1. Introduction
Since Katsaras first introduced the notion of fuzzy norm on a vector space, a study for fuzzy normed spaces has been actively progressed [
1]. In 1992, Felbin introduced a new definition of a fuzzy norm (namely, Felbin-fuzzy norm) related to a specific fuzzy metric [
2,
3]. In 2003, Bag and Samanta defined a more general notion of a fuzzy norm (namely, B-S fuzzy norm) [
4,
5]. Because of their pioneering research, topological properties have been studied according to Felbin type’s fuzzy norms and B-S type’s fuzzy norms, respectively [
6,
7,
8]. Cho et al. investigated the classical and recent results of fuzzy normed spaces and fuzzy operators in their book [
9].
The approximation property (AP) is an essential concept in researching functional analysis because the AP has been studied for the Shauder basis and operator theory. The AP means that the identity operator on an Banach space can be approximated in the compact open topology by finite rank operators (please see References [
10,
11,
12,
13]). In 2010 and 2016, Yilmaz et al. introduced the approximation property in B-S fuzzy normed spaces [
14,
15]. In 2019, the approximation property in Felbin fuzzy normed spaces was introduced [
16]. Related works emerged from fuzzy theory. We would refer the reader to where intuitionistic fuzzy Banach space theory is outlined in Reference [
17].
In this paper, we study approximation properties in Felbin-fuzzy normed spaces. Moreover, we will develop topological tools related approximation properties in Felbin-fuzzy normed spaces. We first identify approximation properties in Felbin-fuzzy normed spaces in terms of infinite sequences. We provide dual problems for such approximation properties. The advantage of our context is to characterize a compact subset in Felbin-fuzzy normed spaces.
Our paper is organized as follows—
Section 2 is comprised of some preliminary results. In
Section 3, we develop the representation of topological dual elements related approximation properties in Felbin-fuzzy normed spaces.
Section 4 is devoted to identifying approximation properties in Felbin-fuzzy normed spaces in terms of infinite sequences. In
Section 5, we apply this identification to provide some results about dual problems for approximation properties in Felbin-fuzzy normed spaces.
2. Preliminaries
Definition 1 (See Reference [
5])
. 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. Define a
partial ordering by
in
if and only if
for all
. Since each
can be considered as the fuzzy real number
denoted by
hence it follows that
can be embedded in
(See [
5]).
Definition 2 (See Reference [
5])
. Let X be a vector space over . Assume the mappings are symmetric and non-decreasing in both arguments, and that and . Let . The quadruple is called a Felbin-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
The following definition gives the notion of strongly fuzzy bounded. Here, we use the arithmetic multiplicative operation ⊗ on
as in [
2]:
Definition 3 (See Reference [
18])
. Let and be Felbin-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 . Then is a vector space. For all we denote bywhere 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
. Then
is a subspace of
. We similarly define
for some
. Now, we provide definitions of the approximation properties in Felbin-fuzzy normed spaces. For the definition and properties of
-level set (
), see Reference [
3,
19].
Note that, if
, the linear space of all reals, we define a function
by
Then is a fuzzy norm on and -level sets of are given by for all .
Definition 4 (See Reference [
20])
. A strongly fuzzy bounded linear operator defined from a Felbin-fuzzy normed space to is called a strongly fuzzy bounded linear functional. We denote the set of all strongly fuzzy bounded linear functionals over by . Definefor all . Remark 1. Definition 4 came from Bag and Samanta [20]. Although they defined a strongly fuzzy bounded linear operator differently from this paper, the two definitions are same in the case of functionals. Definition 5 (See Reference [
21])
. Let be a Felbin-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 . Given a Felbin-fuzzy normed space
, we recall
Definition 6 (See Reference [
21])
. Let be a Felbin-fuzzy normed space and . A point is called a closure of A if A sequence of X is called a cauchy sequence if for all . A subset is said to be complete if every cauchy sequence in A converges in A. Definition 7 (See Reference [
16])
. A Felbin-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 . As an example, having the AP in a Felbin-fuzzy normed space, let us consider Banach space
with
. Moreover,
is another norm on
. Now let us define
Then
has the AP in a Felbin-fuzzy normed space ([
16], Example 1). Moreover, in Reference [
16], we made a comparison study among approximation properties in Felbin fuzzy normed spaces with other fuzzy normed spaces.
Definition 8 (See Reference [
16])
. Let λ be a positive real number. A Felbin-fuzzy normed space is said to have the λ-bounded approximation property, briefly λ-BAP, if for every compact set K in and for each and , there exists an operator such thatfor every . Also we say that has the BAP if has the λ-BAP for some . 3. The Dual Space of
In this section, we give a representation of the dual space of endowed with the topology of uniform convergence on compact subsets of where and are Felbin-fuzzy normed spaces. We recall the following definition.
Definition 9 (See Reference [
16])
. Let and be Felbin-fuzzy normed spaces. For a compact , , , and we putLetbe the collection of all suchs. Then the τ-topology onis the topology generated by. Clearly, τ is locally convex topology. Moreover,is the vector spaces of all continuous fucntionals of
The -topology is important tool to study approximation properties as followings. The proofs are trivial.
Theorem 1. Letbe a Felbin-fuzzy normed space andbe an identity operator and λ be a positive real number.
(a) if and only if X has the AP.
(b) if and only if X has the λ-BAP.
Now we establish a representation of , which will be applied to characterize the approximation property and bounded approximation property in fuzzy normed spaces. The proof shall be given later.
Theorem 2. Letandbe complete Felbin-fuzzy normed spaces. Thenconsists of all functionals φ of the formwhereandfor somesuch that By the proof of (Reference [
15], Lemma 4.2), we have the following.
Lemma 1. Letbe a Felbin-fuzzy normed space and K be a compact subset in. Then there exists a finite setin K such that for, we havefor some.
The following Theorem shows that a relatively compact subset of a fuzzy normed space contained in the convex hull of a null sequence. Here, the convex hull of a subset A of a topological vector space X, denoted by co(A), is the smallest convex set that includes A and we denote the closure of co(A) by .
Theorem 3. Letbe a Felbin-fuzzy normed space. Suppose that K is a relatively compact subset of. Then there is a sequencein X converging to 0 insuch that Proof. First, we recall that
for each
and
and
and
whenever
,
, and
([
21], Lemma 3.1). It may be assumed that
. Since
is relatively compact, ([
21], Theorem 4.2),
is fuzzy bounded in
. Then there exists
such that
Also, by Lemma 1, there exists a finite set
in
such that
. Let
Then
. Since
is nonempty and relatively compact, there exists a finite set
in
such that
. Let
Then
. Since
is nonempty and relatively compact, there exists a finite set
in
such that
. Let
The construction of is continued in the obvious fashion.
Now, let
and
. Then we choose
such that
and
. By the above claim and
’s selection, we obtain that for each
hence we have
. Thus, for all
, we have
hence
converges to 0 in
.
Finally,
. Since
, there is a positive integer
such that
such that
, so there is a positive integer
such that
such that
, and so forth. If follows that for each
Hence we have
for each
, and therefore we have
. □
Corollary 1. Letbe a fuzzy normed space. Then there exists a sequenceinsuch thatwhereis the sequence selected in Theorem 3. Proof. As we see in the proof of Theorem 3, we have a sequence
in
with
,
satisfying
Clearly, we have
in
and
□
The following lemma can be obtained from the proof of (Reference [
15], Lemma 5.7).
Lemma 2. Letandbe Felbin-fuzzy normed spaces. If, then there exists a finite subsetof X andandsuch thatimplies.
Proof of Theorem 2. If a linear functional
on
is given by
where
and
for some
such that
then we shall show that there are a compact subset
K of
and
such that
for all
, which proves that
is in
. First, let
be a sequence of positive reals so that
Put
and
.
Now, we shall show that
K is a compact subset of
. We are enough to prove that
Take any
. Since
,
, we have
,
hence
Now we take any
. By the above argument, for
,
hence we have
By continuing this process for
, we prove our claim. On the other hand, we have for all
Now, for the converse, assume that
. Then, by Lemma 2, there are compact sets
and
and
such that
,
,
implies
. Put
and
. Then it is easy to check that
for all
. By Theorem 3, there is a sequence
in
X converging to 0 in
such that
Moreover, Corollary 1, there exists a sequence
in
such that
There exists
such that
. Put
Then we may assume
such that for all
n,
and
Also, we can observe
for all
.
Now we put
Let us consider the linear map
Since for each
, there exists
such that
the linear map
is well defined. Now we define
if
and
. We claim that
is well-defined and linear continuous functional on
. Indeed, observe that
implies
, hence
. Thus
is well-defined. Also, we obtain that
it proves our claim.
Now, by applying Hahn-Banach extension theorem, the functional
can be extended to a
. Then there is uniquely the sequence
in
such that
and
Thus, for each
we have
4. Characterizations of Approximation Properties
In this section, we establish new characterizations of approximation properties in Felbin-fuzzy normed spaces in terms of infinite sequences by applying the representation of
Section 5. The following includes our main theorems. The proof shall be given later.
Theorem 4. A complete Felbin-fuzzy normed spacehas the AP if and only if for every,andsuch thatwe have Theorem 5. A complete Felbin-fuzzy normed spacehas the BAP if and only if for every,andsuch thatwe have Remark 2. Theorems 4, 5 are very important tools to solve dual problems for approximation properties in Felbin-fuzzy normed spaces (see Section 5). Lemma 3. Letbe a Felbin-fuzzy normed space. Suppose thatis a subspace of. Let. Then the following are equivalent.
(a) T belongs to .
(b) For every such that for all , we have .
Proof. This can be directly derived from a result of the locally convex space version of the Hahn-Banach theorem. One may refer to Reference [
22], Corollary 2.2.20. □
Moreover, we need the following lemma [
23].
Lemma 4. Letbe a Felbin-fuzzy normed space. Suppose thatis a balanced convex subset of. Let. Then the following are equivalent.
(a) T belongs to .
(b) For every such that for all , we have .
Proposition 1. Let Y be a finite dimensional subspace of a Felbin-fuzzy normed space. Suppose that. Then there is a strongly fuzzy bounded linear functional f on X such that,and.
Proof. The proof comes from Reference [
16], Theorem 2. □
The following theorem gives the representation of finite rank strongly fuzzy bounded operators ([
16], Corollary 2).
Theorem 6. Letandbe Felbin-fuzzy normed spaces. Ifis a finite rank strongly fuzzy bounded linear operator, then there existandsuch thatfor all. Now, we are ready to prove Theorems 4 and 5.
Proof of Theorem 4. Let
,
and
such that
Now a linear functional
on
is given by
By Theorem 2, we have
. Since
has the AP, we have
Then, by Lemma 3, we are enough to show that
We can observe that for each
and
, by assumption, we have
because
is sequentially fuzzy continuous. Let
S be in
. By Theorem 6, there exist
and
such that
Then, by the above observation, we have
Conversely, let
be in
and
Then, by Theorem 2, we have
where
and
for some
such that
By Lemma 3 and the assumption, we are enough to show that
First, we claim that for each
,
exists in
. Indeed, take any
and
and
. There is
with
such that for all
,
Then we have
so
is Cauchy sequence. Since
is complete,
exists. Also, we claim that for each
and
, we have
Indeed, let
and
. Put
Then
, by assumption, we have
Since
is sequentially fuzzy continuous, we have
It follows that
because if not, that is,
, then, by Proposition 1, there exists
such that
it is a contradiction. □
Proof of Theorem 5. Since is a balanced convex, by Lemma 4 and similar arguments of Theorem 4, we can prove it. □
5. Application to the Dual Problems
In this section, we provide dual problems for approximation properties in Felbin-fuzzy normed spaces and their partial solutions.
Dual problems. Let be a Felbin-fuzzy normed space. If has the AP, then does have the AP? Conversely, if has the AP, then does have the AP?
In Banach space theory, dual problems were completely solved (see Reference [
13]). However, in fuzzy normed spaces, we have partial solutions.
Proposition 2. Letbe a fuzzy normed space and. We define Thenis a fuzzy normed space. Also, for allwe have Proof. The proof is clear. □
We have a partial solution for dual problems by giving the specific condition of a dual space as the following.
Theorem 7. Letbe a complete Felbin-fuzzy normed space. If for every,has the AP, thenhas the AP.
Proof. We shall use Theorem 4. Let
. Also we fix
and
such that
Now let us consider
. Since,
,
converges to 0 in
, we have
We can observe that
because for each
, by Proposition 2, we have
Also, for each
, by again Proposition 2, we obtain
Now we define that for each
,
Then,
, we can regard
as the element of
because
Thus it follows
, hence we have
Since
has the AP, by Theorem 4, we obtain
so
, by again Theorem 4,
has the AP. □
The following proposition is the solution for the other part of the dual problems.
Proposition 3. There exists a Felbin-fuzzy normed spacewhich has the AP butdoes not have the AP. does not have the AP.
Proof. Let us consider a Banach space
X having the AP whose dual fails to have the AP (see Reference [
13]). We denote this Banach space by
. Let us define
It is clear that for all . Clearly, is complete fuzzy normed space and hast the AP.
Now we claim that
as vector spaces. Let
. Then there exists a positive
such that
it follows
i.e.
. The converse can be obtained the same way. Moreover, we have
Then we have that does not have the AP. □
6. Conclusions and Further Works
In this paper, we have studied approximation properties in Felbin-fuzzy normed spaces and developed topological tools to analyse such approximation properties. We have identified approximation properties in Felbin fuzzy normed spaces in terms of infinite sequences. By using this, we provided partial solutions about dual problems for approximation properties in Felbin fuzzy normed space. We hope that our approach may present topological foundations to research topological objects including compact subsets and weakly compact subsets in Felbin-fuzzy normed spaces. Moreover, dual problems for approximation properties can be answered completely.