Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces

It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are dened on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we rst developed the algebraic structure of the class of fuzzy sets F(R) and gave denitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. en, with special norms, namely, ‖u‖q (∫ 1 0 (supx∈[u]α‖x‖) qdα)1/q where 1≤ q≤∞, we stated that (F(R), ‖u‖q) is a complete normed space. Furthermore, we introduced an inner product in this space for the case q 2.e inner product must be in the form〈u, v〉 ∫10〈[u] α, [v]〉K(Rn)dα ∫ 1 0 〈a, b〉Rndα : a ∈ [u] α, b ∈ [v]α { }. For u, v ∈ F(R). We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F(R). Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.


Introduction
Investigation of fuzzy sets was given by Zadeh [1] and then notions of fuzzy number, fuzzy metric, fuzzy norm, and their applications have been introduced by several authors. For example, Katsaras [2] rst introduced the notions of fuzzy seminorm and norm on a vector space. Independently, Felbin [3] gave the concept of fuzzy normed space (brie y, FNS) by applying the notion of fuzzy distance of Kaleva and Seikkala [4] on vector spaces. Xiao and Zhu [5] improved Felbin's de nition of the fuzzy norm of a linear operator between FNSs [6]. e notion of fuzzy quasilinear space is introduced in [6] depending on the notion of quasilinear space which was de ned by Aseev in [7].
Aseev rst introduced the concept of quasilinear space which allows us to investigate both linear spaces and some nonlinear spaces such as special classes of sets in some Banach spaces, such as special classes of multivalued mappings and fuzzy sets. He followed a similar way to methods in linear algebra and in functional analysis. Furthermore, he presented some results which are "quasilinear" counterparts of fundamental de nitions and theorems in linear functional analysis and di erential calculus in Banach spaces. is pioneering work has motivated a lot of authors to introduce new results on multivalued mappings, fuzzy quasilinear spaces and operators, and set-valued analysis [8][9][10][11][12][13][14]. In this way, Rojas Medar et al. [6] introduced the concept of fuzzy quasilinear spaces and de ned the notion of a norm on these spaces.
In this study, we rst constructed the algebraic structure of the class of fuzzy sets F(R n ) and gave de nitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. en, with special norms, e. g., u q ( 1 0 (sup x∈[u] α x ) q dα) 1/q where 1 ≤ q ≤ ∞, we stated that (F(R n ), u q ) is a complete normed space. Furthermore, we introduced an inner product in this space for the case q 2.
e inner product must be in the following form: for u, v ∈ F(R n ). Furthermore, we showed that the inner product norm is just We also prove that the parallelogram law can only be provided in the regular subspace, not in the entire of F(R n ). Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.

Preliminaries and Some New Results
In the universe of R n of generalization of fuzzy numbers defined by F(R n ) � u: R n ⟶ { [0, 1]: u satisfies (i)-(iv) below}, where (i) u is normal, that is, there exists an x 0 ∈ R n such that u(x 0 ) � 1 (ii) u is fuzzy convex, that is, for x, y ∈ R n and 0 ≤ λ ≤ 1, u(λx + (1 − λ)y) ≥ min[u(x), u(y)] (iii) u is upper semicontinuous (iv) e closure of x ∈ R n : u(x) > 0 { }, denoted by [u] 0 is compact.
For 0 < α ≤ 1, the α-level set [u] α is defined by [u] α � x ∈ R n : u(x) ≥ α { }. en, from (i) to (iv), it follows that the α -level set [u] α ∈ K C (R n ) and [u] 0 � ∪ α∈ (0,1] [u] α where K C (R n ) denote the family of all nonempty, compact, convex subsets of R n . e set of fuzzy numbers forms an important algebraic structure called quasilinear space. Now, let us give the definition of quasilinear space.
A set X is called a quasilinear space (briefly, QLS) [7], on the field R, if a partial order relation " ⪯ ," an algebraic sum operation, and an operation of multiplication by real numbers are defined in it in such a way that the following conditions hold for all elements x, y, z, v ∈ X and for all α, β ∈ R: x � y if x ⪯ y and y ⪯ x (Q4) x + y � y + x (Q5) x + (y + z) � (x + y) + z (Q6) there exists an element (zero) θ ∈ X such that Any linear space is a QLS with the partial order relation e inverse is unique whenever it exists. An element x possessing an inverse is called a regular, otherwise it is called a singular. Suppose that each element x in a QLS X has an inverse element x ′ ∈ X. en, the partial order in X is determined by equality, the distributivity conditions hold, and consequently X is a linear space [7]. It will be assumed in what follows that − x � (− 1) · x and sometimes − x may not be the inverse of x. An element x has an inverse element x ′ if and only if x ′ � − x. Suppose that X is a QLS and Y⊆X. en, Y is called a subspace of X whenever Y is a QLS with the same partial order and with the restriction of the operations on X to Y. Furthermore, we saw that Y is a subspace of a QLS X if and only if for every x, y ∈ Y and α, β ∈ R, α · x + β · y ∈ Y, [11]. Now let us denote by X r the set of all regular elements and by X s the sets of all singular elements and the zero element in X, respectively. Furthermore, it can be easily shown that X r and X s are subspaces of X.
ey are called regular and singular subspaces of X, respectively. Furthermore, it is not hard to prove the fact that the summation of a regular element with a singular element results in a singular element, and the regular subspace of X is a linear space while the singular subspace is not.
K(R n ), the set of all compact subsets of R n is also quasilinear space with the inclusion and K C (R n ) is a sub- e distance between A and B is defined by the Hausdorff metric.
It is well known that K C (R n ) is complete with this metric. Furthermore, K C (R n ) is a quasilinear space with the inclusion relation. After the introduction of normed quasilinear spaces, we will say that this metric comes from a norm.
Definition 1 (see [7]). Let X be a QLS. A real function ‖ · ‖ X : X ⟶ R is called a norm if the following conditions hold: if for any ε > 0 there exists an element x ε ∈ X such that x ⪯ y + x ε and ‖x ε ‖ X ≤ ε then x ⪯ y A quasilinear space X with a norm defined on it is called normed quasilinear space, briefly, normed QLS. It follows from [7] that if any x ∈ X has an inverse element x ′ ∈ X, then the concept of normed QLS coincides with the concept of real normed linear space. e Hausdorff metric or norm metric on X is defined by the equality.
Since x ⪯ y + (x − y) and y ⪯ x + (y − x), the quantity h X (x, y) is well-defined for any elements x, y ∈ X, and it is not hard to see that the function h X satisfies all the metric axioms [7]. Also, we should note that h X (x, y) may not equal to ‖x − y‖ X if X is not a linear space. However, h X (x, y) ≤ ‖x − y‖ X for every x, y ∈ X.
We should note the following useful properties of the Hausdorff metric which are given by Aseev [7]. e operations of algebraic sum and multiplication by real numbers are continuous with respect to the Hausdorff metric. e norm is continuous and the following properties are satisfied: (a) Suppose that x n ⟶ x 0 and y n ⟶ y 0 , and that x n ⪯ y n for any positive integer n. en, x 0 ⪯ y 0 . (b) Suppose that x n ⟶ x 0 and z n ⟶ x 0 . If x n ⪯ y n ⪯ z n for any n, then y n ⟶ x 0 . (c) Suppose that x n + y n ⟶ x 0 and y n ⟶ θ.
Furthermore, for each α ∈ R and for every x, y, u, v ∈ X, is the family of all nonempty compact subsets of R n , then ‖A‖ � sup a∈A ‖a‖ defines a norm on these quasilinear spaces. In this case the Hausdorff metric is defined by where S r (x) � y ∈ R n : ‖x − y‖ ≤ r .
Let u, v ∈ F(R n ) and α ∈ (0, 1]. e algebraic sum and scalar multiplication by a real number λ ∈ R are defined by  [6,15], the relation is obtained. is a partial order on F(R n ) and hence it is a quasilinear space [6] with the zero element χ 0 { } . Now let us determine two important subspace of F(R n ).

Theorem 1. All regular elements in
for any u ∈ F(R n ), and so, Hence, for x ≠ 0, (u − u)(x) � sup y∈X min u(y), u(y− x)} � 0meaning that min u(y), u(y − x) � 0. We know that y − x ≠ y for x ≠ 0. erefore, for c ∈ (0, 1] and y ∈ R n we can consider the fuzzy set u defined by the following: Moreover, for x � 0, (u − u)(0) � sup y∈R min u(y), u(y)} � sup y∈R u(y) � c. By the definition of χ 0 { } for x � 0, (u − u)(0) � 1 and so c must be 1. us, there exists an inverse of u ∈ F(R n ) if and only if u must be defined by is contradicts with the assumption u ≠ χ a { } . Hence, the regular subspace And [v] α is compact convex for all α ∈ [0, 1] in R. e function v is given in [1]. Let us try to find the inverse of v.
We has seen that two important subspaces of F(R n ) are the regular subspace F(R n ) s and the singular subspace F(R n ) r . ey are only intersect at the zero χ 0 { } . Another important subspace of F(R n ) is the symmetric subspace F(R n ) sym . An element x in a QLS X is called symmetric whenever x � − x, and X sym denotes the set of all symmetric elements. It is a subspace of X s and hence of X. For example,

On the Algebra of the Fuzzy Sets
In this section, we will establish the necessary infrastructure to define the concepts of base and dimension in fuzzy quasilinear spaces. In this context, the dimension of a fuzzy quasilinear space can be expressed as a binary number (a, b) where a and b are natural numbers [10].
Let u k m k�1 be a subset of F(R n ) where m ≤ n and m is a positive integer. A (linear) combination of the set u k m k�1 is an element z of F(R n ) in the form α 1 u 1 + α 2 u 2 + · · · + α m u m � z where the coefficients α 1 , α 2 , . . . , α m are real scalars. On the other hand, a quasilinear combination of the set u k m k�1 is an element z ∈ F(R n ) such that α 1 · u 1 + α 2 · u 2 + · · · + α m · u m ⪯ z for some real scalars α 1 , α 2 , . . . , α m . Hence, the quasilinear combination, briefly ql-combination, is defined by the partial order relation on F(R n ). In fact, the definition of linear combination in F(R n ) is also depended on the partial order relation, and it can be defined as in the following expanded form: a linear combination of the set u k m k�1 is an element z of F(R n ) such that where the coefficients α 1 , α 2 , . . . , α m are real scalars. Clearly, a linear combination of u k m k�1 , is a quasilinear combination of u k m k�1 , but not conversely. In quasilinear spaces, there are two kinds of combinations of the set u k m k�1 , namely, linear combination and quasilinear combination. If our quasilinear space is a linear space, we do not encounter a combination called a quasilinear combination. According to the definition of quasilinear combination, linear combination of u k m k�1 corresponding to α k m k�1 is unique but quasilinear combination is not unique. For any nonempty subset A of F(R n ), span of A is given by following known definition: However, QspA, the quasispan (q-span, for short) of A, is defined by the set of all possible quasilinear combinations of A, that is, Obviously, SpA⊆QspA. Furthermore, SpA � QspA for a linear QLS (linear space), hence, the notion of QspA is redundant in linear spaces. Moreover, we say A quasispans F(R n ) whenever QspA � F(R n ).

Let us give an example.
Example 2. Take A � χ [1,5] , a singleton in F(R). e q-span of A is 4 Journal of Mathematics is means, for example, χ [2,13] ∈ Q spA since 2 · χ [1,5] ⪯ χ [2,13] . However, χ [2,13] ∉ SpA since there is no λ ∈ R satisfying λ · χ [1,5] � χ [2,13] . Note that Furthermore, χ [2,3] ∉ QspA since we cannot find any λ ∈ R satisfying the condition λ · χ [1,5] ⪯ χ [2,3] . Hence, QspA ≠ F(R) and we say in this case A or the element χ [1,5] cannot q-span F(R). Let us consider another singleton of a regular element in is example has given us important clues as to how we can define the concept of the dimension of the quasilinear space of fuzzy sets. Let us give a result that we can easily see the proof: en, QspA is a subspace of F(R n ).
If we recall again that every linear space is a QLS with the relation " � ," it can be seen that the notions of quasilinear independence and dependence coincide with linear independence and dependence, respectively. e set A in the previous example is ql-independent since χ 0 is is an unusual case since a nonzero singleton is obviously linearly independent in linear space. On the other hand, the set χ [2,3] , χ [− 1,2] is ql-dependent. In general, we can see from the definition that any subset including an element related to is is a generalization of the well-known result: a subset including zero must be linearly dependent in linear spaces.

Example 3. Let us consider two dimensional vector space
where e 1 � (1, 0) and e 2 � (0, 1) are unit basis vectors in two dimensional vector space R 2 . en, v 1 and v 2 are horizontal and vertical bars of equal length intersecting at zero in R 2 . en, for On the other hand, let us consider u � (t, s): and similarly we can write the basis of F(R n ). is shows that the standard basis of F(R n ) is just the copy of those of R n . e following example is extraordinary since it presents an example of QLS which has no basis. is is an unusual case since all linear spaces have a (Hamel) basis.
Λ has no basis. Because, Λ is the set of all characteristic functions of nondegenerate real intervals and we cannot find any subset quasispanning Λ. Now let us introduce the notion of dimension of F(R n ). Our investigation shows that it is necessary to split it into two different notions as regular and singular dimension. Previously, let us give a classical definition. Let S be a qlindependent subset of F(R n ). S is called maximal ql-independent subset of F(R n ) providing that S is ql-independent but any superset V S is ql-dependent.
Definition 4. Let X be any quasilinear space. Regular (singular) dimension of F(R n ) is the cardinality of any maximal ql-independent subsets of X r (X s ). If this number is finite, then X is said to be finite regular (singular)-dimensional, otherwise it is said to be infinite regular (singular)dimensional. Regular dimension is denoted by r − dimX and singular dimension is denoted by s-dimX. If r − dimX � a and s − dimX � b then we say that X is an (a, b)-dimensional where a and b are natural numbers or ∞. e above mentioned definition means that r − dimX is just the dimension of the linear subspace X r of X and so r-dim X � dimX r . Notice that a nontrivial singular subspace of a quasilinear space cannot be a linear space. Furthermore, we can see that any quasilinear space is (n, 0)-dimensional if and only if it is n-dimensional linear space. In this respect, the trivial linear subspace χ 0 Proof. For the set e 1 , e 2 , . . . , e n , the standard basis of the linear space R n , is just the basis of the linear space (F(R n )) r . is means r − dimF(R n ) � n. Now let us consider bars.
en, the set of its α-levels 1] in the quasilinear space (K C (R n )) s of singular subspace of all compact convex subsets of R n . is means if then, λ 1 � λ 2 � . . . � λ n+1 � 0. None of the set [x k ] α can contain the zero element of the vector space R n . Otherwise, its ql-independence would be broken. By the abovementioned assumption, we can find a 1 ∈ [x 1 ] α , a 2 ∈ [x 2 ] α , . . . , a n+1 ∈ [x n+1 ] α such that 0 � λ 1 a 1 +λ 2 a 2 + . . . +λ n+1 a n+1 . We conclude the set a 1 , a 2 , . . . , a n+1 is a linearly independent subset of R n since In this respect, all n -dimensional vector spaces on the field real numbers are (n, 0) -dimensional quasilinear space.
If X � K C (R 2 ) s ∪ (x, y) : y � 0, x ∈ R then X is a subspace of K C (R 2 ), the quasilinear space of all compact convex subsets of R 2 with the inclusion relation, and r − dimX � 1 since X r � (x, y) : y � 0, x ∈ R . Furthermore, the set v 1 , v 2 in Example 3 is ql-independent and there is Consider the QLSX � K C (c 0 ), the quasilinear space of all bounded closed convex subsets of c 0 , the set of all sequences convergent to zero. X r is equivalent to c 0 and so r − dimX � ∞. Let us define the set is ql-independent in X s , where e k 's are coordinate vectors of c 0 , k � 1, 2, . . . erefore, s − dimX � ∞ and so X � K C (c 0 ) is an (∞: ∞)-dimensional QLS. In general, an infinite-dimensional linear space E is an (∞, 0)− dimensional QLS.
Definition 5. Let X be a QLS and y ∈ X. e set of all regular elements preceding from y is called floor of y, and F y denotes the set of all such elements. erefore, F y � x ∈ X r : x ⪯ y . e floor of any subset M of X is the union of floors of all elements in M and is denoted by F M .
Hence, F M � ∪ y∈M F y and it is clear that F x � x { } for some x ∈ X r . is means that F X r � X r , and so, the notion of floor is redundant in linear spaces. Furthermore, F X s is a subspace of the linear space X r in the QLS X.
Example 6. Consider (K C (R)) sym � [− a, a]: a ∈ R { }, the symmetric subspace of K C (R). It is interesting that (K C (R)) sym is a (0, 0) -dimensional QLS, just similar to the trivial linear space.

Norm and Inner Product on F(R n )
In [6], it was shown that fuzzy quasilinear spaces have a normed QLS structure with various norms. In this study, we will say that F(R n ) is a complete normed QLS structure with these special norms. Furthermore, in the next section, we will show that a particular one of these norms comes from an inner product. For 1 ≤ q < ∞, the function defines a metric on F(R n ) and it is a complete metric space by this metric. Moreover, this metric comes from the norm.
Let us give and prove the following theorem.
Proof. Consider the function g: [0, 1] ⟶ R, g(α) � sup x∈[u] α ‖x‖. en, from the compactness of each α -level set in R n , sup x∈[u] α ‖x‖ exists, and from the properties of fuzzy sets the required integral also exists. Hence, the norm is welldefined. Now, let us verify the norm axioms: (3) If u ∈ F(R n ) and λ ∈ R, then such that u ⪯ v + w ε and w ε ≤ ϵ for any ε > 0. en, there exists an element erefore, F(R n ) is a normed quasilinear space with norm ‖u‖ q .
(F(R n ), ‖ · ‖ q ) is a complete with ‖u‖ q � D q (u, θ) since K C (R n ) is complete. Hausdorff metric for this norm can be computed by the following formula: We should note here that the norm metric (Hausdorff metric) cannot be obtained by the way ‖u − v‖ q � D q (u, v). Only we can write D q (u, v) ≤ ‖u − v‖ q .
After this stage, we will show that the norm ‖u‖ q is an inner product norm for q � 2. Now, in order to describe this inner product, let us present some of the concepts about Journal of Mathematics normed quasilinear spaces that we have obtained earlier [8,9,16]. □ Definition 6 (see [8]). X is called consolidate (solid-floored) QLS whenever y � sup x ∈ X r x ⪯ y for each y ∈ X. Otherwise, X is called a nonconsolidate QLS, briefly, nc-QLS. e supremum in this definition is taken on the order relation " ⪯ " in the definition of a QLS. Abovementioned definition assumes sup x ∈ X r : x ⪯ y exists for each y ∈ X. Implicitly, we say that X is consolidate if and only if y � supF y , for each y ∈ X.
Let us first note that any linear space is a consolidate QLS. Indeed, X r � X for any linear space X. So, for any element y ∈ X.

Theorem 5. F(R n ) is a consolidate QLS, that is, for each
is shows y ≠ supF y � χ 0 { } . It may not be possible to perform many important functions in nonconsolidated quasilinear spaces. In order to eliminate these negative situations, we have introduced a new definition under the name of consolidation of quasilinear spaces. e consolidation concept we have given is unnecessary for linear spaces because every linear space is consolidated. If we did not have this concept, it would not be possible for us to define the concept of the inner-product for some important quasilinear spaces. Let us give a definition.
Definition 7. For some two quasilinear spaces (X, ≤ ) and (Y, ⪯ ), we say Y compatible contains X whenever X⊆Y and the partial order relation " ≤ " on X is the restriction of the partial order relation ⪯ on Y. We briefly use the symbol X⊆Y in this case. We write X ≡ Y whenever X⊆Y and Y⊆XX ≡ Y means X and Y are the same sets with the same partial order relations which make them a quasilinear space. We may write X � Y for X ≡ Y whenever the relations are clear from the context.
Consolidation of X is the smallest consolidate QLS X which compatibly contains X, that is, if there exists another consolidate QLS Y which compatibly contains X then X⊆Y.
Clearly, for some consolidate QLS X, X � X. We do not yet know whether every quasilinear space has a consolidation or not. Furthermore, each linear space is a consolidate QLS. Now, let us show K C (R) s � K C (R).
Proof. Obviously, K C (R) compatible contains (K C (R)) s . Suppose that Z is another consolidate QLS which contains (K C (R)) s . For an arbitrary element x of K C (R) we will show that x ∈ Z. If x ∈ (K C (R)) s , then the proof is clear. If for any ε > 0. is means there exists an element u ε ∈ Z r such that u ε ⊆[a − ε, a + ε] in Z. erefore, we have a { } ∈ Z r . Otherwise, the set [a − ε, a + ε] cannot be a closed set. So, this conflicts with the fact that [a − ε, a + ε] is an element of (K C (R)) s . us, the assumption a { } ∉ Z is incorrect. Similarly, one can see that K C (R) sym � K C (R). We can also prove that F(R n ) s � F(R n ) and FR n ) sym � F(R n ) with a slightly more difficult proof technique.
For any element y of a QLS X, the set F X y � z ∈ (X) r : z ⪯ y denotes the floor of y in X and sometimes F X y is said to be the floor of y in the consolidation. For a consolidate QLS, this notion is unnecessary. But, this concept is important in a nonconsolidate QLS.  5] . Now, let us give the definition of inner product that we previously define on quasilinear spaces [8,17]. 8 Journal of Mathematics Definition 9. Let X be a quasilinear space having a consolidation X. A mapping, : X × X ⟶ K(R) is called an inner product on X if for any x, y, z ∈ X and α ∈ R the following conditions are satisfied: (IPQ1) If x, y ∈ X r then 〈x, y〉 ∈ K(R) r ≡ R (IPQ2) 〈x + y, z〉⊆〈x, z〉 + 〈y, z〉 x ⪯ y and u ⪯ v then 〈x, u〉⊆〈y, v〉 (IPQ8) if for any ε > 0 there exists an element x ε ∈ X such that x ⪯ y + x ε and 〈x ε , x ε 〉⊆S ε (θ) then x ⪯ y where F X x denotes the floor of x in the consolidation X of X. A quasilinear space with an inner product is called an inner product quasilinear space, briefly, IPQLS [8]. An IPQLS is called a Hilbert QLS whenever it is complete with the following inner product metric. Every IPQLS X is a normed QLS with the norm defined by ‖x‖ � ���������� � ‖〈x, x〉‖ K(R) for every x ∈ X. is norm is called inner product norm. e inner-product metric is obtained by the following formula: For A, B ∈ K C (R) or K(R), 〈A, B〉 � ab: a ∈ A, b ∈ B { } is an inner product and they are Hilbert quasilinear space by this inner product norm. Furthermore, if x n ⟶ x and y n ⟶ y in a IPQLS, then 〈x n , y n 〉 ⟶ 〈x, y〉 [9].

Theorem 7.
For some A, B ∈ K(R n ), 〈A, B〉 K(R n ) � 〈a, b〉 R n : a ∈ A, b ∈ B defines an inner product on K(R n ). Furthermore, K(R n ) is a Hilbert quasilinear space by this inner product norm [9]. Definition 10 (see [8]). An element x of an IPQLS X is said to be orthogonal to an element y ∈ X if ‖〈x, y〉‖ K(R) � 0. We also say that x and y are orthogonal, and we write x⊥y. An orthonormal set M ⊂ X is an orthogonal set in X whose elements have norm 1.
Definition 11 (see [8]). Let A be a nonempty subset of an inner product quasilinear space X. An element x ∈ X is said to be orthogonal to A, denoted by x⊥A, if ‖〈x, y〉‖ K(R) � 0 for every y ∈ A. e set of all elements of X orthogonal to A, denoted by A ⊥ , is called the orthogonal complement of A and is indicated by Theorem 8. Let X be a quasilinear space having a consolidation X.
You can see the proof of this theorem in cites [8,9,16].

The Inner Product on F(R n )
Now, let us give some basic concepts which are used to define an inner-product on F(R n ). Let B(R k ) and B(K C (R n )) denote the σ− algebras of Borel subsets of R k and K C (R n ), respectively, where k, n are positive integers. Let F: T ⟶ K C (R n ) be a function and T⊆R k where we use again the symbol big F for a set-valued function. If for any

Proposition 1.
[15] e following results are equivalent: for every closed subset C of R n (v) e function d(x, F(·)): T ⟶ R is measurable for every x ∈ R n (vi) e function ‖F(·)‖: T ⟶ R is measurable (vii) e function s(x, F(·)): T ⟶ R is measurable for every x ∈ R n . e statements (v)-(vii) explain the measurability of the single-valued mapping defined from T to R regarding to the Borel σ -algebras B(R k ) and B(R). If these functions are continuous, then such mappings are measurable. For some set-valued mappings we have the following informations [15]. Any function F: T ⟶ K C (R n ) is measurable if it is upper semicontinuous or lower semicontinuos and hence if it is continuous. Let F i : T ⟶ K C (R n ) be a sequence of measurable functions for i � 1, 2, . . . and suppose that lim t⟶∞ D(F i (t), F(t)) � 0 for every t ∈ T. en, the limit function F: T ⟶ K C (R n ) is also measurable. A selector of a set-valued mapping F: T ⟶ K C (R n ) is a single-valued mapping f: T ⟶ R n such that f(t) ∈ F(T) for every t ∈ T, [15]. If the function F is measurable, then it's all selector functions are also measurable. e following theorem, known as the "Castaing Representation eorem" gives an additive characterization of the measurability of a set-valued mapping.
Theorem 9 (see [15]). e function F: T ⟶ K C (R n ) is measurable if and only if there exist a sequence f i of measurable function F such that for each t ∈ Tf

Journal of Mathematics
For any lower semicontinuous function F: T ⟶ K C (R n ) there exist a continuous selector f of F such that f(t) � x for every x ∈ F(t) and t ∈ T. Namely, if a set-valued mapping is lower semicontinuous, then it has a continuous selector. Now, let us give the definition of "Aumann Integral" [15]. Suppose that F: Theorem 10 (see [15]). If F: For a set-valued mapping F as in eorem 10, the Castaing Representation eorem 9 applies and provides a sequence f i of integrable selectors which are pointwise dence in F. Moreover, and so we need only consider these selectors to evaluate 1 0 F(t)dt.
Theorem 11 (see [15]). e Aumann integral satisfies the following properties for all Aumann integrable functions F, G: In addition, the Aumann integral uniquely determines its integrand.

a measurable and integrably bounded then it is Aumann integrable
Let us give a main result.

Theorem 13. F(R n ) is an inner product quasilinear space by the function
for u, v ∈ F(R n ).
Proof. First, we must show that the equality (5.2) is welldefined. We know from the cites [8,9,16], the function 〈, 〉 K(R n ) is an inner product on the QLS K(R n ), and we proved that an inner product is a continuous function. is means it can be Aumann integrable on [0, 1]. Hence, the function 〈, 〉 is well-defined. Now, let us verify the inner-product axioms: (2) From eorem 11, we write (3) If we use eorem 11, then (42) Let us remember that R n is a linear inner product space. erefore, we say that 〈a, a〉 R n ≥ 0. So, 〈u, u〉 ≥ 0 since 〈u, u〉 ∈ (K(R)) r ≡ R for u ∈ F(R n ) r . (6) We shall prove that ‖〈u, From eorem 5 and from the definition of the Aumann integral we can write where the supremum is taken over the partial order relation ⊆ on K(R). us, we have (7) Suppose that u 1 ≤ v 1 and u 2 ≤ v 2 . en, and by eorem 11 (8) We suppose that there exists an element for any ε > 0. en, there exists an element en, we get [u] α ⊆[v] α for any α ∈ [0, 1] by eorem 7.
is means that u ≤ v.
Now, let us determine the inner product norm. For u ∈ F(R n )

Journal of Mathematics
(49) □ Example 9. Let us consider the set A � χ v 1 , χ v 2 , . . . , χ v n in the proof of eorem 2. For m ≠ r and m, r ≤ n From eorem 7. Hence, A is an orthogonal subset of F(R n ). It cannot quasispan F(R n ).
Consider the standard basis e 1 , e 2 , . . . , e n of R n . en, Furthermore, ere exists a copy of R n in F(R n ), and this copy is just the regular subspace of F(R n ). In order for a set to be a basis of F(R n ), all its elements must first be regular elements. None of the elements of A are regular, whereas all elements of B are regular.
A striking feature of F(R n ) is that some element pairs may not be able to satisfy the parallelogram law. Especially, in the singular subspace of F(R n ), this law may not be valid. Let us give a striking example that proves this situation. Now, And so, (57) Hence, the parallelogram law is not valid in F(R n ).
In this example u � v are singular elements of F(R n ). Let us say that the parallelogram law is valid in the regular subspace of F(R n ).

Theorem 14.
e parallelogram law is satisfied on the regular subspace F(R n ) r , which is a Hilbert (linear) subspace of F(R n ).
Let us give another important result: the Schwarz inequality.

Classical Fuzzy Sequence Spaces
A sequence U � (u k ) ∞ k�1 of fuzzy numbers is a function U from positive integers into F(R). e fuzzy number u k is called the k-th term of the sequence. Let U �(u k ) be a sequence of fuzzy numbers. e sequence U of fuzzy numbers is said to be bounded if the set ‖u k ‖ is bounded in R where ‖u k ‖ � ( 1 0 (sup a∈[u k ] α ‖a‖) 2 dα) 1/2 . Furthermore, U � (u k ) ∞ k�1 is said to be convergent to some u ∈ F(R) whenever where D 2 is the inner-product metric which comes from the above inner-product norm ‖.‖. In this case, we write u k ⟶ u or limu k � u in F(R) Note that ‖u k − u‖ ⟶ 0 implies u k ⟶ u but not conversely, in general. For example, the constant sequence u k � χ [0,1] , for each k ∈ N, is convergent to u � χ[0, 1]. But  More details about sequence spaces of fuzzy numbers are in [18][19][20][21][22][23] and references therein. Now, ℓ F ∞ and c F 0 denote the set of all bounded and convergent (to zero) sequences of fuzzy numbers, respectively. Hence, Furthermore, for 1 ≤ p < ∞, is the set of all p th order absolute summable sequences of fuzzy numbers. Furthermore, the set of all fuzzy numbers are denoted by w F and so w F � U � (u k ) ∞ k�1 : u k ∈ F(R) .

Theorem 16.
By the coordinate wise operations and by the coordinate wise relation of fuzzy numbers, w F is a consolidate quasilinear spaces.
1 0 (sup x∈[u] α ‖x‖) q dα) 1/q where 1 ≤ q ≤ ∞. After that, we introduced an inner product on this space for the case q � 2. We proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F(R n ). Finally, we obtained a special class of fuzzy number sequences that is a Hilbert quasilinear space. We hope that our presented idea herein will be a source of motivation for other researchers to extend and improve these findings for their applications. In addition, we intend to develop a method to approximate the autocorrelation of nondeterministic signals containing some uncertainties with the help of the fuzzy inner product that we have defined in this article. Our plan for the next study is to define the concept of fuzzy signal.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.