The Hahn-Banach Extension Theorem for Fuzzy Normed Spaces Revisited

Abstract

This paper deals with fuzzy normed spaces in the sense of Cheng and Mordeson. We characterize fuzzy norms in terms of ascending and separating families of seminorms and prove an extension theorem for continuous linear functionals on a fuzzy normed space. Our result generalizes the classical Hahn-Banach extension theorem for normed spaces.

1. Introduction

The Hahn-Banach extension theorem is without doubt one of the most important theorems in the whole theory of normed spaces. A classical formulation of such theorem is as follows.

Theorem 1. Let be a normed space and let be a continuous linear functional on a subspace of . There exists a continuous linear functional on such that and .

The fact that the class of normed spaces is strictly included in the class of fuzzy normed spaces motivates the following natural question: is it possible to give a theorem of Hahn-Banach type in the frame of fuzzy normed spaces which generalizes the classical one for normed spaces? In this paper, we will give an affirmative answer to this question by proving the following (the notation and terminology can be found along the paper).

Theorem 2. Let be a fuzzy normed space such that the -seminorms corresponding to the fuzzy norm are norms, and let be a continuous linear functional on a subspace of . Then, there exists for which the following two conditions are satisfied: (1)for all , there is a continuous linear functional on such that and ;(2) for all .

The antecedents of our study are in the paper by Bag and Samanta [1], where the authors obtain a theorem of Hahn-Banach type for a special class of fuzzy normed spaces, using in its proof the classical Hahn-Banach theorem for normed spaces. The scope of our result is a more general class of fuzzy normed spaces which allows us to deduce the classical theorem for normed spaces as a consequence.

It is well known that the fuzzy normed spaces are topological vector spaces, and hence the existence of continuous linear extension for each continuous linear functional defined on a linear subspace is guaranteed. Nevertheless, we do not use this fact in the proof of our result, but we give an explicit proof of the existence of a continuous linear extension because in this way we can also establish how to compute the fuzzy norm of the extension as stated in part (2) of Theorem 2 above. Some illustrative examples are also presented.

The first definition of fuzzy norm on a linear space was given by Katsaras [2] in 1984. Following this work, Felbin [3] offered in 1992 an alternative definition of a fuzzy norm on a linear space with an associated metric of Kaleva and Seikkala's type [4]. In 1994, Cheng and Mordeson [5] gave another definition of fuzzy norm that corresponds with the notion of a fuzzy metric as defined by Kramosil and Michálek in [6]. Bag and Samanta considered in [7] a fuzzy norm slightly different from this one and they proved a series of results that have been used in many subsequent works in this context. Although this definition is less restrictive than the one given by Cheng and Mordeson, the more interesting results given in the mentioned paper require the use of two very restrictive additional conditions which leave out of the scope of these results important examples of fuzzy normed spaces.

In this paper, we consider fuzzy normed spaces in the sense of Cheng and Mordeson. The organization of the paper is as follows. Section 2 comprises the basic notions on fuzzy normed spaces and some preliminary results. In Section 3, we study the relation between seminorms and fuzzy normed spaces. Theorems 8 and 12 of this section generalize the corresponding results given by Bag and Samanta in [7]. In this section, we also deduce from our approach the well-known relationship between fuzzy normed spaces and locally convex spaces (see Theorem 14). Finally, in Section 4, we prove Theorem 2 above and deduce Theorem 1 from it.

2. Terminology and Basic Notions

According to [8], a binary operation is a continuous -norm if satisfies the following conditions: (i) is associative and commutative; (ii) is continuous; (iii) for every ; (iv) whenever and , with .

Three paradigmatic examples of continuous -norms are , , and (the Lukasiewicz -norm), which are defined by ,  , and , respectively. Recall that . In fact, for every continuous -norm .

Definition 3. If is a real vector space, a fuzzy norm on is a pair such that is a continuous -norm and is a fuzzy set in satisfying the following conditions for every , and :(FN1);(FN2) for all ;(FN3) for every ;(FN4);(FN5);(FN6) is left continuous.

The triple is called a fuzzy normed space.

If condition (FN5) is omitted we say that is a weak fuzzy norm on , and the triple will be called a weak fuzzy normed space. These spaces will play a crucial role in the paper.

Recall that if in Definition 3 we put , then one has the notion of a fuzzy norm as given by Cheng and Morderson [5].

The class of fuzzy normed spaces is equivalent to a suitable subclass of Serstnev spaces in the case of continuous -norms (see Remark 1 of [9]).

The following well-known example shows that every normed space can be considered as a fuzzy normed space.

Example 4. Let be a normed space. Then,(a)let given by for all and let for all and . Then, is a fuzzy norm on , where is any continuous -norm. This fuzzy norm is called the standard fuzzy norm induced by .(b)Let given by if and if . Then, is a fuzzy norm on , where is any continuous -norm. This fuzzy norm will be called the 01-fuzzy norm induced by .

If is a weak fuzzy normed space, the open ball with center , radius , , and is defined as follows:

We note that , for all and . The closed ball with center , radius , , and is defined as follows:

It is clear that if is a weak fuzzy normed space, the fuzzy set in given by is a fuzzy metric on in the sense of Kramosil and Michálek [6]. This fuzzy metric induces a topology on , which has as a base the collection . Moreover, is metrizable and the countable collection of balls forms a fundamental system of neighborhoods of , for all .

It is well known, and easy to see, that if is a normed space, then the topology agrees with the topology induced by the norm , when is the standard fuzzy norm or the 01-fuzzy norm on .

It is interesting to note that if is a fuzzy normed space, the open (closed) balls are absorbent, balanced, and convex sets (Propositions 1 and 3 of [9]). However, if is a weak fuzzy norm, it can happen that the open (closed) balls are not absorbent sets.

By omitting the condition of left continuity of the real function in the definition of fuzzy norm given by Cheng and Morderson [5], Bag and Samanta gave in [7] the following notion.

Definition 5 (see [7]). A BS-fuzzy norm on a real vector space is the pair such that is a fuzzy set in satisfying the following conditions for every and (N1), for all ;(N2) for all ;(N3) for every ;(N4);(N5) is nondecreasing and .

We point out that since (N2) and (N4) imply that is nondecreasing, this condition may be dropped from (N5).

A BS-fuzzy normed space satisfies the condition (N6) (page 691 of [7]) if , for all , implies .

A BS-fuzzy normed space satisfies the condition (N7) (page 693 of [7]) if, for ,   is a continuous function of which is strictly increasing on the subset .

The standard fuzzy norm and the 01-fuzzy norm induced by a norm are BS-fuzzy norms. The standard fuzzy norm satisfies (N7) but not (N6) while the 01-fuzzy norm satisfies (N6) but not (N7).

3. Seminorms and Fuzzy Normed Spaces

Bag and Samanta proved in Theorem 2.1 of [7] that if is a BS-fuzzy normed space which satisfies condition (N6), then is an ascending family of norms on , where From the proof of this result, it follows that if the condition (N6) is not required, then each is a seminorm on .

In the next proposition, we give a shorter alternative proof of this result.

Proposition 6. Let be a fuzzy normed space and let . Then, the following hold. (a)The function given by is a seminorm on . In fact, it is the Minkowski functional of the ball . (b)The family is separating.

Proof. (a) Let be the Minkowski functional of the closed ball . Clearly, is a seminorm because is an absolutely convex absorbent set. Then,
(b) If for every , then , for every and . Therefore, for all and so .

Example 7. (a) Let be a normed space and let be the standard fuzzy norm induced by . Then, . Note that does not satisfy condition (N6) but is a norm for all .
(b) Let be a normed space and let be the 01-fuzzy norm induced by . Then, , for all .

If is a fuzzy normed space, the family will be called the -seminorms corresponding to the fuzzy norm .

Bag and Samanta stated in Theorem 2.2 of [7] that given an ascending family of norms on a real linear space , then the pair is a BS-fuzzy norm on , where is defined by , and , otherwise.

Next, we generalize this result for the case that is a family of extended separating seminorms on . In fact, our extension requires an explicit proof because Bag-Samanta's construction of presents some slight disarrangement. Indeed, if is a nontrivial normed space and put for all , then for each it follows that is not defined whenever .

Theorem 8. Let be an ascending family of separating seminorms on a real linear space , and let be given by , for all . Then, the pair is a fuzzy norm on , where is given by , for all , and for all and .   will be called the fuzzy norm induced by the seminorms .

Proof. (FN1) Consider that , by definition of .
(FN2) If for all , then for all . Therefore, for all . Consequently, , since the family of seminorms is separating.
(FN3) Let ,   , and . Then,
(FN4) Let and let . Suppose that . Then, whenever . Since , it follows that whenever for all . Consequently, .
(FN5) Let and let . There exits such that . Let be such that . Then, . Hence, .
(FN6) Let and let .
If , then , for all .
Suppose . Then, given , there is such that and . Let be such that . Then, and so
Therefore, is left continuous.

Remark 9. If in the above theorem we define instead of , the left continuity of is not guaranteed. Indeed, if is a nontrivial normed space and put for all , then if and if , and this function is not left continuous at for every

Let be a linear space and let . If satisfies the conditions of a seminorm, we say that is an extended seminorm.

If is a family of extended seminorms on , we say that the family is separating if for all ,  , there are such that and .

If we consider an ascending family of separating extended seminorms on , mimicking the proof of Theorem 8, we obtain the following result.

Theorem 10. Let be an ascending family of separating extended seminorms on a real linear space , and let be given by , for all . Then, the pair is a weak fuzzy norm on , where is given by , for all , and for all and .   will be called the weak fuzzy norm induced by the seminorms .

Bag and Samanta stated in Theorem 2.3 of [7] that if is a BS-fuzzy normed space which satisfies (N6) and (N7), then coincides with the BS-fuzzy norm induced by the -norms corresponding to the fuzzy norm . Next, we generalize this theorem to the case that is a fuzzy normed space. To prove this result, first we prove the following lemma.

Lemma 11. Let be a fuzzy normed space. Let be the -seminorms corresponding to the fuzzy norm .(a)If , then .(b)If , then .

Proof. (a) Suppose that . Since is nondecreasing, it follows that for every such that . Therefore, which provides a contradiction because .
(b) If , then , by definition of . Suppose that . Since is left continuous and , there exists such that . Now, if , then which provides a contradiction.

Theorem 12. Let be a fuzzy normed space. Let be the -seminorms corresponding to the fuzzy norm . If is the fuzzy norm induced by the seminorms then .

Proof. Let and let . Let such that . Then, by Lemma 11 (a), . Now, if for every such that , then Suppose that there exist and such that . Let such that . By Lemma 11 (b), , so , which provides a contradiction.
By using known properties of locally convex vector spaces, the family of -seminorms corresponding to the fuzzy norm induces a topology on such that is a Hausdorff locally convex space. Since the family of -seminorms is ascending, a base of neighborhoods of the origin in this topology consists of the sets of the form

Proposition 13. Let be a fuzzy normed space. Let be the topology induced on by the fuzzy norm and let be the topology induced on by the family of -seminorms corresponding to the fuzzy norm . Then, .

Proof. Take the open ball and let it be with and . Then, because if , then, by Lemma 11 (a), . Consequently, is coarser than .
Now, given with and then, by Lemma 11 (b), . Therefore, is coarser than .

As a consequence of the above results, it is possible to obtain the relationship between fuzzy normed spaces and locally convex spaces. This result was obtained by Radu [10] in the realm of random normed spaces of Serstnev [11]. A timely update of this fact was presented in [9].

Theorem 14 (see [9, 10]). (a) Let be a fuzzy normed space. Then, is a metrizable locally convex space.
(b) Let be a metrizable locally convex space. Then, there is a fuzzy norm on such that .

Proof. (a) It is immediate by Proposition 13.
(b) If is a metrizable locally convex space, is determined by a separating family of seminorms on . Let be the ascending family of seminorms such that , and then is also determined by this family, and the sets form a base of neighborhoods of the origin in .
Let be the ascending family of separating seminorms such that if , for all . By Theorem 8, this family induces a fuzzy norm on . Next, we show that . To this end, take the open ball with . We claim that . Indeed, if , then for all . Therefore, On the other hand, if we take ,  ,  , then . Indeed, if , then , and hence, .

Remark 15. If is a weak fuzzy normed space, then is a metrizable topological space, but it is not a topological vector space since the open (closed) balls are not absorbent sets, in general.

The following example shows that the family of -seminorms corresponding to a fuzzy norm compatible with a nonnormable metrizable locally convex space can be a family of norms.

Example 16. Let be a matrix of nonnegative real numbers such that for all . Let and let be the Köthe echelon space of order ,
It is well known (Lemma 27.1 of [12]) that is a locally convex space for the topology generated by the seminorms and that has a compatible complete metric. Since for all , then is a norm for all .

By Theorem 14 (b), there is a fuzzy norm on such that . Moreover, the proof of Theorem 14 (b) shows that is induced by the ascending family of norms given by , where ,  . Finally, if we consider the fuzzy normed space , the -seminorms corresponding to the fuzzy norm are the norms , by Theorem 12.

4. Continuous Linear Functionals and the Hahn-Banach Theorem

Let be a fuzzy normed space and let be the standard fuzzy norm on , that is; for and for all and (see Example 4 (a)).

Denote by the set of all continuous linear mappings from to . (Note that is the usual topology of

Proposition 17. Let be a fuzzy normed space and let be the -seminorms corresponding to the fuzzy norm . Then, if and only if there exist and such that for every .

Proof. Let . By Proposition 13, there are and such that , where . If and , then and so If , then for all . Therefore, for each and then . Consequently,
Conversely, suppose there exist and such that for every . Then, given , we have that . Since is a neighborhood of the origin in the topology , we have that is continuous at the origin and, by linearity, is continuous at each point of .
For each define and whenever .

Proposition 18. Let be a fuzzy normed space. Then, is an ascending family of extended norms on .

Proof. It is easy to show that is an extended seminorm on for each . Now, if for all , then for all , where . Since is absorbent, given , there exists such that . Then, and so .
If , then and then . On the other hand, by Proposition 17, for every , there is such that . Therefore, is an ascending family of extended norms on .

The following example shows that can be infinity.

Example 19. Let be the linear space of all sequences of real scalars. is a metrizable locally convex vector space for the topology generated by the seminorms ,  .

By Theorem 14 (b), there is a fuzzy norm on such that . Moreover, if , following the proof of this result, is induced by the ascending family of separating seminorms given by if , for all .

By Theorem 12, this family of seminorms is the -seminorms corresponding to the fuzzy norm .

Let be the linear function given by . Since , we have that is continuous and .

For each , let be the sequence given by if and . Since and , for all , we have for all . Therefore, and , where .

By Theorem 10 and Proposition 18, the pair is a weak fuzzy norm on , where for all , and for all .

Lemma 20. Let be a fuzzy normed space. If and , then where .

Proof. If , since , for all , we have If , then , for all . Hence,

In order to prove Theorem 2, we will also use the following terminology.

If is a linear subspace of a fuzzy normed space , we denote by the set of all continuous linear mappings from to . If and , we denote by the extended norm on given by

By , we will denote the weak fuzzy norm on given by and for all and .

Lemma 21 (Theorem 5, p. 132 of [13]). Let be a convex functional on a real linear space and let be a linear functional on a subspace of such that for all . Then, there is a linear functional on such that and for all .

Proof of Theorem 2. Since , by Proposition 17, there exists such that . Let . If , we define by for all . Clearly, is a convex functional on .
Let . Since , it follows that for all and . Therefore, by Lemma 21, for each , there is a linear functional on such that and for all . By Proposition 17, is continuous on for all . Moreover, since for every such that , we obtain that .
Next, we show that for all .
By definition of , it immediately follows that , for all . Hence, On the other hand, if and , we have that and then . Therefore,

As we indicated in Section 1, the classical Hahn-Banach theorem for normed spaces (Theorem 1) can be obtained from Theorem 2.

Indeed, let be a normed space and let be a continuous linear functional on a subspace of . Let be the 01-fuzzy norm on induced by . The -seminorms corresponding to are for every . Moreover, for every and for all . Therefore, by applying Theorem 2, we have that for each there is such that and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the reviewers for their valuable suggestions. They also acknowledge the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2012-37894-C02-01.