The Hahn-Banach Extension Theorem for Fuzzy Normed Spaces Revisited

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.


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.
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).
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 3 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 wellknown 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.
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.
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  × R satisfying the following conditions for every ,  ∈  and ,  ∈ R : We point out that since (N2) and (N4) imply that (, ⋅) is nondecreasing, this condition may be dropped from (N5).
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).

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 {‖ ⋅ ‖  :  ∈ (0, 1)} 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  ∈ (0, 1).Then, the following hold.
Let  be a linear space and let  :  → R + ∪ {∞}.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  ∈ ,  ̸ = 0, there are ,  ∈  such that   () ̸ = 0 and   () ̸ = ∞.If we consider an ascending family of separating extended seminorms on , mimicking the proof of Theorem 8, we obtain the following result.
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 {  :  ∈ N} on .Let {  :  ∈ N} be the ascending family of seminorms such that   = max{ 1 , . . .,   }, and then  is also determined by this family, and the sets form a base of neighborhoods of the origin in (, ).
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  = ( , ) ,∈N be a matrix of nonnegative real numbers such that 0 <  , ≤  ,+1 for all ,  ∈ N. Let  ∈ [1, ∞) 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 {  :  ∈ N} and that  has a compatible complete metric.Since  , > 0 for all ,  ∈ N, then   is a norm for all  ∈ N.
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   () = |  |,  ∈ N.