Abstract and Applied AnalysisVolume 2011 (2011), Article ID 762361, 41 pagesdoi:10.1155/2011/762361

Review Article

On Random Topological Structures

Ravi P. Agarwal,¹ Yeol Je Cho,² and Reza Saadati³

¹Department of Mathematics, Florida Institute of Technology, FL 32901, USA²Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea³Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Received 27 January 2011; Accepted 6 April 2011

Academic Editor: Alexander I. Domoshnitsky

Copyright © 2011 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present some topics about random spaces. The main purpose of this paper is to study topological structure of random normed spaces and random functional analysis. These subjects are important to the study of nonlinear analysis in random normed spaces.

1. Introduction

The purpose of this paper is to give a comprehensive text to the study of nonlinear random analysis such as the study of fixed point theory, approximation theory and stability of functional equations in random normed spaces. The notion of random normed space goes back to Šherstnev [1] and Hadžić [2–4] who were dulled from Menger [5], and Schweizer and Sklar [4] works. Some authors [6–11] considered some properties of probabilistic normed and metric spaces also fuzzy metric and normed spaces [12–21]. Fixed point theory [3, 22–26], approximation theory [27–31], and stability of functional equations [32–38] are studied at random normed space and its depended space that is, fuzzy normed space.

This paper is introduced as a survey of the latest and new results on the following topics.(i)Basic theory of triangular norms.(ii)Topological structure of random normed spaces.(iii)Random functional analysis.(iv)Relationship between random normed spaces and fuzzy normed spaces.

2. Triangular Norms

Triangular norms first appeared in the framework of probabilistic metric spaces in the work Menger [5]. It also turns out that this is a crucial operation in several fields. Triangular norms are an indispensable tool for the interpretation of the conjunction in fuzzy logics [39] and, subsequently, for the intersection of fuzzy sets [40]. They are, however, interesting mathematical objects themselves. We refer to some papers and books for further details (see [2, 4, 41–44]).

Definition 2.1. A triangular norm (shorter -norm) is a binary operation on the unit interval , that is, a function such that for all the following four axioms are satisfied:(T1) (commutativity);(T2) (associativity);(T3) (boundary condition);(T4) whenever (monotonicity).

The commutativity of (T1), boundary condition (T3), and the the monotonicity (T4) imply that for each -norm and for each the following boundary conditions are also satisfied: and therefore all -norms coincide on the boundary of the unit square .

The monotonicity of a -norm in its second component (T3) is, together with the commutativity (T1), equivalent to the (joint) monotonicity in both components, that is, to Basic examples are the Łukasiewicz -norm : and the -norms ,,, where

If, for two -norms and , the inequality holds for all , then we say that is weaker than or, equivalently, that is stronger than .

As a result of (2.2) we obtain, for each , Since, for all , trivially , we get for an arbitrary -norm , That is, is weaker and is stronger than any other -norms. Also since , we obtain the following ordering for four basic -norms

Proposition 2.2 (see [2]). (i) The minimum is the only -norm satisfying for all . (ii) The weakest -norm is the only -norm satisfying for all .

Proposition 2.3 (see [2]). A -norm is continuous if and only if it is continuous in its first component, that is, if for each the one-place function is continuous.

For example, the minimum and Łukasiewicz -norm are continuous but the t-norm defined by for , is not continuous.

Definition 2.4. (i) A -norm is said to be strictly monotone if (ii) A -norm is said to be strict if it is continuous and strictly monotone.

For example, the -norm is strictly monotone but the minimum and Łukasiewicz -norm are not.

Proposition 2.5 (see [2]). A -norm is strictly monotone if and only if

If is a t-norm, then is defined for every and by 1, if and , if .

Definition 2.6. A -norm is said to be Archimedean if for all there is an integer such that

Proposition 2.7 (see [2]). A -norm is Archimedean if and only if for each one has

Proposition 2.8 (see [2]). If -norm is Archimedean, then for each one has

For example, the product , Łukasiewicz -norm , and weakest -norm are all Archimedean but the minimum is not an Archimedean -norm.

A t-norm is said to be of Hadžić-type (we denote by ) if the family is equicontinuous at , that is, is a trivial example of a t-norm of Hadžić type, but is not of Hadžić type.

Proposition 2.9 (see [2]). If a continuous -norm is Archimedean, then it cannot be a -norm of Hadžić type.

Other important triangular norms are [see [45]]:(i)the Sugeno-Weber family is defined by and if ;(ii)the Domby family , defined by , if , if and if ;(iii)the Aczel-Alsina family , defined by , if , if and

if .

A -norm can be extended (by associativity) in a unique way to an -array operation taking for any the value defined by

can also be extended to a countable operation taking for any sequence in the value The limit on the right side of (2.20) exists since the sequence is nonincreasing and bounded from below.

Proposition 2.10 (see [45]). (i) For , the following implication holds: (ii) If is of Hadžić-type, then for every sequence in such that . (iii) If , then (iv) If , then

Definition 2.11. Let and be two continuous -norms. Then dominates , denoted by , if for all ,

Now, we extended definitions and results on the triangular norm on lattices. Let be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and , .

Definition 2.12 (see [46]). A -norm on is a mapping satisfying the following conditions:(a) (boundary condition);(b) (commutativity);(c) (associativity);(d) and (monotonicity).

Let be a sequence in converges to (equipped order topology). The -norm is said to be a continuous -norm if for each .

Note that we put whenever .

Definition 2.13 (see [46]). A continuous -norm on is said to be continuous -representable if there exist a continuous -norm and a continuous -conorm on such that, for all ,

For example, for all are continuous –representable.

Define the mapping from to by:

A negation on is any decreasing mapping satisfying and . If for all , then is called an involutive negation. In the following is endowed with a (fixed) negation .

3. Distribution Functions and Fuzzy Sets

We denote, , the space of all distribution functions, that is, the space of all mappings , such that is left-continuous and nondecreasing on , and . is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual point-wise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by

Example 3.1. The function , defined by is a distribution function. Since , then . Note that for each .

Example 3.2. The function , defined by is a distribution function. Since , then . Note that for each .

Example 3.3 (see [6]). The function , , defined by is a distribution function. Since , then . Note that for each .

Definition 3.4. If is a collection of objects denoted generically by , then a fuzzy set in is a set of ordered pairs: is called the membership function or grade of membership of in which maps to the membership space . Note that, when contains only the two points 0 and 1, is nonfuzzy and is identical to the characteristic function of a non-fuzzy set. The range of the membership function is or a complete lattice.

Example 3.5. Consider the fuzzy set which is real numbers considerably larger than 10, where

Example 3.6. Consider the fuzzy set which is real numbers close to 10, like (3.6) where Note that, in this paper, in short, we apply membership functions instead of fuzzy sets.

Definition 3.7 (see [47]). Let be a complete lattice and let be a nonempty set called the universe. An - fuzzy set in is defined as a mapping . For each in , represents the degree (in ) to which is an element of .

Lemma 3.8 (see [46]). Consider the set and operation defined by for all . Then is a complete lattice.

Definition 3.9 (see [48]). An intuitionistic fuzzy set in the universe is an object , where and for all are called the membership degree and the nonmembership degree, respectively, of in and, furthermore, satisfy .

Example 3.10. Consider the intuitionistic fuzzy set which is real numbers considerably larger than 10 for first place and real numbers close to 10 in second place, where As we said above, we apply .

4. Random Normed Spaces

Random (probabilistic) normed spaces were introduced by Šerstnev in 1962 [1] by means of a definition that was closely modelled on the theory of (classical) normed spaces and used to study the problem of best approximation in statistics. In the sequel, we will adopt usual terminology, notation, and conventions of the theory of random normed spaces, as in [4, 6–8].

Definition 4.1. A Menger probabilistic metric space (or random metric spaces) is a triple , where is a nonempty set, is a continuous -norm, and is a mapping from into such that, if denotes the value of at the pair , the following conditions hold:(PM1) for all if and only if ;(PM2) ;(PM3) for all and .

Definition 4.2 (see [1]). A random normed space (briefly, RN-space) is a triple , where is a vector space, is a continuous -norm, and is a mapping from into such that the following conditions hold:(RN1) for all if and only if ;(RN2) for all , ;(RN3) for all and .

Example 4.3. Let be a linear normed space. Define Then is a random normed space. (RN1) and (RN2) are obvious and we show (RN3). for all and . Also, is a random normed space.

Example 4.4. Let be a linear normed spaces. Define Then is a random normed space. (RN1) and (RN2) are obvious and we show (RN3). for all and . Also, is a random normed space.

Example 4.5 (see [36]). Let be a linear normed space. Define Then is an RN-space (this was essentially proved by Mušthari in [49], see also [50]). Indeed, for all for all and obviously and . Next, for every and we have

Let be a function defined on the real field into itself with the following properties:(a) for every ;(b);(c) is strictly increasing and continuous on , and .

Examples of such functions are ; , ; , .

Definition 4.6 (see [51]). A random -normed space is a triple , where is a real vector space, is a continuous –norm, and is a mapping from into such that the following conditions hold: (-RN1) for all if and only if ;(-RN2) for all in , and ;(-RN3) for all and .

Example 4.7 (see [37]). An important example is the space , where is a p-normed space and (-RN1) and (-RN2) are obvious and we show (-RN3). Let , then for each we have Now, if , then we have . Now, since we have then it follows that which implies that . Hence for all and .

By a non-Archimedean field we mean a field equipped with a function (valuation) from into such that if and only if , , and for all . Clearly and for all . By the trivial valuation we mean the mapping taking everything but 0 into 1 and . Let be a vector space over a field with a non-Archimedean non-trivial valuation , that is, that there is an such that is not in .

The most important examples of non-Archimedean spaces are -adic numbers. In 1897, Hensel [52] discovered the -adic numbers as a number theoretical analogue of power series in complex analysis. Fix a prime number . For any nonzero rational number , there exists a unique integer such that , where and are integers not divisible by . Then defines a non-Archimedean norm on . The completion of with respect to the metric is denoted by , which is called the -adic number field.

A function is called a non-Archimedean norm if it satisfies the following conditions:(i) if and only if ;(ii)for any , ;(iii)the strong triangle inequality (ultrametric); namely,

Then is called a non-Archimedean normed space. Due to the fact that a sequence is a Cauchy sequence if and only if converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent.

Definition 4.8. A non-Archimedean random normed space (briefly, non-Archimedean RN-space) is a triple , where is a linear space over a non-Archimedean field , is a continuous -norm, and is a mapping from into such that the following conditions hold:(NA-RN1) for all if and only if ;(NA-RN2) for all , ;(NA-RN3) for all and .It is easy to see that, if (NA-RN3) holds then so will (RN3) .

Example 4.9. As a classical example, if is a non-Archimedean normed linear space, then the triple , where is a non-Archimedean RN-space.

Example 4.10. Let be a non-Archimedean normed linear space. Define Then is a non-Archimedean RN-space.

Definition 4.11. Let be a non-Archimedean RN-space. Let be a sequence in . Then is said to be convergent if there exists such that for all . In that case, is called the limit of the sequence .A sequence in is called a Cauchy sequence if for each and there exists such that for all and we have .If each Cauchy sequence is convergent, then the random normed space is said to be complete and the non-Archimedean RN-space is called a non-Archimedean random Banach space.

Remark 4.12 (see [53]). Let be a non-Archimedean RN-space, then So, the sequence is a Cauchy sequence if for each and there exists such that for all we have

5. Topological Structure of Random Normed Spaces

Definition 5.1. Let be an RN-space. We define the open ball and the closed ball with center and radius , as follows:

Theorem 5.2. Let be an RN-space. Every open ball is an open set.

Proof. Let be an open ball with center and radius with respect . Let . Then . Since , there exists such that . Put . Since , there exists such that . Now for given and such that , there exists such that . Consider the open ball . We claim . Now, let . Then . Therefore, we have Thus and hence .

Different kinds of topologies can be introduced in a random normed space [4]. The -topology is introduced by a family of neighborhoods In fact, every random norm on generates a topology (-topology) on which has as a base the family of open sets of the form

Remark 5.3. Since is a local base at , the -topology is first countable.

Theorem 5.4. Every RN-space is a Hausdorff space.

Proof. Let be an RN-space. Let and be two distinct points in and . Then . Put . For each , there exists such that . Consider the open balls and . Then clearly . For, if there exists then we have which is a contradiction. Hence is a Hausdorff space.

Definition 5.5. Let be an RN-space. A subset of is said to be R-bounded if there exists and such that for all .

Theorem 5.6. Every compact subset of an RN-space is R-bounded.

Proof. Let be a compact subset of an RN-space . Fix and . Consider an open cover . Since is compact, there exist such that Let . Then and for some . Thus we have and . Now, let Then . Now, we have Taking , we have for all . Hence is R-bounded.

Remark 5.7. In an RN-space every compact set is closed and R-bounded.

Theorem 5.8 (see [4]). If is an RN-space and is a sequence such that , then almost everywhere.

Theorem 5.9. Let be an RN-space such that every Cauchy sequence in has a convergent subsequence. Then is complete.

Proof. Let be a Cauchy sequence and let be a subsequence of that converges to . We prove that . Let and such that Since is a Cauchy sequence, there is such that for all . Since , there is positive integer such that and Then, if , then we have Therefore, and hence is complete.

Lemma 5.10. Let be an RN-space. If one defines then is a random (probabilistic) metric on , which is called the random (probabilistic) metric induced by the random norm .

Lemma 5.11. A random (probabilistic) metric which is induced by a random norm on a RN-space has the following properties for all and every scalar : (i), (ii).

Proof. We have Also, we have

Lemma 5.12. If is an RN-space, then (i) the function is continuous, (ii) the function is continuous.

Proof. If and , then as This proves (i).Now, if , and , then as and this proves (ii).

Definition 5.13. The RN-space is said to be a random Banach space whenever is complete with respect to the random metric induced by random norm.

Lemma 5.14. Let be an RN-space and define by for each and . Then we have (i) for every and ; (ii) if satisfies (2.15), then for any , there is such that for any ; (iii) the sequence is convergent with respect to a random norm if and only if . Also the sequence is a Cauchy sequence with respect to a random norm if and only if it is a Cauchy sequence with .

Proof. For (i), we find For (ii), by (2.15), for every we can find such that Thus we have for every , which implies that Since is arbitrary, we have For (iii), note that since is continuous, is not an element of the set as soon as . Hence we have for every . This completes the proof.

Definition 5.15. A function from an RN-space to an RN-space is said to be uniformly continuous if for given and , there exist and such that implies

Theorem 5.16 (uniform continuity theorem). If is continuous function from a compact RN-space to an RN-space , then is uniformly continuous.

Proof. Let and be given. Then we can find such that Since is continuous, for each , we can find and such that implies But and then we can find such that Since is compact and is an open covering of , there exist in such that Put and , . For any , if then Since , there exists such that Hence we have Now, since we have it follows that Now, we have Hence is uniformly continuous.

Remark 5.17. Let be a uniformly continuous function from RN-space to RN-space . If is a Cauchy sequence in , then is also a Cauchy sequence in .

Theorem 5.18. Every compact RN-space is separable.

Proof. Let be the given compact RN-space. Let and . Since is compact, there exist in such that In particular, for each , we can choose a finite subset such that in which . Let Then is countable. We claim that . Let . Then for each , there exists such that . Thus converges to . But since for all , . Hence is dense in and thus is separable.

Definition 5.19. Let be any nonempty set and be an RN-space. Then a sequence of functions from to is said to be uniformly convergent to a function from to if for given and , there exists such that for all and .

Definition 5.20. A family of functions from an RN-space to a complete RN-space is said to be equicontinuous if for any and , there exist and such that for all .

Lemma 5.21. Let be an equicontinuous sequence of functions from an RN-space to a complete RN-space . If converges for each point of a dense subset of , then converges for each point of and the limit function is continuous.

Proof. Let and be given. Then we can find such that Since is equicontinuous family, for given and , there exist and such that, for each , for all . Since is dense in , there exists and converges for that . Since is a Cauchy sequence, for any and , there exists such that for all . Now, for any , we have Hence is a Cauchy sequence in . Since is complete, converges. Let . We claim that is continuous. Let and be given. Then we can find such that Since is equicontinuous, for given and , there exist and such that for all . Since converges to , for given and , there exists such that Also since converges to , for given and , there exists such that for all . Now, for all , we have Hence is continuous.

Theorem 5.22 (Ascoli-Arzela Theorem). Let be a compact RN-space and be a complete RN-space. Let be an equicontinuous family of functions from to . If is a sequence in such that is a compact subset of for each , then there exists a continuous function from to and a subsequence of such that converges uniformly to on .

Proof. Since be a compact RN-space, by Theorem 5.18, is separable. Let be a countable dense subset of . By hypothesis, for each , is compact subset of . Since every RN-space is first countable space, every compact subset of is sequentially compact. Thus by standard argument, we have a subsequence of such that converges for each . By Lemma 5.21, there exists a continuous function from to such that converges to for all .Now we claim that converges uniformly to on . Let and be given. Then we can find such that Since is equicontinuous, there exist and such that for all . Since is compact, by Theorem 5.16, is uniformly continuous. Hence for given and , there exist and such that for all . Let and . Since is compact and is dense in , for some . Thus for each , there exists , , such that But since and , we have, by the equicontinuity of , and we also have, by the uniform continuity of , Since converges to , for any and , there exists such that for all . Now, for each , we have Hence converges uniformly to on .

Lemma 5.23. A subset of is R-bounded in if and only if it is bounded in .

Proof. Let be a subset in which is R-bounded in , then there exist and such that for each and therefore we have Now, . If we put , then we have for each , that is, is bounded in . The converse is easy to see.

Lemma 5.24. A sequence is convergent in the RN-space if and only if it is convergent in .

Proof. Let in , by Lemma 5.14 (i), we have Then, by Lemma 5.14 (iii), .Conversely, let , then, by Lemma 5.14, we have Now, and so in .

Corollary 5.25. If the real sequence is R-bounded, then it has at least one limit point.

Definition 5.26. The 3-tuple is called a random Euclidean normed space if is a continuous -norm and is a random Euclidean norm defined by where , , , , and is a random norm.For example, let , , and . Then we have or equivalently .

Lemma 5.27. Suppose that the hypotheses of Definition 5.26 are satisfied. Then is an RN-space.

Proof . The properties of (RN1) and (RN2) follow immediately from the definition. For the triangle inequality (RN3), suppose that and . Then

Lemma 5.28. Suppose that is a random Euclidean normed space and is an infinite and R-bounded subset of . Then has at least one limit point.

Proof. Let be an infinite sequence. Since is R-bounded and so is . Therefore there exist and such that for each , which implies that . However, we have for each . Therefore in which , that is, the real sequences for all are bounded. Hence there exists a subsequence which converges to in w.r.t. the random norm . The corresponding sequence is bounded and so there exists a subsequence of which converges to with respect to the random norm . Continuing like this, we find a subsequence converging to .

Lemma 5.29. Let be a random Euclidean normed space. Let be a countable collection of nonempty subsets in such that , each is closed and is R-bounded. Then is nonempty and closed.

Proof. Using the above lemma, the proof proceeds as in the classical case [see Theorem 3.25 in [54]].

We call an -dimensional ball a rational ball if , , and .

Theorem 5.30. Let be a random Euclidean normed space in which satisfies (2.15). Let be a countable collection of -dimensional rational open balls. If and is an open subset of containing , then there exists a such that , for some .

Proof. Since and is an open, there exist and such that . By (2.15), we can find such that . Let be a finite sequence such that and and then we can find such that . Therefore, we have and so . Now we prove that . Let . Then and hence On the other hand, there exists such that and . Now and the proof is complete.

Corollary 5.31. In a random Euclidean normed space in which satisfies (2.15), every closed and R-bounded set is compact.

Proof. The proof is similar to the proof of Theorem 3.29 in [54].

Corollary 5.32. Let be a random Euclidean normed space in which satisfies (2.15) and . Then is compact set if and only if it is R-bounded and closed.

Corollary 5.33. The random Euclidean normed space is complete.

Proof. Let be a Cauchy sequence in the random Euclidean normed space . Since the sequence in which is a Cauchy sequence in and convergent to then, and by Lemma 5.14, the sequence is convergent in RN-space . We prove that is convergent to and

6. Random Functional Analysis

Theorem 6.1. Let be a linearly independent set of vectors in vector space and be an RN-space. Then there is and an RN-space such that for every choice of the real scalars we have

Proof. Put . If , all s must be zero and so (6.1) holds for any . Let . Then (6.1) is equivalent to the inequality which we obtain from (6.1) by dividing by and putting , that is, where and . Hence it suffices to prove the existence of a and random norm such that (6.2) holds. Suppose that this is not true. Then there exists a sequence of vectors, where , such that as for every . Since , we have and then, by Lemma 5.23, the sequence of is R-bounded. According the Corollary 5.25, has a convergent subsequence. Let denote the limit of that subsequence, and let denote the corresponding subsequence of . By the same argument, has a subsequence for which the corresponding of real scalars convergence; let denote the limit. Continuing this process, after steps, we obtain a subsequence of such that where , and as . By the Lemma 5.14 (ii) for any there exists such that as . By the Lemma 5.14 (iii), we conclude where , so that not all can be zero. Put . Since is a linearly independent set, we thus have . Since by assumption, we have Hence it follows that and so , which is a contradiction.

Definition 6.2. Let and be two RN-spaces. Then two random norms and are said to be equivalent whenever in if and only if in .

Theorem 6.3. On a finite dimensional vector space , every two random norms and are equivalent.

Proof. Let and be a basis for . Then every has a unique representation . Let in , but, for each , has a unique representation, that is, By Theorem 6.1, there is and a random norm such that (6.1) holds. So Now, if , then for every and hence in . On the other hand, by Lemma 5.14 (ii), for any , there exists such that Since , then we have in . With the same argument in imply in .

Definition 6.4. A linear operator is said to be random bounded if there exists a constant such that for every and

Note that, by Lemma 5.14 and last definition, we have

Theorem 6.5. Every linear operator is random bounded if and only if it is continuous.

Proof. By (6.15) every random bounded linear operator is continuous. Now, we prove that the converse it. Let the linear operator be continuous but not random bounded. Then, for each there is in such that . If we let then it is easy to see but do not tend to 0.

Definition 6.6. A linear operator is called a random topological isomorphism if is one-to-one and onto and both and are continuous. RN-spaces and for which such a exists are said to be random topologically isomorphic.

Lemma 6.7. A linear operator is random topological isomorphism if is onto and there exists constants such that .

Proof. By hypothesis is random bounded and, by last theorem, is continuous and, since , we have and consequently . So is one-to-one. Thus exists and, since is equivalent to or where , we see that is random bounded and, by last theorem, is continuous. Hence is an random topological isomorphism.

Corollary 6.8. Random topologically isomorphism preserves completeness.

Theorem 6.9. Every linear operator where but other, not necessarily finite dimensional, is continuous.

Proof. If we define where , then is an RN-space because (RN1) and (RN2) are immediate from definition, for the triangle inequality (RN3), Now, let , then, by Theorem 6.3, but since, by (6.15), then . Hence is continuous.

Corollary 6.10. Every linear isomorphism between finite dimensional RN-spaces is topological isomorphism.

Corollary 6.11. Every finite dimensional RN-space is complete.

Proof. By Corollary 6.10, and are random topologically isomorphic. Since is complete and random topological isomorphism preserves completeness, is complete.

Definition 6.12. Let be an RN-space, be a linear manifold in and be the natural mapping, . For any , we define

Theorem 6.13. Let be a closed subspace of an RN-space . If and , then there is in such that , .

Proof. By the properties of , there always exists such that . Now, it is enough to put .

Theorem 6.14. Let be a closed subspace of RN-space and be given in the above definition. Then (1) is an RN-space on , (2), (3) if is a random Banach space, then so is .

Proof. It is clear that . Let . By definition, there is a sequence in such that . Thus or, equivalently, and since is closed, and , the zero element of . Now, we have for , , and . Now, if we take the , then we have Therefore, is random norm on . (2) By Definition 6.12, we have Note that, by Lemma 5.14, (3) Let be a Cauchy sequence in . Then there exists such that, for every , . Let . Choose such that However, and so .Now, suppose that has been chosen and so choose such that Hence we have However, for every positive integer and , by Lemma 5.14, there exists such that By Lemma 5.14, is a Cauchy sequence in . Since is complete, there is in such that in . On the other hand, Therefore, every Cauchy sequence is convergent in and so is complete. Thus is a random Banach space.

Theorem 6.15. Let be a closed subspace of an RN-space . If two of the spaces , , and are complete, then so is the third one.

Proof. If is a random Banach space, then so are and . Hence all that needs to be checked is that is complete whenever both and are complete. Suppose that and are random Banach spaces and let be a Cauchy sequence in . Since for each , the sequence is a Cauchy sequence in and so it converges to for some . Thus there is such that, for every , Now, by the last theorem, there exist a sequence in such that and Thus we have and, by Lemma 5.14, for every , that is, . Therefore, is a Cauchy sequence in and thus is convergent to a point . This implies that converges to and hence is complete.

Theorem 6.16 (open mapping theorem). If is a random bounded linear operator from an RN-space onto an RN-space , then is an open mapping.

Proof. The theorem will be proved in several steps.Step 1. Let be a neighborhood of the 0 in . We show . Let be a balanced neighborhood of 0 such that . Since and is absorbing, it follows that and so there exists an such that has nonempty interior. Therefore, . On the other hand, So, the set includes the neighborhood of 0.Step 2. We show . Since and is an open set, there exist and such that . However, and so a sequence can be found such that in which . On the other hand, , where and so, by Step 1, there exist and such that Since the set is a countable local base at zero and as , so and can be chosen such that and as .Now, we show that Suppose . Then and so for and the ball intersects . Therefore, there exists such that , that is, or equivalently By the similar argument, there exist in such that If this process is continued, then it leads to a sequence such that and Now, if and , then we have where . Put . Since , there exists such that for . Therefore, for we have Hence it follows that That is, for all . Thus the sequence is a Cauchy sequence and consequently the series converges to some point because is a complete space.By fixing , there exists such that for because . Thus and thus Therefore, we have But it follows that Hence .Step 3. Let be an open subset of and . Then we have Hence is open, because it includes a neighborhood of each of its point.

Corollary 6.17. Every one-to-one random bounded linear operator from a random Banach space onto random Banach space has a random bounded inverse.

Theorem 6.18 (closed graph theorem). Let be a linear operator from the random Banach space into the random Banach space . Suppose that, for every sequence in such that and for some elements and , it follows that . Then is random bounded.

Proof. For any , and , define where .First we show that is a complete RN-space. The properties of (RN1) and (RN2) are immediate from the definition. For the triangle inequality (RN3), suppose that , , and , then Now, if is a Cauchy sequence in , then for every and there exists such that for . Thus for , Therefore, and are Cauchy sequences in and , respectively, and there exist and such that and and consequently . Hence is a complete RN-space. The remainder of the proof is the same as the classical case.

7. Fuzzy Normed Spaces

Now, we define the fuzzy normed spaces and give an example of these spaces. Here the -norms notation is showed by .

Definition 7.1. The triple is said to be a fuzzy metric space if X is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions: for all and ,(FM1) ;(FM2) for all if and only if ;(FM3) ;(FM4) for all ;(FM5) is continuous.

Definition 7.2. The triple is said to be a fuzzy normed space if is a vector space, is a continuous -norm, and is a fuzzy set on satisfying the following conditions: for every and ;(FN1) ;(FN2) if and only if ;(FN3) , for all ;(FN4) ;(FN5) is continuous;(FN6) .

Lemma 7.3. Let be a fuzzy norm, then we have (i) is nondecreasing with respect to for each , (ii).

Proof. Let then and we have and this proves (i). To prove (ii) we have

Example 7.4. Let be a normed space, define or and for all . Then is a fuzzy normed space. In particular, if , then we have which is called the standard fuzzy norm induced by norm .

Lemma 7.5. Let be a fuzzy normed space. If one defines then is a fuzzy metric on , which is called the fuzzy metric induced by the fuzzy norm .

We can see both definition and properties fuzzy normed spaces are very similar to random normed spaces. Then equipped with and can be regarded as an RN-space. Now, we extend the definition of fuzzy metric space. In fact we extend the range of fuzzy sets to arbitrary lattice.

Definition 7.6. The triple is said to be an -fuzzy normed space (briefly, -normed space) if is a vector space, is a continuous -norm on , and is an -fuzzy set on satisfying the following conditions: for every in and in ;(a) ;(b) if and only if ;(c) for each ;(d) ;(e) is continuous;(f) .

In this case, is called an -fuzzy norm (briefly, -norm). If is an intuitionistic fuzzy set and the -norm is t-representable then the triple is said to be an intuitionistic fuzzy normed space (briefly, -normed space).

Example 7.7. Let be a normed space. Denote for all and let and be fuzzy sets on defined as follows: for all . Then is an -normed space.

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