 1 = N (x, t) ∗ N (0, k) ≤ N (x, s). This proves the (i). To prove (ii) we have N (x − y, t) = N ((−1)(y − x), t) t = N y − x, | − 1| = N (y − x, t). Lemma 3. Let (X, N, ∗) be a fuzzy normed space. If we define M (x, y, t) = N (x − y, t), then M is a fuzzy metric on X, which is called the fuzzy metric induced by the fuzzy norm N . Lemma 4. A fuzzy metric M which is induced by a fuzzy norm on a fuzzy normed space (X, N, ∗) has the following properties for all x, y, z ∈ X and every scalar α 6= 0: (i) M (x + z, y + z, t) = M (x, y,t), t (ii) M (αx, αy, t) = M x, y, |α| Proof. M (x + z, y + z, t) = N ((x + z) − (y + z), t) = N (x − y, t) = M (x, y, t) Also, M (αx, αy, t) = N (αx − αy, t) t = N x − y, |α| t = M x, y, . |α|
478
R. Saadati and S. M. Vaezpour
Example 1. Let (x, k.k) be a normed space. We define a ∗ b = ab or a ∗ b = min(a, b) and ktn k, m, n ∈ R+ . N (x, t) = n kt + mkxk Then (X, N, ∗) is a fuzzy normed space. In particular if k = n = m = 1 we have t N (x, t) = , t + kxk which is called the standard fuzzy norm induced by norm k.k. Example 2. Let (x, k.k) be a normed space. We define a ∗ b = ab and −1 kxk N (x, t) = exp , t for x ∈ X and t ∈ (0, ∞). Then (X, N, ∗) is a fuzzy normed space.
Definition 5. Let (X, N, ∗) be a fuzzy normed space. We define the open ball B(x, r, t) and the closed ball B[x, r, t] with center x ∈ X and radius 0 < r < 1, t > 0 as follows: B(x, r, t) = {y ∈ X : N (x − y, t) > 1 − r} B[x, r, t] = {y ∈ X : N (x − y, t) ≥ 1 − r}.
Lemma 5. If (X, N, ∗) is a fuzzy normed space, then (a) The function (x, y) −→ x + y is continuous, (b) The function (α, x) −→ αx is continuous.
Proof. If xn −→ x and yn −→ y, then as n −→ ∞, t t N ((xn + yn ) − (x + y), t) ≥ N xn − x, ∗ N yn − y, −→ 1. 2 2 This proves (a). Now if xn −→ x, αn −→ α and αn 6= 0 then N (αn xn − αx, t) = N (αn (xn − x) + x(αn − α), t) t t ∗ N x(αn − α), ≥ N αn (xn − x), 2 2 t t ∗ N x, −→ 1, = N xn − x, 2αn 2(αn − α)
Some results on fuzzy Banach spaces
as n −→ ∞, and this proves (b).
479
Definition 6. The fuzzy normed space (X, N, ∗) is said to be a fuzzy Banach space whenever X is complete with respect to the fuzzy metric induced by fuzzy norm.
3. Quotient spaces
Definition 7. Let (X, N, ∗) be a fuzzy normed space, M be a linear manifold in X and let Q : X −→ X/M be the natural map, Qx = x + M . We define N (x + M, t) = sup{N (x + y, t) : y ∈ M }, t > 0. Theorem 8. If M is a closed subspace of fuzzy normed space X and N (x+M, t) is defined as above then (a) N is a fuzzy norm on X/M . (b) N (Qx, t) ≥ N (x, t). (c) If (X, N, ∗) is a fuzzy Banach space, then so is (X/M, N, ∗).
Proof. It is clear that N (x + M, t) ≥ 0. Let N (x + M, t) = 1. By definition there is a sequence {xn } in M such that N (x + xn , t) −→ 1. So x + xn −→ 0 or equivalently xn −→ (−x) and since M is closed so x ∈ M and x + M = M , the zero element of X/M . On the other hand we have, N ((x + M ) + (y + M ), t) = N ((x + y) + M, t) ≥ N ((x + m) + (y + m), t) ≥ N (x + m, t1 ) ∗ N (y + m, t2 ) for m, n ∈ M , x, y ∈ X and t1 + t2 = t. Now if we take sup on both sides, we have, N ((x + M ) + (y + M ), t) ≥ N (x + M, t1 ) ∗ N (y + M, t2 ). Also we have, N (α(x + M ), t) = N (αx + M, t) = sup{N (αx + αy, t) : y ∈ M } = sup{N (x + y, t/|α|) : y ∈ M } = N (x + M, t/|α|) Therefore (X, N, ∗) is a fuzzy normed space.
480
R. Saadati and S. M. Vaezpour
To prove (b) we have, N (Qx, t) = N (x + M, t) = sup{N (x + y, t) : y ∈ M } ≥ N (x, t). Let {xn + M } be a Cauchy sequence in X/M . Then there exists n > 0 such that n −→ 0 and, N ((xn + M ) − (xn+1 + M ), t) ≥ 1 − n . Let y1 = 0. We choose y2 ∈ M such that, N (x1 − (x2 − y2 ), t) ≥ N ((x1 − x2 ) + M, t) ∗ (1 − 1 ). But N ((x1 − x2 ) + M, t) ≥ (1 − 1 ). Therefore, N (x1 − (x2 − y2 ), t) ≥ (1 − 1 )(1 − 1 ). Now suppose yn−1 has been chosen, yn ∈ M can be chosen such that N ((xn−1 + yn−1 ) − (xn + yn ), t) ≥ N ((xn−1 − xn ) + M, t) ∗ (1 − n−1 ), and therefore, N ((xn−1 + yn−1 ) − (xn + yn ), t) ≥ (1 − n−1 ) ∗ (1 − n−1 ). Thus, {xn + yn } is a Cauchy sequence in X. Since X is complete, there is an x0 in X such that xn + yn −→ x0 in X. On the other hand xn + M = Q(xn + yn ) −→ Q(x0 ) = x0 + M. Therefore every Cauchy sequence {xn + M } is convergent in X/M and so X/M is complete and (X/M, N, ∗) is a fuzzy Banach space. Theorem 9. Let M be a closed subspace of a fuzzy normed space X. If x ∈ X and ∈ [0, N (x + M, t)), then there is an x0 in X such that, x0 + M = x + M and N (x0 , t) > N (x + M, t) ∗ .
Proof. By (1.4) there always exists y ∈ M such that, N (x + y, t) > N (x + M, t) ∗ . Now it is enough to put x0 = x + y.
Theorem 10. Let M be a closed subspace of a fuzzy normed space (X, N, t). If a couple of the spaces X, M , X/M are complete, so is the third one.
Some results on fuzzy Banach spaces
481
Proof. If X is a fuzzy Banach space, so are X/M and M . Therefore all that needs to be checked is that X is complete whenever both M and X/M are complete. Suppose M and X/M are fuzzy Banach spaces and let {xn } be a Cauchy sequence in X. Since N ((xn − xm ) + M, t) ≥ N (xn − xm , t) whenever m, n ∈ N , the sequence {xn + M } is Cauchy in X/M and so converges to y + M for some y ∈ M . So there exists a sequence {n } such that n −→ 0 and N ((xn − y) + M, t) > 1 − n for each t > 0. Now by last theorem there exists a sequence {yn } in X such that yn + M = (xn − y) + M and N (yn , t) > N ((xn − y) + M, t) ∗ (1 − n ). So limn N (yn , t) ≥ 1 and limn yn = 0. Therefore {xn − yn − y} is a Cauchy sequence in M and thus is convergent to a point z ∈ M and this implies that {xn } converges to z + y and X is complete. Theorem 11. (Open mapping theorem) If T is a continuous linear operator from the fuzzy Banach space (X, N1 , ∗) onto the fuzzy Banach space (X, N2 , ∗), then T is an open mapping. Proof. The theorem will be proved in several steps. ¯ o. Step1: Let E be a neighborhood of 0 in X. We show that 0 ∈ ((T (E)) Let V be a balanced neighborhood of 0 such T that V + V ⊂ E. Since T (X) = Y and V is absorbing, it follows that Y = n T (nV ). So by Theorem 3.17 in [3] there exists a positive integer n0 such that T (n¯0 V ) has nonempty interior. ¯ ))o − (T (V ¯ ))o . On the other hand, Therefore, 0 ∈ (T (V ¯ ))o − (T (V ¯ ))o ⊂ (T (V ¯ )) − (T (V ¯ )) (T (V = T (V ) ¯ − T (v) ¯ ⊂ T (E). ¯ includes the neighborhood (T (V ¯ ))o − (T (V ¯ ))o of 0. So the set T (E) o Step 2: It is shown 0 ∈ (T (E)) . Since 0 ∈ E and E is an open set, then there exist 0 < α < 1 and t0 ∈ (0, ∞) such that B(0, α, t0 ) ⊂ E. But for 0 < α < 1 a sequence {n } can be found such that n −→ 0 and 1 − α < lim[(1 − 1 ) ∗ · · · ∗ (1 − n )]. n
On the other hand 0 ∈ T (B(0,¯n , t0n )), where t0n = 21n t0 , so by step 1 there exist σn ∈ (0, 1) and tn > 0 such that B(0, σn , tn ) ⊂ T (B(0,¯ n , t0n ). Since the set
482
R. Saadati and S. M. Vaezpour
{B(0, 1/n, 1/n)} is a countable locally base at zero and t0n −→ 0 as n −→ ∞, so tn and σn can be chosen such that tn , σn −→ 0 as n −→ ∞. Now it is shown that B(0, σ1 , t1 ) ⊂ (T (E))o . Suppose y0 ∈ B(0, σ1 , t1 ). Then y0 ∈ T (B(0,¯1 , t0 )) and so for σ2 > 0 and t2 > 0 the ball B(y0 , σ2 , t2 ) intersects T (B(0, 1 , t0 )). Therefore there exists x1 ∈ B(0, 1 , t0 ) such that T x1 ∈ B(y0 , σ2 , t2 ), i.e., N2 (y0 − T x1 , t2 ) > 1 − σ2 or equivalently y0 − T x1 ∈ B(0, σ2 , t2 )⊂ T (B(0,¯1 , t0 )) and by the similar argument there exists x2 in B(0, 2 , t0 ) such that N2 (y0 − (T x1 + T x2 ), t3 ) = N2 ((y0 − T x1 ) − T x2 , t3 ) > 1 − σ3 . If this process is continued, it leads to a sequence {xn } such that xn ∈ B(0, n , t0n ) and ! n−1 X N2 y0 − T xj , tn > 1 − σn . j=1
Now if n ∈ N and {pn } is a positive and increasing sequence, then ! ! n+p n+p n X Xn Xn xj − x j , t = N1 xj , t N1 j=1
j=1
j=n+1
≥ N1 (xn+1 , t1 ) ∗ · · · ∗ N1 (xn+pn , tpn ). where t1 + t2 + · · · + tpn = t. By putting t0 = min{t1 , t2 , · · · , tpn }, since t0n −→ 0 so there exists n0 such that 0 < t0n ≤ t0 for n > n0 . Therefore, N1 (xn+1 , t0 ) ∗ · · · ∗ N1 (xn+pn , tpn ) ≥ N1 (xn+1 , t0n+1 ) ∗ · · · ∗ N1 (xn+pn , t0n+pn ) ≥ (1 − n+1 ) ∗ · · · ∗ (1 − n+pn ). Hence, lim N1
n−→∞
that is, N1
n+p Xn
xj , t
j=n+1 n+p Xn
xj , t
!
!
≥ lim [(1 − n+1 ) ∗ · · · ∗ (1 − n+pn )] = 1, n−→∞
−→ 1 for all t > 0. So the sequence
j=n+1
Cauchy sequence and cosequently the series
X ∞ j=1
x0 ∈ X, because X is a complete space.
xj
X n
xj
is a
j=1
converges to some point
Some results on fuzzy Banach spaces
483
By fixing t > 0, there exists n0 such that t > tn for n > n0 , because tn −→ 0. So it follows: ! ! ! ! n−1 n−1 X X N2 y0 − T xj , t ≥ N2 y0 − T x j , tn j=1
and thus N2 y0 − T
n−1 X
j=1
! !
xj , t
≥ (1 − σn ) −→ 1. Therefore,
j=1
y0 = lim T n
n−1 X
xj
!
= T x0 .
j=1
But, N1 (x0 , t0 ) ≥ lim sup N1 n
n X
x j , t0
!
j=1
≥ lim sup(N1 (x1 , t01 ) ∗ · · · ∗ N1 (xn , t0n )) n
≥ lim sup((1 − 1 ) ∗ · · · ∗ (1 − n )) n
= 1 − α. Hence x0 ∈ B(0, α, t0 ). Step 3: Let G be an open subset of X and x ∈ G. Then, T (G) = T x + T (−x + G) ⊃ T x + (T (−x + G))o . Hence T (G) would be open, because it includes a neighborhood of each of its point. Theorem 12. (Closed graph theorem) Let T be a linear operator from the fuzzy Banach space (X, N1 , ∗) into the fuzzy Banach space (Y, N2 , ∗). Suppose for every sequence {xn } in X such that xn −→ x and T xn −→ y for some elements x ∈ X and y ∈ Y it follows T x = y. Then T is continuous. Proof. At first it is proved that the fuzzy norm N which is defined on X × Y by, N ((x, y), t) = N1 (x, t) ∗ N2 (y, t) is a complete fuzzy norm. For each x, z ∈ X , y, u ∈ Y and t, s > 0 it follows: N ((x, y), t) ∗ N ((z, u), s) = [N1 (x, t) ∗ N2 (y, t)] ∗ [N1 (z, s) ∗ N2 (u, s)] = [N1 (x, t) ∗ N1 (z, s)] ∗ [N2 (y, t) ∗ N2 (u, s)]
484
R. Saadati and S. M. Vaezpour
≤ N1 (x + z, t + s) ∗ N2 (y + u, t + s) = N ((x + z, y + u), t + s). Now if {(xn , yn )} is a Cauchy sequence in X × Y , then for every > 0 and t > 0 there exists n0 ∈ N such that for m, n > n0 , N ((xn , yn ) − (xm , ym ), t) > 1 − . So for m, n > n0 , N1 (xn − xm , t) ∗ N2 (yn − ym , t) = N ((xn − xm , yn − ym ), t) = N ((xn , yn ) − (xm , ym ), t) > 1 − . Therefore {xn } and {yn } are Cauchy sequences in X and Y respectively and there exist x ∈ X and y ∈ Y such that xn −→ x and yn −→ y and consequently (xn , yn ) −→ (x, y). Hence (X × Y, N, ∗) is a complete fuzzy normed space.
References 1. Deng Zi-ke, Fuzzy pseudo metric spaces, J. Math. Anal. Appl. 86 (1982), 74-95. 2. Fang, Jin-xuan, On the completion of fuzzy normed spaces, J. Fuzzy Math. 9(1) (2002), 139-149. 3. A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395-399. 4. M. Gerbiec, Fixed points in fuzzy spaces, Fuzzy Sets and Systems 27 (1988), 385-389. 5. O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984), 215-229. 6. R. E. Megginson, An introduction to Banach space theory, Springer Verlag, New York, 1998. 7. W. Rudin, Functional Analysis, Mc Graw-Hill, New York, 1973. 8. L. A. Zade, Fuzzy sets, Information and Control 98 (1965), 338-353. Reza Saadati received his BS and MS from Yazd University under the direction of professor S. Mansour Vaezpour. Since 2000 he has been at Amol Azad University. His research interests center on the Fuzzy normed spaces and Fuctional analysis. No 1, Fajr 4 Allay, Nour St. Amol, Iran S. Mansour Vaezpour received his BS, MS and Ph. D from Shiraz University under the direction of professor Karim Seddighi. Since 1988 he has been at Yazd University. At present he is associate prof. in the dept. of Mathematics and the president of the School of Sciences at Yazd University. His research interests center on the Fuzzy normed spaces, Operator theory and Fuctional analysis. Department of Mathematics, University of Yazd, P. O. Box: 89195-741, Yazd, Iran e-mail:
[email protected]
