1. Introduction
Fuzzy normed linear spaces, briefly FNL spaces, were first introduced by Katsaras, who introduced some general types of fuzzy topological linear spaces [
1,
2]. In fact, a fuzzy norm of Katsaras’s type is associated to each absolutely convex and absorbing fuzzy set. In 1992, Felbin [
3] introduced another concept of fuzzy norm defined on a vector space by putting in correspondence to each element of the linear space, a fuzzy real number. Inspired by Cheng and Mordeson [
4], in 2003, Bag and Samanta [
5] defined a more suitable notion of fuzzy norm, even if it could be more refined, made simpler or even made more general (see [
6,
7,
8,
9,
10]).
In this context, there are two concepts of boundedness, one of them introduced by Bag and Samanta [
5] and the other one introduced by Sadeqi and Kia [
11] in 2009. On the other hand, as any fuzzy norm induces naturally a fuzzy metric, for studying boundedness we can also use the notion of F-bounded introduced by George and Veeramani for fuzzy metric spaces (see [
12]). The notion of fuzzy totally bounded set was first dealt with by Sadeqi and Kia [
11]. Numerous applications have emerged from fuzzy sets theory. To name a few recent ones, we would refer to where the fuzzy set theory merged with chaos theory [
13]. This approach may potentially improve some recent results in chaos theory application, e.g., designing chaotic sensors, see [
14]. Also, applications of fuzzy set theory may be considered within the actual scope of neuroscience like in [
15].
In this paper we emphasize different properties of such bounded sets. Moreover, we will make a comparative study among these concepts of boundedness. We establish the implications between them and we illustrate by examples that these concepts are not similar. Our context is very general because we work with fuzzy normed linear space of type
, where ∗ is an arbitrary t-norm, as they were considered by Nădăban and Dzitac in paper [
9].
Structurally, the paper comprises the following: we begin with the preliminary section, then, in
Section 2, we study fuzzy bounded sets. This concept of boundedness corresponds to the classical boundedness, as it is shown in Theorem 4. In
Section 3, we present bounded sets. We prove that the union and the sum of two bounded sets are also bounded and so is the closure of a bounded set. We characterize the boundedness of a set of Carthesian product of FNL spaces. F-bounded sets are considered in
Section 4 and in the next section we present different properties of fuzzy totally bounded sets. We highlight that in Theorem 6 it is proved that any compact set is fuzzy totally bounded. The last section is very important. In Theorem 8 we obtain the implications between these types of boundedness. In Theorem 10 is presented an example of a F-bounded set which is not fuzzy bounded. Finally, in Proposition 12 is given an example of a fuzzy bounded set that is not fuzzy totally bounded.
2. Preliminaries
Definition 1 ([
16])
. A binary operationis called triangular norm (t-norm) if it satisfies the following condition:- 1.
;
- 2.
;
- 3.
;
- 4.
If and , with , then .
Remark 1. Three basic examples of continuous t-norms are , which are defined by (the minimum t-norm), (usual multiplication in ) and (the Lukasiewicz t-norm). Our basic reference for fuzzy metric spaces and related structures is [17], while for t-norms, is [18]. Definition 2 ([
18])
. A t-norm ∗ is strictly monotonic ifA t-norm is strict if it is continuous and strictly monotonic.
Remark 2. We note that the usual multiplication · is a strict t-norm but the minimum t-norm ∧ is continuous but not strictly monotonic. This remark leads us to the following more general definition.
Definition 3. A t-norm is called almost strictly monotonic if A t-norm is called almost strict if it is continuous and almost strictly monotonic.
Remark 3. The usual multiplication · and the minimum t-norm ∧ are almost strict.
Definition 4 ([
19])
. The triple 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 in satisfying the following conditions:- (M1)
;
- (M2)
if and only if for all ;
- (M3)
;
- (M4)
;
- (M5)
is left continuous and .
Definition 5 ([
12])
. Let be a fuzzy metric space. A subset A of X is said to be F-bounded if Definition 6 ([
9])
. Let X be a vector space over a field (where is or ) and ∗ be a continuous t-norm. A fuzzy set N in is called a fuzzy norm on X if it satisfies:- (N1)
;
- (N2)
if and only if ;
- (N3)
;
- (N4)
;
- (N5)
, is left continuous and .
The triple will be called fuzzy normed linear space (briefly FNL space).
Example 1 ([
20])
. Let be a normed linear space. Let defined byThen is a FNL space.
Theorem 1 ([
9])
. Let be a FNL space.- 1.
We define by . Then M is a fuzzy metric on X.
- 2.
For we define the open ball Thenis a topology on X and is a metrizable topological vector space.
Recall [
21] that considering
two FNL spaces, the application
is a fuzzy norm on the Carthesian product
, named the fuzzy product norm.
We denote by the projection function from onto , defined by for
The next result deals with the Carthesian product of fuzzy normed linear spaces.
Theorem 2. Let be FNL spaces with the topologies and , respectively. If N is the fuzzy product norm, then is the product topology on .
Proof. We first prove that
such that
. Consider
From Lemma 3.6 [
22],
such that
By taking
,
, we obtain
.
Conversely, we have to prove that such that . Consider . Then for and , , we get and . Indeed from it results (otherwise ) and analogously , hence and . □
Definition 7 ([
5])
. Let be a FNL space and be a sequence in X. The sequence is said to be convergent if such that In this case, x is called the limit of the sequence and we denote or . Definition 8 ([
5])
. Let be a FNL space. A subset B of X is called the closure of the subset A of X if for any , such that . We denote the set B by .A subset A of X is called closed if .
Remark 4. As any FNL space is a fuzzy metric space, the notion of F-bounded set can be used in the context of FNL spaces. More precisely, a subset A of a FNL space X will be called F-bounded if We will denote by the family of all F-bounded subset of X.
Definition 9 ([
5])
. A subset A of a FNL space X is said to be bounded ifWe will denote by the family of all bounded subset of X.
Definition 10 ([
11])
. A subset A of a FNL space X is called fuzzy bounded ifWe will denote by the family of all fuzzy bounded subset of X.
Definition 11 ([
11])
. A subset A of a FNL space X is called fuzzy totally bounded ifWe will denote by the family of all fuzzy totally bounded subsets of X.
3. Fuzzy Bounded Sets
One might think by looking at the concepts of boundedness presented above, that there is still one concept missing, namely the one in which the boundedness of a set
A is defined as follows:
In fact it is not missing because it coincides with one above, as the following theorem shows.
Theorem 3. Let be a FNL space. A subset A of X is fuzzy bounded if and only if Proof. Let
. Then there exists such that . Since A is fuzzy bounded, for there exists such that .
Let
and
. We have that
Let
. Using Lemma 3.6 [
22] we obtain that there exist
such that
.
Let
be fixed. As
, we have that there exits
such that
. From our hypothesis, for
there exists
such that
. Let
. Then, for all
, we have
□
Remark 5. One can observe that a subset A of a topological linear space X is called bounded if for each neighbourhood V of , there exists a positive number k such that .
Theorem 4. Let be a FNL space. A subset A of X is fuzzy bounded if and only if A is bounded in topology .
Proof. Let
V be a neighbourhood of
. Then there exist
,
such that
. Since
A is fuzzy bounded, for
,
such that
. Let
. We have that
. Thus
Let . Since is a neighbourhood of , there exists such that . Thus . Hence A is fuzzy bounded. □
Remark 6. Previous result was mentioned by Sadeqi and Kia (see [11]) in the context of FNL spaces of type . Corollary 1 ([
23])
. Let be a FNL space. Then:- 1.
If are fuzzy bounded, then and are fuzzy bounded;
- 2.
If A is fuzzy bounded, then is fuzzy bounded.
Corollary 2. Let , be two FNL spaces. Then if and only if .
Proposition 1. Let be a FNL space and be fuzzy bounded subsets of X. Then there exist such that is a fuzzy bounded subset of X.
Proof. Let
be the metric of
X. Let
and
. Since
is a neighbourhood of
, there exists
such that
As
are fuzzy bounded subsets of
X, there exists
such that
. Let
. Then
□
4. Bounded Sets
Proposition 2. Let be a FNL space and be two bounded subsets of X. Then is bounded.
Proof. Since are bounded subsets of X, there exist such that and . Let and . Let . If , then . Similarly, if , we obtain that . Thus . □
Proposition 3. Let be a FNL space, where ∗ is almost strict. If are two bounded subsets of X, then is a bounded subset of X.
Proof. Since
are bounded subsets of
X, there exist
such that
and
. Let
such that
and
. Let
. Then there exist
such that
. We have that
□
Proposition 4. Let be a FNL space and . Then .
Proof. As
A is bounded we have that there exist
,
such that
Let
such that
and
such that
. Let
and
. Thus
such that
. Hence
. Thus there exists
such that
. Therefore, for
, we have that
Hence is bounded. □
Proposition 5. Let , be two FNL spaces where is almost strict and let N be the fuzzy product norm. Then if and only if .
Proof. Let . Then there exist such that . Following the proof of Theorem 2, it results that and . Therefore .
Conversely, since , there exist such that . Since “∗” is almost strict there exist such that . Consider . Then , hence . It follows . □
6. Fuzzy Totally Bounded Sets
Theorem 5. Let be a FNL space. The following statements are equivalent:
- 1.
A is fuzzy totally bounded;
- 2.
;
- 3.
;
- 4.
.
Proof. . Let
. Then there exists
such that
Indeed, if we suppose that
by passing to the limit, for
, we obtain that
, which is a contradiction.
As
A is fuzzy totally bounded,
Let
. We show that
Let
. Then there exists
such that
, namely
. We have that
Thus .
. Let
and
. Let
. By our hypothesis,
. It is obviously.
. Let . For , by our hypothesis, such that . □
Proposition 8. Let be a FNL space. If are fuzzy totally bounded subsets of X, then and are fuzzy totally bounded.
Proof. Let
and
. Then there exist
such that
. Let
. Then
Indeed, if
and
, then
and
. Thus
Hence
. If
are fuzzy totally bounded, then there exist
and
such that
and
. Therefore
Hence is fuzzy totally bounded.
Let now
. As
are fuzzy totally bounded, there exist
such that
and
. Thus
Hence is fuzzy totally bounded. □
Lemma 1. Let such that . Then .
Proof. If , then there exists such that , namely and .
Let such that . The existence of results by the continuity of the mapping . Indeed, for , as and , there exists such that , namely .
Finally, for
such that
, as
there exists
such that
. Thus
Hence . □
Proposition 9. Let be a FNL space. If A is fuzzy totally bounded, then is fuzzy totally bounded.
Proof. Let
. Let
. As
A is fuzzy totally bounded,
such that
. Thus, by Lemma 1, it follows
Hence is fuzzy totally bounded. □
Proposition 10. Let be a normed linear space and let be the FNL space defined bywhere ∗ is an arbitrary t-norm. Then coincides with the norm topology on X. Moreover: - 1.
A set M is bounded in if and only if M is fuzzy bounded in ;
- 2.
A set M is totally bounded in if and only if M is fuzzy totally bounded in .
Proof. Let
and
. We show that there exist
and
such that
. Let
and
. Let
. Then
Thus .
Conversely, let
. We show that there exists
such that
. Let
and
. Thus
. Hence
Therefore . □
Theorem 6. Let be a FNL space and be a compact set in . Then K is fuzzy totally bounded.
Proof. Let and . As and K is compact, such that . By Theorem 5 we obtain that K is fuzzy totally bounded. □
Theorem 7. Let and let N be the fuzzy product norm. Then if and only if
Proof. Let . Since is continuous from onto , it results the inverse image of through , is a neighbourhood of in . Therefore, there exist and such that . By Theorem 5 it follows that there exist such that whence . Therefore . Conversely, suppose that . Let and . By Theorem 2, it results that there exist such that . Since is fuzzy totally bounded, it follows that there exist such that . Thus , hence A is fuzzy totally bounded. □
7. A Comparative Study among Different Types of Boundedness
Theorem 8. Let be a FNL space. We have that .
Proof. We prove first that
. Let
. Then there exist
such that
. As
A is fuzzy totally bounded,
. As
, we have that there exist
such that
. Let
. Then
. Let
. Let
. We show that
. Indeed, as
, there exists
such that
, i.e.,
. Thus
Now we prove that
. Let
A be a fuzzy bounded subset of
X. Let
such that
. As
A is fuzzy bounded, we have that there exist
such that
and
. Let
such that
and
. Then, for all
, we have
Finally, we prove that
. Let
A be a F-bounded subset of
X. Then there exist
and
such that
. Let
be fixed. Then there exists
such that
. Let
and
. Then, for all
, we have
□
Theorem 9. Let be a FNL space, where ∗ is almost strict. Then .
Proof. By previous theorem we have that . For inverse inclusion, let A be a bounded subset of X. Then there exist and such that . As ∗ is almost strict, such that . Let and . Then Thus . □
Corollary 3. Let be a FNL space, where ∗ is almost strict and let be F-bounded subsets of X. Then and are F-bounded subsets of X.
Corollary 4. Let , be two FNL spaces where is almost strict and let N be the fuzzy product norm. Then if and only if .
Theorem 10. The inclusion is strict.
Proof. Let
and
. Let
defined by
, where
. It is easy to show that
is a sufficient and ascending family of semi-norms on the linear space
X. Let
, defined by
Then, by Theorem 8 of [
9], we have that
is a FNL space.
It is obvious that is bounded in X and using previous theorem it is F-bounded. We will prove that M is not fuzzy bounded. Let . We show that such that .
Let and defined by
It is obvious that
is continuous. We show that
. But
Let
such that
. As
we have that
Finally, we prove that . Indeed, for , as , we have that . Hence . On the other hand, for , as we have that . Hence . Thus . □
Proposition 11. Let be a normed linear space and defined by A subset A of the FNL space is bounded if and only if A is fuzzy bounded.
Proof. We have that
. It remains to prove that
. Let
A be a bounded set. Then
Thus . Therefore . Hence A is bounded in . This means that A is fuzzy bounded. □
Proposition 12. The inclusion is strict.
Proof. Let
.
with the norm
became a normed linear space. If ∗ is an arbitrary t-norm and
then
is a FNL space.
Let . As M is bounded in , by Proposition 10 we obtain that M is fuzzy bounded in . If we suppose that M is fuzzy totally bounded in , then M is totally bounded in . Thus in we have a totally bounded set which is a neighborhood of the origin. Hence is finitely dimensional, which is absurd. Hence M is not fuzzy totally bounded. □