1. Introduction
Since Katsaras first introduced the notion of fuzzy norm on a vector space [
1], a study for fuzzy normed spaces has actively progressed. In 1992, Felbin introduced a new definition of a fuzzy norm (namely, Felbin-fuzzy norm, in this paper, fuzzy norm) related to a specific fuzzy metric [
2,
3]. Because of his pioneering researches, topological properties have been studied according to Felbin type’s fuzzy norms [
4,
5]. Especially, the authors recently introduced the approximation properties in fuzzy normed spaces [
6,
7].
A study of spaces of sequences in vector spaces is a very important concept to research functional analysis, because these spaces have been investigated for Shauder basis and operator theory [
8]. In 1989, Nanda introduced sequences of fuzzy numbers [
9]. In 2000, Savaş introduced summable sequences of fuzzy numbers [
10,
11,
12]. Felbin investigated sequences and their limits in the sense of fuzzy normed spaces [
2]. In Felbin’s sense, Sencimen and Pehlivan [
13] introduced the notions of a statistically convergent sequence and statistically Cauchy sequence in a fuzzy normed linear space. Hazarika [
14,
15] provided the concepts of I -convergence, I -convergence, and I -Cauchy sequence in a fuzzy normed linear space in terms of general framework. For more researches of sequences in fuzzy normed spaces, we refer to [
16,
17,
18].
In previous studies, many researchers concentrated on investigating the convergency of sequences in fuzzy normed spaces and their topological properties. However, until now, no research for spaces of sequences in fuzzy normed spaces itself has been done, because it is difficult to define fuzzy norm for spaces of sequences in fuzzy normed space. Because the space of sequences in fuzzy normed spaces itself is very important in fuzzy functional analysis, we need to investigate that space in terms of fuzzy norms.
In this paper, we study spaces of sequences in fuzzy normed spaces. We establish a well-defined fuzzy norm for spaces of sequences in fuzzy normed spaces and its completeness. This is an important contribution of our paper. Moreover, we investigate the fuzzy dual of spaces of sequences in fuzzy normed spaces. We characterize the approximation property for spaces of sequences in fuzzy normed spaces. The contribution of our paper is to make tools for fuzzy analysis, because we characterize the duality and approximation property in the sense of sequences in fuzzy normed spaces.
Our paper is organized, as follows.
Section 2 gives some preliminary results. In
Section 3, we define fuzzy norms for spaces of sequences in fuzzy normed spaces. Furthermore, we provide their completeness and several examples. In
Section 4, we give the representation of the dual space of the space of sequences in a fuzzy normed space. In
Section 5, we provide the results of the approximation property for spaces of sequences in fuzzy normed spaces (for reference, see [
19]).
2. Preliminaries
Definition 1 (See [
2]).
A mapping is called a fuzzy real number with α-level set , if it satisfies the following conditions:- (i)
there exists a such that
- (ii)
for each , there exist real numbers , such that the α-level set is equal to the closed interval
The set of all fuzzy real numbers is denoted by
. If
and
whenever
, then
is called a non-negative fuzzy real number and
denotes the set of all non-negative fuzzy real numbers. We note that the real number
for all
and all
. Because each
can be considered as the fuzzy real number
denoted by
hence, it follows that
can be embedded in
.
Lemma 1 (See [
20]).
Let and . Then for all , Definition 2 (See [
20]).
Let and for all . Define a partial ordering by in if and only if for all . The strict inequality in is defined by if and only if for all . Definition 3 (See [
21]).
Let X be a vector space over . Assume that the mappings are symmetric and non-decreasing in both arguments, and that and . Let . The quadruple is called a fuzzy normed space with the fuzzy norm , if the following conditions are satisfied:(F1) if , then
(F2) if and only if ,
(F3) for and ,
(F4) for all ,
(F4L) whenever and
(F4R) whenever and
Remark 1 (See [
2]).
If and for all , then the triangle inequality (F4) shown in Definition 4 is equivalent to In this paper, we fix and for all because of the triangle inequality, and we write .
Definition 4 (See [
2]).
Let be a fuzzy normed space. A sequence of X is said to converge to () if for all . A subset A of X is called compact in if each sequence of elements of A has a convergent subsequence in . Remark 2. Let be a fuzzy normed space. Subsequently, for all , and are normed spaces.
Definition 5 (See [
21]).
Let and be fuzzy normed spaces. The linear operator is said to be a strongly fuzzy bounded if there is a real number , such that for all . We will denote the set of all strongly fuzzy bounded operators from to by . Afterwards, is a vector space. For all , we denote by where M is a positive real number. is called a bounded in if for some . Moreover, we denote the set of all finite rank strongly fuzzy bounded operators from to by . Subsequenty, is a subspace of . Similarly, we define for some .
3. Spaces of Bounded Sequences and Null Sequences in Fuzzy Normed Spaces
In this section, we provide definitions of fuzzy norms for spaces of bounded sequences and null sequences in fuzzy normed spaces. Recall that a sequence
of a fuzzy normed space
X is said to be bounded if there exists a fuzzy real number
, such that
([
9]).
Definition 6. Let be a fuzzy normed space. We denote, by , the set of all bounded sequences in . A set is clearly vector space with respect to componentwise summation and componentwise multiplication by a scalar. We define by,We shall write and . Lemma 2 (See [
3])
Let be a given family of non-empty intervals. Assume- (a)
for all .
- (b)
whenever is an increasing sequence in converging to α.
Subsequently, the family represents the α-level sets of a fuzzy number. Conversely, if are the α-level sets of a fuzzy number the conditions (a) and (b) are satisfied.
Theorem 1. Let be a fuzzy normed space and a bounded sequence in . Afterwards, is a fuzzy real number.
Proof. Let
be given. There exists a fuzzy real number
such that
. Subsequently, we have
thus we obtain
Hence, is a nonempty interval for all . We show that the interval satisfies conditions (a) and (b) in Lemma 2.
- (a)
Let
. Afterwards,
Because
, we have
hence
.
- (b)
Let (
) be a increasing sequence in (0,1] converging to
. Subsequently,
and, thus,
Let
. Afterwards, there exists
such that
. By Lemma 2 (b), we have
. Subsequently, there exists
, such that
On the other hand, by the proof of [
5] (Theorem 5.4), we can show that
hence the interval
satisfies conditions (a) and (b) in Lemma 2.
□
Remark 3. Let be a fuzzy normed space and in . LetAfterwards, the fuzzy real number is the element of , such that for every , i.e., is the smallest fuzzy real number of . Indeed, take any and . Subsequently, we haveAdditionally, since , we havehence it follows that and . Theorem 2. The vector space that is equipped with the norm defined in Definition 9 is a fuzzy normed space.
Proof. We shall show that satisfies the conditions (F1)–(F4) of Definition 4.
- (F1)
Let
in
. Afterwards, there exists
such that
. We have for all
so we obtain
.
- (F2)
Let
. It is clear that if
, then
. Conversely, let
. Subsequently, we have for all
so we obtain
. Afterwards, it is clear that
for all
n.
- (F3)
Let
and
. For all
, we have
where
.
- (F4)
Let
be given. It is clear that
Now, let
be given. There are fuzzy real numbers
, such that
where
and
for all
n. Subsequently, we have
Recall that a sequence
is called Cauchy if, for a given
, there exists
, such that, for all
Additionally, a fuzzy normed space is said to be complete if every Cauchy sequence converges in [5].
Theorem 3. Let be a complete fuzzy normed space. Subsequently, is a complete fuzzy normed space.
Proof. Let
be a Cauchy sequence. First, we claim that, for each
j, there exists
, such that
as
in
. Indeed, take any
and
. Subsequently, there exists
, such that
i.e., we have
for all
. Fix
. Afterwards, there exists a fuzzy real number
such that
Thus, , we obtain , so it follows that is a Cauchy sequence in . Because is complete, there exists , such that as . Continuing this process, it proves our claim. Now, we put .
Second, we claim that
as
. Let
and
be given. By results of the first claim, if
then there exists a fuzzy real number
, such that we have
Now, we fix
. There exists
with
, such that
Therefore, if
we have
Hence, if
we obtain for all
j,
so if
Finally, we show that
. By the second claim, there exist a fuzzy real number
and
, such that
we obtain for all
j,
Now, we fix
. Since
, there exists a fuzzy real number
such that
Hence, for all
j, we obtain
□
Definition 7. Let be a fuzzy normed space. A bounded sequence in is null if We denote, by , the set of all null sequences in with the same vector space operations and fuzzy norm as , i.e., Corollary 1. Let be a complete fuzzy normed space. Subsequently, is a complete fuzzy normed space.
Proof. We are enough to show that
is a closed subspace of
. Let us take a sequence
converging to
Z in
and
and
. Put
and
. Afterwards, there exists
, such that if
We fix
. Subsequently, there exists a fuzzy real number
, such that
Because
, there exists
, such that if
,
Hence, if
, we obtain
□
Example 1. Consider the linear space of all real numbers. Let us define It is clear that is a fuzzy normed space andfor all . Consider . Afterwards, for all , we have Example 2. Consider the linear space of all real numbers. Let us define It is clear that is a fuzzy normed space andfor all . Consider . Subsequently, for all , we have 4. The Dual Space of
We note that, if
, we define a function
by
Afterwards,
is a fuzzy norm on
and
-level sets of
are given by
Definition 8 (See [
4]).
Let be a fuzzy normed space. A strongly fuzzy bounded operator from to is called a strongly fuzzy bounded functional. Denote by the set of all strongly fuzzy bounded functionals over . Let . Subsequently, is a nested bounded and closed intervals of real numbers and, thus, it generates a fuzzy interval say , By [4] (Definition 6.3, is a fuzzy norm of . We call the strong fuzzy dual space of . Remark 4. The definition of a strong fuzzy bounded operator on introduced by T. Bag and S. K. Samanta [4] is slightly different from Definition 8. However, they have the same strong fuzzy dual space. Subsequently, it is natural to consider the following question.
Q. Let be a fuzzy normed space. What is the representation of ?
The answer is positive. First, we consider the following lemma.
Lemma 3. If is monotonically increasing in ℓ and the sequence is monotonically increasing in n, then we have Proof. This lemma is a well-known result in real analysis. For completeness, we give a proof. By the assumption, we obtain that each
and that the
are monotonically increasing in
n. Afterwards, by the Monotone Convergence theorem with respect to the counting measure, we have
By the property of the counting measure, we have,
so we obtain
Now, we provide the representation of strong fuzzy dual space of space of null sequence in fuzzy normed spaces.
Theorem 4. Let be a fuzzy normed space. If , then there is only one the sequence in , such that and Proof. Let
f be in
. Subsequently, there exists
, such that
For each
, we define
where
x is the
n-th component. Clearly,
is linear functional on
. Additionally, we have
for all
since
for all
First, we claim that
Fix
. We can easily observe that
Additionally, we have
as
, since, for each
and
as
, hence it proves the first claim.
Second, we claim that
Indeed, take any
. Afterwards, we obtain that for each
,
hence, we have
. Now, let us show the converse. By definition of
, for each
, there exists a sequence
in
, such that
Because, for each
, there is a scalar
, such that
and
, we may assume that
for each
and, so, that
and
is monotonically increasing in
k. Put
. Because
is monotonically increasing in
ℓ and the sequence
is monotonically increasing in
k, by Lemma 3, we have
Because
we have
it proves our claim. Additionally, we shall show that
Indeed, for any
and
, we have
so we have
so,
hence, it follows
. For the converse part, we can use the method for showing
. Indeed, by the definition of
, for each
, there exists a sequence
in
, such that
Subsequently, the remain part is similar to the proof of
Finally, to show the uniqueness, suppose that there exist such two
and
in
. Because
we obtain that, for each
and
,
hence we have
.
□
5. Approximation Properties of
In this section, we show that, given a fuzzy normed space
,
has the approximation property, even the bounded approximation property. The authors gave the definitions of approximation properties in fuzzy normed spaces [
7]. Recently, the author also provided the definitions of approximation properties in Felbin-fuzzy normed spaces (see [
6]). The definitions are as follows.
Definition 9. A fuzzy normed space is said to have the approximation property, briefly AP, if, for every compact set K in and for each and , there exists an operator , such thatfor every . Definition 10. Let λ be a positive real number. A fuzzy normed space is said to have the λ-approximation property, briefly λ-BAP, if for every compact set K in and for each and , there exists an operator , such thatfor every . Additionally, we say that has the BAP if has the λ-BAP for some . Lemma 4. Let be a complete fuzzy normed space. If a subset K in is compact, then for every and , there exists , such thatfor all . Proof. Let
and
be given. Because
K is compact in
, by [
22] (Theorem 4.2), there exists a finite subset
of
K such that
,
for some
. Put
for all
.
First, we claim that there exists
, such that
for all
. Consider
. Subsequently, we can generate the following sequence
in
:
We shall show that
is Cauchy sequence in
. Indeed, let
and
. Because
in
, there exists
, such that, if
, then
. Now, take any
. Subsequently, we have
By [
23] (Theorem 2), we can put
and
is a fuzzy real number and
.
Subsequently, we have
so it follows that
is a Cauchy sequence in
. Because
is complete, we have
Afterwards, there exists
such that if
, then
i.e.,
Continuing this process, for each
, there exists
, such that, if
, then
Now, we put . Afterwards, we prove our claim.
Now, we shall show our lemma. By the above claim, there exists a fuzzy real number
with
such that
for all
and
. Now, take any
. Subsequently, there exists
, such that
Afterwards, there exists a fuzzy real number
with
such that
Subsequenty, it follows that for all
Theorem 5. Let be a complete fuzzy normed space. If has the AP, then also has the AP.
Proof. First, we put
. Afterwards,
Y is a subspace of
. We simply denote
We shall show that
Y has the AP. Let
be compact and
and
. We consider
(resp.
) is the operator from
into
Y defined by
(resp.
). Afterwards,
are strong fuzzy bounded operator, since, for each
,
Now, for each
, let us consider the projection
by
It follows that
is a strong fuzzy bounded operator, because, for each
,
We note that
is compact in
for
because
is strong fuzzy bounded operator for
. Subsequently,
there is a
such that
for every
. Put
. Afterwards, for all
, we have
hence,
Y has the AP. Similarly, we can show that for all
,
has the AP.
Finally, we shall show that
has the AP. Let
be compact and
and
. By Lemma 5.3, there exists
, such that
for all
. Let
be the projection from
into
defined by
. Clearly,
is a strong fuzzy bounded operator, because, for each
,
Since
is compact in
and
has the AP, there is a
, such that
for all
. Now,
and for all
Hence,
has the AP.
□
Corollary 2. Let be a complete fuzzy normed space. If has the BAP, then also has the BAP.
Proof. It comes from the proof of Theorem 5. □
6. Conclusions and Further Works
In this paper, we have introduced spaces of sequences in fuzzy normed spaces and investigated their several examples. We have established a well-defined fuzzy norm for a space of sequences in fuzzy normed spaces. The completeness of the fuzzy nom in our context has been proved. We provided the representation of of the dual of a space of sequences in a fuzzy normed space. Moreover, the results of the approximation property for spaces of sequences in fuzzy normed spaces are given. We hope that our approach may provide a key role in fuzzy analysis by applying to fuzzy function spaces, for example, spaces of fuzzy continuous functions.