1. Introduction
Let be a normed space. We denote by the family of all fuzzy sets in U. Given any and , the addition and scalar multiplication are well known. Under these operations, the space cannot be a vector space. The main reason is that the additive inverse does not exist. For the purpose of applications, we expect to treat as a subset of a vector space. Since we have the difficulty of treating the whole family as a subset of a normed space, in this paper, we are going to present some specific subfamilies of which can be treated as the closed convex cone in some Banach spaces.
We denote by
the family of all nonempty subsets of a normed space
. Given any
, the Hausdorff metric of
A and
B is defined by
where
is an extended real number. By convention, we define
We say that
is a fuzzy set in a normed space
when a membership function
is associated with
. For
, the
-level sets
of
are defined by
For
, the strong
-level sets
of
are defined by
Then, we have
The support
of
is given by
In this case, we can define the 0-level set of
by
, which means that the 0-level set of
is the closure of the support of
.
We consider the following function
where
denotes the
-level set of
for
. We say that
is continuous with respect to the Hausdorff metric
when, given any
, there exists
such that
implies
. Then, we adopt the following notations.
We denote by the family of all fuzzy sets in U with nonempty compact and convex -level sets for .
We denote by the family of elements in such that the function is continuous.
In this paper, we are going to use the support functions of fuzzy sets to embed the families and into some useful Banach spaces.
Example 1. We say that is a fuzzy interval when its α-level sets are bounded closed intervals given byIt is clear to see that the bounded closed interval is a compact and convex set in , which means . We also haveWe can define two real-valued functions l and u on byIt is clear to see that, if the functions l and u are continuous on , then the functionis also continuous on , which means . In order to define the differentials of fuzzy functions, Puri and Ralescu [
1] developed an embedding theorem to embed the space
into a normed space by following the approach of Rådström [
2]. Also, Kaleva [
3,
4] and Román-Flores and Rojas-Medar [
5] established the different kinds of embedding theorems to study the fuzzy differential equations. On the other hand, the embedding theorem is also applicable to study the fuzzy random variables by referring to Puri and Ralescu [
6,
7] and Klement et al. [
8]. Recently, Wu [
9] considered the equivalence classes of Cauchy sequences to study the embedding theorems of some interesting families of fuzzy sets.
To consider the families of fuzzy sets in a normed space as a subset of some Banach spaces is an important issue in applications. There are two ways to study the embedding theorems for some specific families of fuzzy sets in the literature. One is directly constructing some suitable Banach spaces such that the specific families of fuzzy sets can be isometrically and isomorphically embedded into the corresponding Banach space. Another approach is to use the support functions of fuzzy sets. In this case, the continuities of support functions should be studied under some suitable topologies. The weak* topology of the closed unit ball in the dual space of a normed space is considered in this paper for the purpose of constructing smaller Banach spaces. It is better to make the embedded Banach spaces as small as possible, which can be achieved by considering the weak* topology. The concept of a maximal embedding behavior is also presented in this paper. It means that the maximal subset satisfying the completeness in the sense that every Cauchy sequence is convergent.
The support functions of fuzzy sets are defined using the concept of supremum. Considering the supremum is equivalent to considering optimization problems. Solving the optimization problem is not an easy task. However, when the optimization problems have some elegant mathematical structures, the well-known techniques developed in the topic of optimization are useful for solving these well-structured optimization problems. Since the support functions of fuzzy sets own the elegant mathematical structures, studying the embedding theorems using the support functions of fuzzy sets is still worthy for real applications. Diamond and Kloeden [
10,
11], Ma [
12], Puri and Ralescu [
6,
13], Román-Flores and Rojas-Medar [
5], Wu and Ma [
14,
15,
16], and Wu [
17] presented the embedding theorems using the support functions of fuzzy sets. Those approaches were mostly based on the finite-dimensional Euclidean space. In this paper, we are going to study the embedding theorems based on the infinite-dimensional normed space. Although we expect to embed the families of fuzzy sets into some Banach spaces, it will be better to make these Banach spaces as small as possible. The advantage of this paper is to consider the weak* topology for the normed dual space of normed space, which can make the embedded Banach spaces smaller than that of the existing Banach spaces presented in the literature.
In
Section 2, the dual space of normed space is considered. In order to make the embedded Banach space as small as possible, the weak* topology of closed unit ball in the dual space is discussed. Since the embedded Banach spaces studied in this paper consist of continuous functions, many spaces of continuous functions are studied in
Section 3. The continuities of support functions of fuzzy sets with respect to the weak* topology are studied in
Section 4. In
Section 5, many embedding theorems of the families
and
are presented in which the embedded Banach spaces involve the weak* topology such that they can be smaller than that of the existing Banach space presented in the existing literature. Also, the concept of maximal embedding behavior is provided.
2. Topologies of the Closed Unit Ball in Norm Dual Space
Let
be a normed space. The closed unit ball and unit sphere are defined by
respectively. The topology
induced by the norm of
U is called the norm topology or strong topology. In this case, we have a topological space
. Sometimes, we also write
The dual space
is the family of all linear functionals defined on
U such that they are continuous with respect to the norm topology. The weak topology
for
U is the weakest (coarsest) topology for
U such that all linear functionals in
are continuous.
Given any
with
, we write the functional value
for
and define
Suppose that
with
. Then, we take
, which says
and
. Therefore, we obtain
It is well known that
is also a normed space.
The norm topology for induced by the norm is also called a strong topology. The weak topology for is the weakest (coarsest) topology for such that all linear functionals in are continuous. Let be a natural embedding of U into . Then, we have . We say that the normed space U is reflexive when . The weak* topology for is the weakest topology for such that all linear functionals in are continuous. The weak* topology for is weaker than the weak topology for . Therefore, we have that a weak* closed subset of is weakly closed, and that the weak convergence implies weak* convergence. When U is reflexive, the weak and weak* topologies for coincide.
The closed unit ball
is not necessarily a compact subset of
with respect to the norm topology for
. However,
is a weak* compact subset of
by the Alaoglu’s theorem, which says that
is compact with respect to the weak* topology
for
. In this case, we can induce a topology
for
such that
is a compact space, where, for each
, there exists
satisfying
. However, the closed unit sphere
does not need to be weak* compact, since it is not even weak* closed (Aliprantis and Border [
18], p. 250).
The closed unit sphere
cannot be a vector subspace of
, since
is not closed under the scalar multiplication. This says that we cannot define a norm on
. However, we can define a metric on
by
Therefore, we can form a metric space
such that it can also induce a metric topology
.
A topological space
is metrizable when the topology
can be generated by some metric. By considering the closed unit ball
, we can also form a metric space
such that it can induce a metric topology
, where the metric
is also defined in the way of (
1). Let
be the norm topology for
. Then, we can also induce a topological subspace
of
, where
means
for some
. Using the routine argument, we can show that
, which also means that
is metrizable with the metric
defined in the way of (
1). The following lemma can be obtained by using the routine argument.
Lemma 1. We have the following properties.
- (i)
We have the equality and the inclusion .
- (ii)
Let Y be a topological space. Suppose that the function is -continuous on . Then, it is also -continuous on .
Lemma 2 (Aliprantis and Border [
18], p. 254).
Let be a normed space with the dual space . The closed unit ball in is weak*-metrizable if and only if U is separable. Lemma 2 says that, if the normed space
is separable, then there exists a metric
defined on
such that the compact space
can be induced by the metric
, i.e., the metric topology induced by the metric
coincides with the topology
. More precisely, the metric can be taken as
where
is a countable dense subset of the closed unit ball in
U and
. Since
for all
n, we have
which implies that
3. The Spaces of Continuous Functions
We denote by
the family of all continuous functions from the topological space
X to the topological space
Y. When
Y is taken to be a normed space, given any
and
, the vector addition
and scalar multiplication
are defined by
Then,
is a vector space. For each
, we define
We also denote by
the space of all continuous functions
satisfying
. The following results can be obtained by the routine argument.
Proposition 1. Let be a normed space. The spaces and with the normare normed spaces. We further assume that Y is a Banach space. Then, and are also Banach spaces. Let X and Y be two topological spaces. We denote by the space of all functions from into Y such that the following conditions are satisfied.
Given any fixed , the function is continuous on X.
Given any fixed , the function is left-continuous on , right-continuous at 0, and has the right limit at any .
Let
be a normed space. Given any
, we define
The following results can be obtained by the routine argument.
Proposition 2. Let be a normed space. The space with the normis a normed space. We further assume that Y is a Banach space. Then, is also a Banach space. Let X and Y be two topological spaces.
We denote by
the space of all functions
such that
F is left-continuous on
, right-continuous at 0 and has the right limit at any
.
We denote by
the space of all functions
such that
F is continuous on
.
Let
be a normed space. Given any
or
, we define
Then, we have
The following results can be obtained by the routine argument.
Proposition 3. Let be a normed space. The spaceswith the normare Banach spaces. In Proposition 1, the space X is a general topological space. When X is a compact space, we have the following useful results.
Proposition 4 (Aliprantis and Border [
18], pp. 119–120).
Let X be a compact space, and let be a metric space. Given any , we defineThen, we have the following properties.- (i)
The metric space is complete if and only if the metric space is complete.
- (ii)
Suppose that X is metrizable and Y is separable. Then, the metric space is separable.
If
, we simply write
. Then,
is the space of all continuous function
satisfying
We denote by
the space of all continuous function
such that, for all
, the set
is compact. We denote by
the space of all continuous function
such that the support
is compact.
Proposition 5 (Conway [
19] pp. 65–67).
Let X be a locally compact space. Then, we have is dense in , and that is a closed subspace of . It also says that is a Banach space. Moreover, when X is compact, we have Let
be a normed space with the dual space
. The spaces of continuous functions defined on the closed unit ball
and the unit sphere
in the dual space
will be used to study the embedding theorems for fuzzy sets. Recall
which is the space of all continuous real-valued functions
, where
is given in (
1) and
is the metric topology induced by the metric
. Let
denote the space of all
-continuous real-valued function
Proposition 1 says that
are Banach spaces.
The Alaoglu’s theorem says that
is a compact space. Assume that the normed space
is separable. Then, using Lemma 2, we have
which is the space of all continuous real-valued functions
, where
is given in (
2) such that the metric topology induced by the metric
coincides with the topology
. Let
denote the space of all
-continuous real-valued functions
Proposition 6. Let be a separable normed space. Regarding the closed unit ball in , we have thatis a separable Banach space. Proof. Proposition 1 says that
are Banach spaces. Since
is a compact metric space by Lemma 2, part (iii) of Proposition 4 says that
is separable. Part (ii) of Proposition 5 also says
This completes the proof. □
4. Support Functions of Fuzzy Sets
Let be a subfamily of . In order to embed into a normed space or Banach space. We need to guarantee that the space is closed under the addition and scalar multiplication. In other words, given any and any , we need to assure and . The following proposition provides some of these subfamilies.
Proposition 7. We have the following properties:
- (i)
Let U be a Hausdorff topological vector space. Then, the family is closed under the addition and scalar multiplication.
- (ii)
Let be a normed space. Then, the family is closed under the addition and scalar multiplication.
Moreover, given any , we havefor any and for all . Let
be a normed space. Puri and Ralescu [
1] considered the function
defined on
by
In this paper, we define
where
is considered. Then, we have
Example 2. Continued from Example 1, we assume that the functions l and u are continuous on . This means that . Therefore, we obtainwhere Proposition 8. Let be a Banach space. Then, the spaces and are complete metric spaces.
Let
be a topological vector space, and let
be the collection of all continuous linear functionals on
U. Given a nonempty subset
A of
U, the function
is called the support function of
A. Since each
is also a bounded linear functional, given any bounded set
A, we have
for any
. Given any compact set
A, the supremum in (
4) is attained.
Example 3. For , it follows that . In this case, we have . Therefore, the support function is defined on and is given byLet be a fuzzy interval. Then, the α-level set is a bounded closed interval given byIn this case, we obtainThe norm of is given by By referring to Hu and Papageorgiou [
20] and Aliprantis and Border [
18], the following basic results will be used.
Proposition 9. We have the following properties:
- (i)
Given any subsets A and B of U, we havefor any and . - (ii)
Given any compact and convex subsets A and B in U, suppose that for all with . Then, we have .
- (iii)
Given any compact and convex subsets A and B in U, we haveMoreover, we havefor any . - (iv)
Given any subset A of U, we havefor all . - (v)
Given any bounded set A in U, the support function satisfies the uniform Lipschitz condition, i.e.,for all . This also says that is continuous on .
Proposition 10. Let be a separable normed space. Given any compact set A in U, the support function is weak*-continuous on .
Proof. According to part (iii) of Proposition 9, we just need to show that
is weak*-upper semi-continuous on
, i.e., we want to show that the set
is weak*-closed for each
. Suppose that
is in the weak*-closure of
. Since
is weak*-metrizable by Lemma 2, there exists a sequence
in
satisfying
as
. Since
A is a compact subset of
U and
is continuous on
A, we have
for some
. Therefore, we can form a sequence
. Since
A is compact, i.e., sequentially compact, there exists a convergent subsequence
, i.e., there exists
satisfying
as
. We also have
as
. Since
x is weak*-continuous on
, it follows that
. Now, we have
Therefore, we obtain
as
. Since
for all
by (
7), it follows
which says
. This shows that
is weak*-closed, i.e., the support function
is indeed weak*-upper semi-continuous on
. This completes the proof. □
The concept of a support function of fuzzy sets was considered in Diamond and Kloeden [
10] and Puri and Ralescu [
6] based on the finite-dimensional Euclidean space
. In this paper, we are going to consider the support functions based on the infinite-dimensional normed space.
Definition 1. Let be a fuzzy set in a normed space , and let be the normed dual space of U. The support functionof is defined byThe norm of support function is defined by Sometimes, the support function
is restricted on
or
, where
are the closed unit ball and unit sphere in
, respectively. In this case, the norms of
are given by
respectively.
Proposition 11. Let be a normed space with the dual space , and let be a fuzzy set in U. We have the following properties.
- (i)
Given any , for , we have - (ii)
Given any , suppose that for all and all . Then, we have .
- (iii)
Given any , we have
Proof. To prove part (i), using (
3), we have
for any
. Using part (i) of Proposition 9, we also have
and
for any
and
.
To prove part (ii), suppose that for all and all . Then, we have . Since and are compact and convex sets in U for any , part (ii) of Proposition 9 says for all . This shows .
To prove part (iii), using part (iii) of Proposition 9, we have
This completes the proof. □
4.1. Continuity Regarding the Weak* Topology
Let
be a separable normed space. Lemma 2 says that the closed unit ball
is weak*-compact and weak*-metrizable with the metric
given in (
2). This metric
can induce the topology
such that
is a compact topological space. Since
is a compact subset of
with respect to the usual topology
, this means that we can endow a topology
to
such that
is a compact topological subspace of
. Using the topologies
and
, we can induce a product topology
for the product space
. Tychonoff’s theorem (ref. Royden [
21]) says that the product space
is also a compact topological space.
Now, we consider the metric spaces
where the metrics are given by
and
For the metric
on
, we can induce a metric topology
. For the metric
on
, we can also induce a metric topology
. We want to show that the metric topologies
and
are equal to the product topology
for the product space
.
The open ball
in the metric space
is given by
Also, the open balls
and
of
and
are given by
and
respectively.
Proposition 12. Let be a separable normed space with the dual space . We have the following properties:
- (i)
Given any and , we haveand - (ii)
Let be the product topology for , and let be the metric topology for induced by the metric in (9). Then, we have . This also says that the product topology is metrizable with the metric .
Proof. To prove part (i), given any
, we have
which says t
that is,
Therefore, we obtain the inclusion
On the other hand, given any
we have
Therefore, we obtain the inclusion
To prove part (ii), since
is the product topology for
, according to the concept of product topology, it means that
if and only if, given any
, there exist neighborhoods
of
and
of
satisfying
. There also exist
satisfying
Let
. Then, we have
Using part (i), we have
which says
. This shows the inclusion
. On the other hand, given any
and
, there exists
satisfying
. Using part (i), we also have
which says
. Therefore, we obtain the inclusion
, and the proof is complete. □
Suppose that we consider the metric
on
. The open ball
in the metric space
is given by
According to the similar arguments of Proposition 12, we also have the following results.
Proposition 13. Let be a separable normed space with the dual space . Then, the following statements hold true.
- (i)
Given any and , we have - (ii)
Let be the metric topology for induced by the metric in (10). Then, we have . This says that the product topology is metrizable with the metric .
The following lemmas are useful for proving the continuity of support functions.
Lemma 3. Let be a fuzzy subset of a universal set U If and , then Lemma 4 (Wu [
22], pp. 5–7).
Let be a sequence of subsets of universal set U satisfying for all n and , and let f be a real-valued function defined on U. Then, we haveand Lemma 5 (Wu [
22], pp. 5–7).
Let be a sequence of subsets of universal set U satisfying for all n and , and let f be a real-valued function defined on U. Then, we haveand Proposition 14. Let be a separable normed space. For , we have the following properties:
- (i)
The support function is -upper semi-continuous on , i.e., -upper semi-continuous on .
- (ii)
The support function is -upper semi-continuous on , i.e., -upper semi-continuous on .
- (iii)
The support function is -upper semi-continuous on .
Proof. To prove part (i), we want to show that the following set
is
-closed for each
. Given any fixed
, suppose that there exists a sequence
in
satisfying
as
. We want to show that
. Since
is a compact subset of
U, we have
for some
. Since
is compact, it is also sequentially compact. Therefore, there exists a convergent subsequence
, i.e., there exists
satisfying
as
. We also have
as
. Since
is weak*-continuous on
, it follows
as
. Now, we have
Therefore, we obtain
as
. Since
for all
j, it follows that
, where
. Therefore, we conclude
which says that
.
Suppose that
. Then, there exists a sequence
in
satisfying
Using (
9), we have
and
as
, which also says that there exists a subsequence
satisfying
or
as
.
We assume that
as
and write
for all
j. Then, we have
by Lemma 3. Using Lemma 4, we obtain
Now, using (
11) and (
12), we also obtain
which shows
, i.e.,
.
We assume
as
. Then, we have
. We write
. Since
is a compact subset of
U, we have
for some
. Since
is compact, there exists a convergent subsequence
, i.e., there exists
satisfying
as
. Using the above same arguments, we can obtain
, where
. Therefore, we have
which says
.
Therefore, we conclude that is -closed.
To prove part (ii) using (
10), we see that
implies
Therefore, we can similarly show that
is
-closed. Finally, part (iii) follows immediately from Proposition 12. This completes the proof. □
Remark 1. Let be a separable normed space. Given any , since is -upper semi-continuous on by Proposition 14 and the space is compact, it follows that the supremum is attained. Therefore, the norm of the support function is given byfor some . Since is a compact set and is a continuous linear functional, we also havefor some . Next, we present the continuity of support functions.
Proposition 15. Let be a separable normed space. Given any , we have the following properties:
- (i)
The support function is -continuous on , i.e., -continuous on .
- (ii)
The support function is -continuous on , i.e., -continuous on .
- (iii)
The support function is -continuous on .
Proof. To prove part (i), according to Proposition 14, we just need to show that
is
-lower semi-continuous, i.e., we want to show that the following set
is
-closed for each
. Given any fixed
, suppose that there exists a sequence
in
satisfying
as
. We want to claim that
. Since
is a compact subset of
U, we have
for some
. Since
is compact, there exists a convergent subsequence
, i.e., there exists
satisfying
as
. Using the same arguments of Proposition 14, we can obtain
Since
for all
j, it follows that
, where
. Since
for some
, we have
On the other hand, since
is weak*-continuous on
, we have
Therefore, using (
14), we obtain
which implies that
by (
16). Using (
15), we also obtain
which shows that
.
Suppose that
. There exists a sequence
in
satisfying
Using (
9), we have
and
as
, which also says that there exists a subsequence
satisfying
or
as
.
We assume that
as
. We write
and let
. Since
is increasing in the sense of set inclusion, using Lemma 5, we have
Using part (iv) of Proposition 9, we also have
The definition of
says
Therefore, using (
19) and (
20), we obtain
From (
18) and (
21), we also obtain
which show
.
We assume
as
. Then, we have
. We write
. Since
is a compact subset of
U, we have
for some
. Since
is compact, there exists a convergent subsequence
, i.e., there exists
satisfying
as
. Using the above same arguments, we can obtain
and
which says
.
Therefore, we conclude that is -closed.
To prove part (ii), from (
10), we have
which implies
Therefore, we can similarly show that
is
-closed. Finally, part (iii) follows from Proposition 13. This completes the proof. □
Proposition 16. Let be a normed space. We have the following properties:
- (i)
For with the support function , given any fixed , the function is lower semi-continuous and weak*-lower semi-continuous on . We also have the following properties.
- (a)
For , the function is continuous on .
- (b)
Suppose that the normed space U is separable. For , the function is weak*-continuous on .
- (ii)
For with the support function , given any fixed , the function is left-continuous on and right-continuous at 0, and has the right limitat any , where
Proof. To prove part (i), given any fixed , part (iii) of Proposition 9 says that the function is lower semi-continuous and weak*-lower semi-continuous on . For , part (v) of Proposition 9 says that the function is continuous on . Suppose that U is separable and . Proposition 10 says that the function is weak*-continuous on .
To prove part (ii), given any fixed
, let
Given any
, Lemma 3 says
for
. Using Lemma 4, we have
which shows that
f is left-continuous at
. Since
it follows that
for
. Using Lemma 5, we have
Since
, part (iv) of Proposition 9 says
which shows that
f is right-continuous at 0. Now, given any
, since
we have
Using Lemma 5, we have
which shows that
f has the right limit
at
. This completes the proof. □
4.2. Continuity Regarding the Norm Topology
Let
be a normed space with the dual space
. Suppose that
is endowed with the usual topology
. Considering the unit sphere
in
, we can form a metric space
where the metric
is defined by
This metric
can induce a metric topology
. Using the topologies
and
, we can form the product topology
for the product space
. We define a metric on
by
which can induce a metric topology
. We want to claim
.
The open ball
in the metric space
is given by
Also, the open balls
and
of
and
are given by
respectively. According to the similar argument of Proposition 12.
Proposition 17. Let be a normed space with the dual space . We have the following properties.
- (i)
Given any and , we haveand - (ii)
Let be the product topology for the product space , and let be the metric topology for with metric defined in (23). We have . This also says that the product topology is metrizable with the metric .
Suppose that we define another metric
on
by
The open ball
in the metric space
is given by
According to the similar arguments of Proposition 12, we also have the following results.
Proposition 18. Let be a normed space with the dual space . We have the following properties:
- (i)
Given any and , we have - (ii)
Let be the metric topology for with metric defined in (24). We have . This says that the product topology is metrizable with the metric .
Proposition 19. Let be a normed space with the dual space . Given any , we have the following properties:
- (i)
The support function is -continuous on , i.e., -continuous on .
- (ii)
The support function is -continuous on , i.e., -continuous on .
- (iii)
The support function is -continuous on .
Proof. The results follow from the similar proof of Proposition 15. □
5. Embedding Theorems
We denote by
the space of all continuous real-valued functions
that is a compact space. We also denote by
the space of all continuous real-valued functions
that is not necessarily a compact space.
For the spaces
we consider the norms
respectively. For the spaces
we mean that
. The following results are needed.
Proposition 20. Let be a normed space with the dual space . We have the following properties.
- (i)
The spaceis metrizable with the metrics and defined in (23) and (24), respectively. - (ii)
Suppose that the normed space is separable. Then, the compact spaceis metrizable with the metrics and defined in (9) and (10), respectively. - (iii)
We have thatconstitute a Banach space. - (iv)
Suppose that the normed space is separable. We have thatis a separable Banach space.
Proof. Part (i) follows from Propositions 17 and 18, and part (ii) follows from Propositions 12 and 13. Part (iii) follows from Proposition 1. To prove part (iv), since
is a compact space and
is separable, Propositions 4 and 5 say that
is a separable Banach space. This completes the proof. □
5.1. Embedding Theorems for the Family
Now, we consider the metric spaces
which can induce the metric topologies
and
, respectively. Propositions 12 and 13 say
Proposition 15 says that if
U is separable, then, given any
, the support function
is continuous on
. Since the space
is closed under the addition and scalar multiplication by Proposition 7, we can consider the following embedding theorem
Theorem 1 (The Embedding Theorem)
. Let be a separable normed space with the dual space . We consider the metric spaceswhereandrespectively. We define the functionwhere the support function is restricted on . Then, we have the following properties. - (i)
The function π is one-to-one and isometric in the sense ofFor , we also have - (ii)
Given any , we haveIn other words, the space is the maximal subset of satisfying
Suppose that is a separable Banach space. Then, the space can be embedded as a closed convex cone into the separable Banach spaceisometrically and isomorphically such that the space is the maximal subset of satisfyingand that the image is complete, i.e., every Cauchy sequence in converges to an element in . Proof. We first note that Proposition 20 says that
is a separable Banach space. Since
, part (ii) of Proposition 11 says that the function
is one-to-one. Using part (iii) of Proposition 11, we also obtain
which shows that
is isometric. Using part (i) of Proposition 11, we have
and
Given any
and
, Proposition 7 says
Since
it follows that
is a convex cone in
.
Now, we assume that
is a separable Banach space. Proposition 8 says that the space
is complete. Let
be a Cauchy sequence in
. Given any
, there exists an integer
satisfying
for
. Then, we have
which says that
is a Cauchy sequence in
. The completeness of
says that there exists
satisfying
as
. Therefore, we obtain
which says that the sequence
converges to
This shows the completeness of
. Next, we want to show that
is a closed subset of the separable Banach space
Given any
, there exists a sequence
in
satisfying that
converges to
, which also says that
is a Cauchy sequence in
. The completeness of
says that there exists
such that
converges to
. The uniqueness of that limit says that
. This shows that the image
is a closed set.
Assume
. We want to claim
Suppose that we have
Since
where
is the metric topology induced by the metric
. The continuity of the support function
says that, given any
, there exists
such that
implies
Since
, it means that the function
is not continuous. Therefore, there exists
such that, given any
, there exist
such that
implies
. Since
and
are compact and convex subsets of
U, part (iii) of Proposition 9 says
Proposition 10 says that the functions
and
are weak*-continuous on
, i.e.,
is weak*-continuous on
. Since
is a weak*-compact subset of
, it follows that the above supremum is attained. Therefore, there exists
satisfying
which contradicts (
26), since
This completes the proof. □
Example 4. Continued from Examples 2 and 3, we can consider the family
. Given any
, the support function
is given in (
5). Since the weak* topology and norm topology in
are identical, we have
Using (
9), we have
By referring to (
25), the embedding function
is given by
Although the family
is not a vector space, Theorem 1 says that the family
can be treated as a closed convex cone of the separable Banach space
via the embedding function
such that the distance between
and
in
is equal to the distance between
and
in the sense of
When the normed space is not separable, Proposition 19 says that, if , then the support function is continuous on . Therefore, we can also have the following embedding theorem by considering the unit sphere instead of the closed unit ball .
Theorem 2 (The Embedding Theorem)
. Let be a normed space with the dual space . We consider the metric spaceswhereandrespectively. We define the functionwhere the support function is restricted on . Then, the function π is one-to-one and isometric in the sense ofFor , we also haveSuppose that is a Banach space. Then, the space can be embedded as a closed convex cone into the Banach spaceisometrically and isomorphically such that the image is complete. Proof. The results follow from the similar proof of Theorem 1. □
Remark 2. We have some observations regarding Theorems 1 and 2.
We can also consider the embedding Theorem 2 by using the closed unit ball
. In other words, we can similarly obtain the embedding theorem by considering the following embedding function
in Theorem 2. Since
, we have the following inclusion
Lemma 1 also says that we have the inclusion
Remark 3. The inclusions shown in (29) and (30) suggest the usage of embedding theorems depending on the separability of normed space U.
Suppose that the normed space U is separable. Then, Theorem 1 is more preferable by considering the unit closed ball and the weak* topology for , since it can be embedded into a smaller space by referring to the inclusion (30). Suppose that the normed space U is not necessarily separable. Then, we must consider the strong (norm) topology for . In this case, we can consider the embedding theorems using the unit sphere given in Theorem 2 or using the closed unit ball by similarly applying the embedding functions (28) to the embedding Theorem 2, respectively. By referring to the inclusion (29), we may prefer to use the closed unit ball by taking the smaller space . However, there is a dilemma in this situation, since checking the continuity on the smaller set is easier than that of checking on the larger set . In this case, the usage of the unit sphere or the closed unit ball may depend on the problems that planned to be studied.
5.2. Embedding Theorems for the Family
We consider the embedding theorem for family
. We denote by
the space of all real-valued functions
defined on
such that the following conditions are satisfied.
Given any fixed , the function is continuous on (with respect to the strong topology).
Given any fixed , the function is left-continuous on , right-continuous at 0, and has the right limit at any .
We also denote by the space of all real-valued functions f defined on such that the following conditions are satisfied.
Given any fixed , the function is weak*-continuous on .
Given any fixed , the function is left-continuous on , right-continuous at 0, and has the right limit at any .
Proposition 2 says that the spaces
with the norms
respectively, are Banach spaces.
Each compact set in a normed space is also closed and bounded. According to Proposition 16, we can consider the following function
where the support function
is restricted on
. Assume that the normed space
is separable. Proposition 16 also says that we can consider the following function
where the support function
is restricted on
.
Theorem 3 (The Embedding Theorem)
. Let be a separable normed space with the dual space . We define a functionwhere the support function is restricted on . Then, the function π is one-to-one and isometric in the sense ofFor , we also have - (i)
The space can be embedded as a convex cone into the Banach spaceisometrically and isomorphically. - (ii)
Suppose that is a separable Banach space. Then, the space can be embedded as a closed convex cone into the Banach spaceisometrically and isomorphically such that the image is complete.
Proof. Assume that . This means that, given any fixed , we have for all , which also says for all . Part (ii) of Proposition 9 says for any , which implies . This shows that the function is one-to-one.
Using part (i) of Proposition 11, we have
and
Given any
and
, using Proposition 7, we have
Since
it shows that
is a convex cone in
. Using part (iii) of Proposition 11, we have
This shows that
is isometric.
Now, we assume that
is a separable Banach space. Part (iii) of Proposition 8 says that the space
is complete. Let
be a Cauchy sequence in
. Given any
, there exists an integer
satisfying
for
. Then, we have
which says that
is a Cauchy sequence in
. The completeness of
says that there exists
satisfying
as
. Therefore, we obtain
which says that
converges to
This proves the completeness.
Next, we want to show that is a closed subset of . Given any , there exists a sequence in such that converges to . This also says that is a Cauchy sequence in . The completeness says that there exists such that converges to . The uniqueness of the limit says that . This shows that is a closed subset of , and the proof is complete. □
When the normed space is not separable, the similar arguments of Theorem 3 can also apply to the space , where the unit sphere is considered instead of the closed unit ball . Therefore, we can obtain the following embedding theorem.
Theorem 4. (The Embedding Theorem)
. Let be a normed space with the dual space . We define a functionwhere the support function is restricted on . Then, the function π is one-to-one and isometric in the sense ofFor , we also have - (i)
The space can be embedded as a convex cone into the Banach spaceisometrically and isomorphically. - (ii)
Suppose that is a Banach space. Then, the space can be embedded as a closed convex cone into the Banach spaceisometrically and isomorphically such that the image is complete.
Theorem 3 is based on the weak* topology for and the closed unit ball , and Theorem 4 is based on the norm (strong) topology for and the unit sphere . The usage of Theorems 3 and 4 can also refer to Remark 3.
5.3. Embedding Theorems into Different Banach Spaces
Next, we consider the embedding theorems for the different kinds of Banach spaces. Let
be a separable normed space with the dual space
. Proposition 6 says that
is a separable Banach space with the norm given by
We denote by
the space of all functions
such that
F is left-continuous on
, right-continuous at 0, and has the right limit at any
.
We denote by
the space of all functions
such that
F is continuous on
.
Then, we have the following inclusion
Proposition 1 says that the spaces
with the norm
are Banach spaces.
Proposition 21. Let be a separable normed space with the dual space . Given any , the functionis left-continuous on , right-continuous at 0, and has the right limit at any , where the support function is restricted on . Proof. Let
A be a nonempty subset of
U. Given any
, we define
Let
A and
B be any two nonempty subsets of
U. We define
and
Then, we have
Let
denote the family of all nonempty subsets of
U. Given any
, we define the function
We are going to show that the function
is left-continuous on
and right-continuous at 0, and has the right limit at any
.
Given any fixed
, for
, since
, we have
. Since
decreases to
as
, using Lemmas 4 and 3, we have
Therefore, we obtain
This shows that
is left-continuous at
.
For
, since
we have
. Since
increases to
as
, using Lemmas 5 and 3, we have
Now, we have
Using (
34), we have
This shows that
is right-continuous at 0.
In general, given any
, for
, since
we have
. Since
increases to
as
, using Lemmas 5 and 3, we have
Now, we have
This shows that
has the right limit
at
.
Part (iii) of Proposition 9 says
It follows that the function
is left-continuous on
, right-continuous at 0, and has the right limit at any
. This completes the proof. □
Since the normed space
is separable, given any
, Proposition 21 says that the function
is left-continuous on
, right-continuous at 0, and has the right limit at any
, where the support function
is restricted on
. In this case, we can define a function
Then, we can have the following embedding theorem.
Theorem 5 (The Embedding Theorem)
. Let be a separable normed space with the dual space . We define a functionThen, the function π is one-to-one and isometric in the sense ofFor , we also have - (i)
The space can be embedded as a convex cone into the Banach spaceisometrically and isomorphically. - (ii)
Suppose that is a separable Banach space. Then, the space can be embedded as a closed convex cone into the Banach spaceisometrically and isomorphically such that the image is complete.
Proof. Assume that
. This means that
for all
, i.e.,
Using part (ii) of Proposition 9, we have
for all
, which implies that
. This says that the function
is one-to-one.
Using part (i) of Proposition 9, for
, we have
and
Therefore, we obtain
and
Given any
and
, Proposition 7 says
Since
it follows that
is a convex cone in
. From part (iii) of Proposition 9, we obtain
This shows that
is isometric.
Now, we assume that
is a separable Banach space. Part (iii) of Proposition 8 says that the space
is complete. Let
be a Cauchy sequence in
. Given any
, there exists an integer
satisfying
for
. Then, we have
which says that
is a Cauchy sequence in
. The completeness of
says that there exists
satisfying
as
. Therefore, we obtain
which says that the sequence
converges to
This proves the completeness.
Next, we want to show that is a closed subset of . Given any , there exists a sequence in such that converges to . This also says that is a Cauchy sequence in . Therefore, there exists such that the sequence converges to . The uniqueness of the limit says . This shows that is a closed subset of . This completes the proof. □
Next, we are going to consider the embedding theorem of the family
into the Banach space
Proposition 22. Let be a separable normed space with the dual space . Given any , the functionis continuous on . Proof. Since
, we have
Given any
, using (
34) and (
36), we have
This shows that
is right-continuous at any
. Using Proposition 22, we conclude that
is continuous at
. This completes the proof. □
Given any
, Proposition 22 says that the function
is continuous on
, where the support function
is restricted on
. In this case, we can define a function
Suppose that
is a Banach space. Since
is complete by Proposition 8, from the same arguments of Theorem 5 by considering the space
that is smaller than the space
, we can also have the following embedding theorem.
Theorem 6 (The Embedding Theorem)
. Let be a separable normed space with the dual space . We define a functionThen, the function π is one-to-one and isometric in the sense ofFor , we also have - (i)
The space can be embedded as a convex cone into the Banach spaceisometrically and isomorphically. - (ii)
Suppose that is a Banach space. Then, the space can be embedded as a closed convex cone into the Banach spaceisometrically and isomorphically such that the image is complete.
Example 5. Continued from Examples 4, Theorem 6 says that the family can be embedded into the Banach spacevia the embedding function π given byGiven an open interval , we consider the following fuzzy functionwhich says that each function value is a fuzzy interval for any . In this case, we say that the fuzzy function is differentiable at when the function is Fréchet differentiable at , which means that the functionis differentiable at . This means that there exists a linear bounded functionsatisfyingTherefore, we conclude that the fuzzy function is differentiable at when there exists a linear bounded function given in (37) satisfyingwhere 6. Conclusions
Let be a normed space with the dual space . Many embedding theorems have been presented in this paper to embed the families and into the different Banach space of continuous functions. Let be a normed space with the dual space . Recall that the closed unit ball is a weak* compact subset of ; that is, the closed unit ball is a compact subset of with respect to the weak* topology , which can induce a topology for such that is a compact topological subspace of . Since is a compact subset of with respect to the usual topology for , the Tychonoff’s theorem says that the topology for and the topology for can induce a product topology for the product space such that is a compact topological space. When the normed space is separable, Lemmas 12 and 13 say that the compact space is metrizable, which is also a Hausdorff and locally compact space. The embedding properties are summarized below.
Consider the embedding theorems of the space . Given any , this means that the -level sets are compact and convex sets in U for all , and that the function is continuous with respect to the Hausdorff metric . The separability of the normed space is an important issue.
Suppose that
is a Banach space. Theorem 6 says that the space
can be embedded as a closed convex cone into the Banach space
isometrically and isomorphically such that the image
is complete.
Suppose that
is a Banach space. Theorem 2 says that the space
can be embedded as a closed convex cone into the Banach space
isometrically and isomorphically such that the image
is complete. We can have the same embedding results when the unit sphere
is replaced by the closed unit ball
.
Suppose that
is a separable Banach space. Theorem 1 says that the space
can be embedded as a closed convex cone into the separable Banach space
isometrically and isomorphically such that the space
is the maximal subset of
satisfying
and that the image
is complete, i.e., every Cauchy sequence in
converges to an element in
.
Since
, we have the following inclusion
Lemma 1 also says that we have the inclusion
The above inclusions suggest the usage of embedding Theorems 1 and 2 depending on the separability of normed space
U, which is presented in Remark 3.
Consider the embedding theorems of the space . Given any , it means that the -level sets are compact and convex sets in U for all . The separability of the normed space is an important issue.
Suppose that
is a Banach space. Theorem 4 says that the space
can be embedded as a closed convex cone into the Banach space
isometrically and isomorphically such that the image
is complete. We can have the same embedding results when the unit sphere
is replaced by the closed unit ball
.
Suppose that
is a separable Banach space. Theorem 3 says that the space
can be embedded as a closed convex cone into the Banach space
isometrically and isomorphically such that the image
is complete.
Suppose that
is a separable Banach space. Theorem 5 says that the space
can be embedded as a closed convex cone into the Banach space
isometrically and isomorphically such that the image
is complete.
Theorem 3 is based on the weak* topology for and the closed unit ball , and Theorem 4 is based on the norm (strong) topology for and the unit sphere . The usage of Theorems 3 and 4 can also refer to Remark 3. In the future research, we try to use these embedding theorems to study the practical problems that involve fuzzy uncertainty.