1. Introduction
One of the most fundamental and significant ideas in mathematics is convergence (of sequences). It was generalized in a number of ways. Even so, the first concept of statistical convergence, also known as nearly convergence, initially appeared in the famed Zygmund’s monograph’s first edition in 1935, H. Fast [
1] and H. Steinhaus [
2] independently discovered the concept of statistical convergence of real number sequences. The idea of the asymptotic density of a subset of the natural numbers serves as its basis. Let
denote the cardinality of
and
the cardinality of
for
and
. The
naturaldensity of
is defined by
Many disciplines of mathematics contain multiple applications of statistical convergence (see, for instance, [
3,
4] and references therein). Let us remark that [
5,
6] explored statistical convergence in function spaces. The concept of sequence convergence with respect to a filter
on
was first developed by Bernstein [
7] in 1970. Using the idea of an ideal, Kostyrko et al. [
8] developed the concept of ideal convergence, this is a typical generalization of statistical and ordinary convergence (see also [
9]). The ideal convergence offers a wide framework for analyzing the traits of various types of convergence. It should be noted that in [
8], several findings for the set of ideal cluster points and ideal limit points were discovered. This article also explains the concepts of ideal cluster points and ideal limit points for sequences in metric spaces. Be aware that the interesting generalization of statistical convergence known as ideal convergence (see [
10,
11]) exists.
“The concept of a fuzzy norm on a linear space was initially put forth in 1984 by Katsaras [
12] while researching fuzzy topological vector spaces (see [
10,
11]). By giving each element a fuzzy real number, Felbin [
13] established the concept of a fuzzy norm on a linear space in 1992. This resulted in the related fuzzy metric for this fuzzy norm, which is of the Kaleva and Seikkala type [
14]. In order for the associated fuzzy metric to be of the Kramosil and Michalek type, Cheng and Mordeson in 1994 [
15] suggested an alternative notion of a fuzzy norm on a linear space”.
T. Bag and S. K. Samanta introduced the idea of a fuzzy norm in [
16], whose associated fuzzy metric is similar to the Kramosil and Michalek type [
17]. This theory was developed after Cheng and Mordeson. The distinctiveness of this formulation comes from the fact that this kind of fuzzy norm can be successfully decomposed into a family of crisp norms according to the theory set forth in [
16]. This idea has been applied in numerous papers by numerous writers to create fuzzy functional analysis and its applications (for references, see [
16,
18,
19].
One of Nature’s fundamental characteristics is symmetry. An object or process is said to be symmetric if it is invariant to a particular set of transformations, known as “symmetry operations” (e.g., translation, rotation, reflection, inversion, etc.) that together make up a mathematical group. To estimate the fuzzy solution to the linear equation, symmetric fuzzy numbers and expected intervals are employed. It is also known as a model for solving a fuzzy Fredholm integral equation (FFIE), which has an arbitrary fuzzy function as its input and producesa symmetric or interval fuzzy function as its output. The use of an analytical method, namely the homotopy analysis method, falls within the category of analytical methods. A solution of a linear Fuzzy Fredholm Integral Equation (FFIE) with any Fuzzy Function input and symmetric triangular (Fuzzy Interval) output is one of the significant applications of symmetry in fuzzy systems.
Definition 1. A non-void class is said to be an ideal if the following conditions hold:
- (i)
implies (additive property);
- (ii)
and implies (hereditary property) [20].
An ideal
is non-trivial provided that
and called admissible if its non-trivial and
for all
. For any ideal, there is a filter
corresponding with
, given by
“An admissible ideal
possesses
property (AP) if for any sequence
mutually exclusive sets of
, there is a sequence
of subsets of
such that each symmetric difference
(
) is finite and
[
8,
21]”.
In contrast to fuzzy settings, ideal convergence in normed spaces is the focus of this study. The paper is structured as follows: In
Section 2, we provide some preliminary definitions and conclusions regarding fuzzy normed spaces. In
Section 3 of this essay, we primarily demonstrate that, under a general assumption, the condition (AP) is both required and sufficient for the ideal Cauchy condition and its adjoint Cauchy condition to be equivalent. We also provide an example to demonstrate that the ideal Cauchy condition is typically true. Not every sequence must have adjoint ideal Cauchy relations.
Section 4 looks into some important, little-studied aspects of ideal convergence and adjoint ideal convergence of sequences and nets in a fuzzy normed space, as well as some additional implications in a fuzzy normed space, such as the characterization of compactness in terms of ideal cluster points. Furthermore, we established the definition of ideal sequential compactness and its relation to countable compactness in a fuzzy normed space. The concepts of ideal divergent and adjoint ideal divergent sequences in a fuzzy normed space are introduced and established in
Section 5 of this study. We mainly demonstrate that condition (AP) is the necessary and sufficient condition for the equivalence of the ideal and its adjoint divergence under specific conditions, similar to convergence and the Cauchy condition. This strengthens and validates the role of condition (AP) in the evaluation of summability via ideals.
3. -Convergence in Fuzzy Normed Linear Space
In this part, ideal convergence and ideal Cauchy are defined in terms of the fuzzy normed space , along with several important findings. The ideal limit point and ideal cluster point of a real sequence in fuzzy normed linear space are other concepts that we introduce.
Definition 13. According to the fuzzy norm on , a sequence in is said to be -convergent to if the set belongs to for each and . In this instance, we write . The -limit of in is the name given to the element .
Remark 1. In terms of neighborhoods, we have , provided that for each and , Example 1. (1) If we take , then is a non-trivial admissible ideal of , and the accompanying convergence is ordinarily converging with respect to the fuzzy norm on .
(2) If we take . The accompanying convergence takes place concurrently with statistical convergence with respect to the fuzzy norm on , making a non-trivial admissible ideal of .
Lemma 3. Let be a real norm and be a setting where the fuzzy norm creates its own fuzzy norms , where and . Then for every sequence Proof. Suppose that
. Then, for every
and for every
, there exists a positive integer
such that
We observe that for any specific
,
it is comparable to
Whence, by letting
we have
This suggests that
for each
. □
Definition 14. A sequence in is said to be ideal Cauchy (abbreviation -Cauchy) with respect to the fuzzy norm on if for each and there exists an integer in such that the set belongs to .
For information on
-Cauchy sequences, (see [
23,
24]).
Lemma 4 ([
8])
. Let be an admissible ideal with the property (AP) and let be an ordinary metric space. Then, - if and only if there exists a set , such that . Lemma 5 ([
8])
. For each , , where is an admissible ideal with the property (AP), let be a countable collection of subsets of . The set is then such that and the set are both finite for all . Lemma 6. If a sequence is -convergent in with respect to the fuzzy norm, then -limit is unique.
Proof. For now, let us assume
and
where
, select
,
such that
and
are disjoint neighborhoods of
and
. Since
are both
-limit of the sequence
, we have
and
both belong to
. This allows us to infer that the sets
and
belong to
. This contradicts the notion that the neighborhoods
and
of
and
are disjoint. Hence, we have
. □
Theorem 1. Let and be sequences in such that and , where . Then
- (a)
;
- (b)
for .
Proof. (a) Suppose that
and
. Given
. Choose
such that
. Let
By assumption,
and
belong to
and so,
. Since
is an ideal, it sufficient to show that
. Let
. Then
and
and so,
and this implies that
. Similarly,
implies that
. Consequently,
Hence,
and so,
, i.e.,
. Therefore,
and hence,
.
(b) It is trivial for
. Now let
,
. Since
, we have
This implies that
Let
. Then we have
So
. Hence
for
. □
Theorem 2. If possesses property (AP), then a sequence in is an ideal convergent sequence in if and only if there is a sequence such that .
Proof. Let us say
. For each
in
and
in (
, let
then
for each
.
Considering that the admissible ideal has the property (AP), Lemma 5 provides us with There exists an integer such that and implies implies .
Define a sequence
in
as
This shows that the sequence is convergent to with respect to the fuzzy norm on . Thus, we have .
Next, suppose that
and
. Let
be given. Then, for each
, we can write
As the second set contains a constant number of integers, it too belongs to
, just like the first set on the right side of Equation (
1). This implies that
belongs to
. Hence, the proof is obtained. □
Theorem 3. Let be a sequence in and let be an admissible ideal with the property (AP). Hence, the following claims are equivalent:
- (i)
- (ii)
There exist and in such that ; and , where is the zero element of the linear space .
Proof. (i) (ii). Suppose Thus, we can deduce that there is a set by using Lemma 4, , such that .
Define the sequence
in
as
This shows that
. Further, we get
for each
. Since
. So, we have
. It follows that
and by Equation (
2), we get
.
(ii) (i). Suppose that there exist two sequences and in such that ; and . We proved that Let . Since , we have ; therefore, , if . Thus, we conclude that there exists a set , , such that . By Lemma 4, it follows that This accomplishes the proof. □
Corollary 1. Suppose that is an admissible ideal and assume that is a sequence in . Then, if and only if in such that ; and
Proof. Let
, where
is the sequence defined by (
2). Then,
and for the inclusion
, we conclude that
Conversely, let , where . By the inclusion , we get . This implies This accomplishes the proof. □
Remark 2. “It is evident from the Theorem 3 argument that if (ii) is satisfied, the ideal need not possess the property (AP). In fact, let and , where is an admissible ideal which does not have the property (AP). Since . For each , we have . Then we have ”
By Theorem 3 and Remark 2, We discover the subsequent theorem.
Theorem 4. Let be the set of all sequences which are -convergent to the zero element of the fuzzy norm and let be the set of all sequences with . Then, for each admissible ideal .
Definition 15. A sequence of elements in is said to be -Cauchy sequence in if for every and , there exists such that Definition 16. We say that a sequence of elements in is called -Cauchy sequence in if for every and , there exists a setsuch that and is an ordinary -Cauchy in . According to the following theorem, the notions of -Cauchy sequence and -Cauchy sequence coincide.
Theorem 5. If is an -Cauchy sequence in , then is an -Cauchy sequence in too.
Proof. Let
be an
-Cauchy sequence. Then, for every
and
, there exists
and a number
such that
for every
. Now, fix
. Then, for every
and
, we have
Let
. It is obvious that
and
Therefore, for every
and
, we can find
such that
, i.e.,
is a
-Cauchy sequence. □
Theorem 6. If a sequence is an ideal convergent in , then its ideal Cauchy is .
Proof. Suppose that
. Let
be given. Then we have
This implies that
Choose
such that
. Then, for every
,
Hence,
. This implies that
that is,
is an ideal Cauchy sequence. □
Theorem 7. Let be a sequence in and denote , where . If is an -Cauchy sequence, then for every , there exists with such that for all .
Proof. Let
be given and set
and choose
such that
. Since
is an
-Cauchy sequence, we have
and for all
, we get
Thus, we have
, for all
. This accomplishes the proof. □
Definition 17. A sequence in is said to be -convergent to with respect to the fuzzy norm if there exists a set , such that
Theorem 8. If a sequence is -convergent in , then it is an -Cauchy sequence in .
Proof. Considering that there exists a set
such that
and
, i.e., there exists
such that
for every
and
and
. Choose
such that
.
for every
,
, and
, we have
for every
, i.e.,
in
is an
-Cauchy sequence in
. Then, by Theorem 5,
is an
-Cauchy sequence in
. □
Theorem 9. If the sequence is -convergent to , then is -convergent to , i.e.,
Proof. Letting
. Hence, by definition, there exists
such that
. Let
and
be given. Since
, there exists
such that
for every
Since
is contained in
and the ideal
is admissible, we have
. Hence,
for every
and
. Whence, we draw the conclusion that
. □
The converse of Theorem 9 need not be true, as the following example demonstrates.
Example 2. Consider with and let for all . For and . ConsiderThen, is a fuzzy normed space. Have a look at the decomposition of as where and for . Let be the class of all subsets of which intersect at the most finite number of . Then, is an admissible ideal. We define a sequence as follows: if . Then, we haveas . Hence, -. Now, we show that -. Suppose that - Then, by definition, there exists a subsetsuch that and -. Since , there exists such that . Then, there exists positive integer such thatThus, and so for infinitely many values in . Thus, it contradicts the assumption that -. Hence, -. Therefore, the converse of Theorem 9 need not be true. If the ideal meets the requirement (AP), the following theorem demonstrates that the converse is true.
Theorem 10. If the admissible ideal satisfies the condition (AP), then the sequence in such that implies .
Proof. Since
, so for every
and
, the set
We define the set
for
as
Then, it is clear that
is a countable family of sets that belong to
that are mutually disjoint, and by the property (AP), there is a countable family of sets
such that the symmetric difference
is a finite set for each
and
. Since
, there is a set
such that
. Now we prove that the subsequence
is convergent to
with respect to the fuzzy norm
. Let
and
. Choose a positive
such that
. Then
Since
is a finite set for each
, there exists
such that
If
and
, then
and
. Hence, for every
and
we have
. Since this holds for every
and
, so we have
-
. This accomplishes the proof. □
Theorem 11. Assume that has at least one accumulation point. If for every sequence , -Cauchy condition implies -Cauchy condition then possesses property (AP).
Proof. Assume that
is accumulated up to
. Then, a sequence of distinct points in
called
occurs such that
converges to
and
for all
. Assume that the sequence of mutually exclusive non-empty sets from
is
. Define a sequence
by
For any
. Let
. Choose
such that
. Then, there exists
such that
for all
. Then,
and
. Now, clearly,
implies that
and
. So,
By doing so, it can be seen that the sequence
is a
-Cauchy sequence. Our assumption is that
is
-Cauchy. Since
and
are Cauchy,
exists. For
, let
. After that, each
. Additionally,
. Therefore,
. In the sets
,
, the following three situations could occur:
- Case (1)
Each is included in a finite subset of .
- Case (2)
In a finite subset of , only one of the ’s; let us say , is excluded.
- Case (3)
The finite subset of does not include more than one ’s.
If scenario (1) occurs, then a finite subset of
includes the
which indicates that
has the (AP) condition.
If case (2) occurs, we redefine
and
for
. Then,
Additionally, if
and
,
holds. The prerequisites for the (AP) condition are thus satisfied, just as in Case (1).
If Case 3 occurs, then with exists such that and are not included in any finite subset of . Allow . Given that is a Cauchy sequence, exists such that for all and . We can choose and with since and are not included in any finite subset of . However, since and , (there are an unlimited number of indices with that property). The Cauchy character of is violated by this. Therefore, Case (3) is ruled out. Additionally, satisfies the possession condition (AP) in light of Cases (1) and (2). □
The following theorem is implied by the following Theorem 3.
Theorem 12. Let be a sequence in . If there exist two sequences and in such that ; and then .
Definition 18. Let be a sequence in . An element is said to be an -limit point of provided that for each there is a set such that and .
Definition 19. Let be a sequence in . An element is said to be an -cluster point of provided that for each and , the set
“We denote and the set of all -limit points and -cluster points of a sequence in ”.
Proposition 2. Let be an admissible ideal. Then, for any sequence in a fuzzy normed space , we have .
Proof. Suppose that
. Then, for each
there exists a set
such that
and
Let
be given. (
3) states that there is an integer
such that for
, we get
Thus, we have
This implies that
Hence,
. □
Proposition 3. Let be a sequence in . If , then
Proof. Suppose
. Then, for each
and
, the set
which implies that
.
We assume that there exists at least one
such that
. Then, there exists
and
such that
However,
implies that
, which contradicts that
. Thus, we have
.
On the other hand , by Theorem 2 and Definition 19, we have . By Proposition 2, we have the result. □
Proposition 4. Let be an admissible ideal. Then, the set is closed in , for each sequence in .
Proof. Let
. Let
and
be given. Then, there exists an
. Choose
such that
. Obviously, we have
This indicates that
. Therefore,
. Hence,
is closed in
. □
4. -Convergence and -Convergence of Nets in Normed Fuzzy Space
In this section, some important topological properties are investigated using the notation of ideal and its adjoint convergence of sequences and nets in a normed fuzzy space. These concepts can be combined to better define compactness. We introduce the notion of ideal sequentially compactness and derive some basic topological space characteristics.
The first two of the following definitions are well-known.
Definition 20 ([
25])
. Let be a binary relation on a non-void set such that the binary relation is reflexive, antisymmetric, and transitive, and such that and for any two elements . Then, we call a directed set. Definition 21 ([
25])
. Let be a nonempty set and be a directed set. A mapping from into is said to be a net in , indicated by or just when the set is obvious from the context. “For
let
. Then, the collection
forms a filter in
. Let
. Then,
is also a non-trivial ideal in
”.
Definition 22 ([
26])
. A -admissible is a non-trivial ideal of for which for all . Definition 23. A net in is said to be ideal convergent to provided for each and , Symbolically we write - and we say that is an -limit of the net .
Remark 3. “If is -admissible, a net will converge in , whereas the opposite is true if . Moreover, if with the natural ordering, the words -admissibility and admissibility overlap, and in that case, is the ideal of all finite subsets of ”.
The following gives the definition of an -cluster point of a net with in .
Definition 24. An element is called an -cluster point of a net if for each and , .
Theorem 13. For every net in , there is a filter on such that is a -limit of the net if and only if is the limit of the filter and, is a -cluster point of the net if and only if is the cluster point of the filter .
Proof. Let be the space, and . Let be the associated filter on and be a non-trivial ideal of . Let us create the set for each . Then, a filter based on is formed by the family . In fact, each is not empty since each is not empty. Furthermore, if where , then where , since is a filter, so each is not empty. Hence, our conclusion is correct. Let represent the filter that this filter base produces. We now demonstrate that has the necessary property.
Let the net be -convergent to . Then, for any and ,. This implies that . We write . Then, by our construction . Since we get and since is an arbitrary neighbourhood of , we conclude that for all neighbourhood . Hence, the filter is convergent to .
Again, let the filter be convergent to . Then, the neighbourhood filter of the point is a subfamily of , i.e., . Let for and be an arbitrary. Then, for some . This implies that , which further implies that , i.e., . This shows that the net is also -convergent to .
Now we suppose that is an -cluster point of the net . Then, for any and , we have , i.e., . Hence, we conclude that the set contains no for any . So, for every , there exists some such that , i.e., there exists for each such that . Thus, we get for all so that becomes a cluster point of the filter .
Next let be a cluster point of the filter . Then, for any and , we have for all , i.e., for all . We conclude that . For, if then this would imply that . So, if we write , then and this leads to a contradiction. Hence, so that becomes an -cluster point of the net . □
“If every family of closed sets with the finite intersection property (abbreviated FIP) has an intersection that is not void, a topological space is said to be compact. Last but not least, we support a crucial conclusion about the compactness of a fuzzy normed space with data”.
Theorem 14. In a compact space , each net has a -cluster point that corresponds to every non-trivial ideal of .
Proof. A net in is defined as , where is a compact space. Assume that is a non-trivial ideal of and that is the filter on that is connected to the ideal . Take into account the set for each . In light of the fact that is a filter, the family that includes all such has FIP. As a result, the family is a family of closed sets that possesses FIP. Since is a compact space, . So there is some such that . Then, for every and , we have . Now, we consider the set . If , then the corresponding set does not intersect , i.e., , which runs counter to the previously inferred fact. Hence, , which implies that . Thus, becomes an -cluster point of the net . □
Definition 25. If every sequence in a space has a -cluster point, where is a non-trivial ideal of the set , then the space is said to be ideal sequentially compact.
“The notations of sequential compactness and ideal sequential compactness of a fuzzy normed space are distinct, as shown by the next two instances”.
Example 3. This example shows how a sequence in a space with fuzzy norms can have a cluster point without simultaneously having an -cluster point, which corresponds to a non-trivial ideal of , the set of natural numbers. The set of all even positive integer subsets and the set of all odd positive integer finite subsets will provide , the non-trivial ideal of .
Consider with and let for all . For and , considerThen, is a fuzzy normed space. Let be a sequence in defined bySo, it is obvious that a convergent subsequence exists for . Nevertheless, there is no -cluster point in . Example 4. With the help of this example, we can see that there is a sequence in a fuzzy normed space that lacks a cluster point but has an -cluster point corresponding to a non-trivial ideal of the set . Make a non-trivial ideal of that includes all the sets of even positive integers as subsets.
Consider with and let for all . For and , considerThen, is a fuzzy normed space. Let be a sequence in defined by for all . Now, it is obvious that the sequence does not have a cluster point in , but every odd positive integer turns into a -cluster point for the sequence . “In the following, we show how, in some situations, the countable compactness and ideal sequential compactness of a fuzzy normed space relate to one another. Currently, we can recall the outcome”.
Lemma 7. For a fuzzy normed space . The following are interchangeable.
- (i)
is countably compact.
- (ii)
There is a non-empty intersection for any countable set of closed subsets of satisfying the FIP.
- (iii)
If is a family of non-empty closed subsets that descends from , then .
Theorem 15. Let be fuzzy normed space and let be an admissible ideal. Then, is a countably compact space if and only if is sequentially compact in the ideal sense.
Proof. Assume that is a space that is progressively compact from . Let be a countable open cover of that has no finite subcover for and . Then, we can choose . The sequence must now contain an -cluster point; let us say . For some , let . Accordingly, . The set must be an infinite subset of since is an admissible ideal of . As a result, there exists some such that . However, according to our method of construction, , we arrive at a contradiction. As a result, needs to be countably compact.
Assume that is a countably compact space to demonstrate the opposite. We will first demonstrate that is a first countable. For this, let , . We will show that is a local basis for . Let be open set and . Since is open, then there exists and such that . Choose such that and . Now, we just need to show . Let . Then, . Since we have . Hence, which implies that . Consequently, is a countable local basis for . Therefore, is first countable.
Assume that consists of a sequence of distinct points, . Let us assume that for each positive integer , . Hence, by Lemma 7 is a descending sequence of non-empty closed sets, and we get . Let . Since is a first countable space, suppose that for and , is a countable local base at the point , where . Observe that for all . Now, . So, there exists some such that . Since , we select an integer that is positive, , such that . Again, since , choose a positive integer such that . Suppose have been chosen such that for . Again, since , there is some such that . Consequently, we obtain a subsequence of the sequence , which is such that , for all for every . We demonstrate the convergence of this subsequence to . Given that is an open subset of , let . Hence, if , there exists some positive integer . So, we get for all . Since is an admissible, the sequence converges to under . This implies that we have for any open set containing . Since is a non-trivial ideal, becomes an -cluster point of the sequence when . Now, that . Hence, we have , which means that becomes an -cluster point of the sequence . As a result, is an -sequentially compact space. □
5. -Divergences and -Divergence
In their article [
27], Macaj and Salat expanded the definition of divergent sequences of real numbers to include statistically divergent sequences of real numbers. Notwithstanding the eventual addition of
-convergence and
-convergence to the definition of statistical convergence (as already indicated), no comparable approach has been taken in the case of divergence. Later, in their publication cited as [
28], Das and Ghosal conceived and developed the notion of a sequence diverging in a metric space. That is what this paragraph aspires to achieve. As an alternative to limiting the idea of divergence to only real sequences (note that our definition includes the basic definition of real divergent sequences as a particular case), with the aid of ideals, we establish it in a fuzzy normed space and broaden it. According to our research, the condition (AP), exactly like it was in the cases of
-convergence and
-Cauchy condition, is once more significant.
Definition 26. A sequence in a fuzzy normed space is said to be divergent (or properly divergent) if there exists an element such that for and .
“It should be noted that in a fuzzy normed space, a divergent sequence cannot have any convergent subsequence”.
Definition 27. A sequence in is said to be -divergent if there exists an element such that for each and .
Definition 28. A sequence in is said to be -divergent if there is at least one such that exists. This is defined as there being at least one or .
Theorem 16. Let be a sequence in . If is -divergent, then is -divergent.
Proof. Since is -divergent, so there exists , i.e., such that is divergent, i.e., there exists at least one such that . Then, for any and , there exists such that for all and . Hence, we have . This implies that is -divergent. □
“The opposite of the aforementioned theorem is not always true, as demonstrated by the example below”.
Example 5. Consider with and let for all . For and , considerThen, is a fuzzy normed space. Let be a decomposition of such that each is infinite and for . Let be the class of all subsets of that can only intersect ’s. If so, is a non-trivial ideal of that is admissible ideal of . Let if . Now, for any and , there exists a natural number such that the set and so is -divergent. Next, we will demonstrate that is not -divergent. Assume that it is -divergent if at all possible. Consequently, an such that is divergent. There is an such that since exists. But after that, for any , . More specifically, . However, this means that the constant subsequence is convergent to and is a constant subsequence of . The divergence of is contradicted by this. Theorem 17. Let be a sequence in . If is an admissible ideal with property (AP), then for any sequence in , -divergence implies -divergence.
Proof. Let us assume for a moment that
has property (AP). Since
is
-divergent, there must be at least one
such that
for
and
. For
, we define the set
as follows:
As a result, we have a collection of mutually disjoint sets
with
for all
. According to the property (AP), a family of sets
exists where
and
is finite for all
’s. Let
. Then,
. Let
and
. Select a positive
such that
. Then,
. Since
is finite, so there exists
such that
Obviously, if
and
, then
implies that
. Therefore,
. Thus,
is divergent. □
Theorem 18. Let be an admissible ideal and be a fuzzy normed space with at least one divergent sequence. If -divergence implies -divergence for any sequence , then possesses property (AP).
Proof. Suppose that the divergent sequence
in
. The element
is thus such that
. Assume that
is a list of non-empty sets from
that are mutually disjoint. Define a sequence
by
for any
. Let
and
. Choose a positive
such that
for all
. Now,
. So,
is
-divergent. According to our presumption,
is
-divergent. As a result, there is a
where
and
is divergent. Let’s say
. Next,
. Insert
for all
. Indeed,
We assert that a finite set,
, exists. If not,
must contain a convergent subsequence of
, which is an infinite series of objects
for every
. This, however, contradicts the divergence of
.
is a finite subset of
. This proves that
satisfies the requirement (AP). □