# Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces

Authors: Yılmaz Yılmaz; Hacer Bozkurt; Halise Levent; Ümit Çetinkaya
Journal: Journal of Mathematics
 (2022)

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## Abstract

It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined 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 first developed the algebraic structure of the class of fuzzy setsFℝn and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, uq=∫01supx∈uαxqdα1/q where 1≤q≤∞, we stated that Fℝn,uq is a complete normed space. Furthermore, we introduced an inner product in this space for the case q=2. The inner product must be in the formu,v=∫01uα,vαKℝndα=∫01a,bℝndα:a∈uα,b∈vα. For u,v∈Fℝn. We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of Fℝn. Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.


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## Body

## 1. 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] first introduced the notions of fuzzy seminorm and norm on a vector space. Independently, Felbin [3] gave the concept of fuzzy normed space (briefly, FNS) by applying the notion of fuzzy distance of Kaleva and Seikkala [4] on vector spaces. Xiao and Zhu [5] improved Felbin’s definition of the fuzzy norm of a linear operator between FNSs [6].The notion of fuzzy quasilinear space is introduced in [6] depending on the notion of quasilinear space which was defined by Aseev in [7].Aseev first 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 definitions and theorems in linear functional analysis and differential calculus in Banach spaces. This pioneering work has motivated a lot of authors to introduce new results on multivalued mappings, fuzzy quasilinear spaces and operators, and set-valued analysis [8–14]. In this way, Rojas Medar et al. [6] introduced the concept of fuzzy quasilinear spaces and defined the notion of a norm on these spaces.In this study, we first constructed the algebraic structure of the class of fuzzy setsFℝn and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, e. g., uq=∫01supx∈uαxqdα1/q where 1≤q≤∞, we stated that Fℝn,uq is a complete normed space. Furthermore, we introduced an inner product in this space for the case q=2. The inner product must be in the following form:(1)u,v=∫01uα,vαKℝndα=∫01a,bℝndα:a∈uα,b∈vα,for u,v∈Fℝn. Furthermore, we showed that the inner product norm is just(2)u,uKℝ=∫01supa∈uαa2dα=u2.We also prove that the parallelogram law can only be provided in the regular subspace, not in the entire ofFℝn. Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.


## 2. Preliminaries and Some New Results

In the universe ofℝn of generalization of fuzzy numbers defined by Fℝn=u:ℝn⟶0,1:u satisfies (i)–(iv) below, where(i)
u is normal, that is, there exists an x0∈ℝn such that ux0=1(ii)
u is fuzzy convex, that is, for x,y∈ℝn and 0≤λ≤1,uλx+1−λy≥minux,uy(iii)
u is upper semicontinuous(iv)
The closure ofx∈ℝn:ux>0, denoted by u0 is compact.For0<α≤1, the α-level set uα is defined by uα=x∈ℝn:ux≥α. Then, from (i) to (iv), it follows that the α -level set uα∈KCℝn and u0=∪α∈0,1uα¯ where KCℝn denote the family of all nonempty, compact, convex subsets of ℝn.The set of fuzzy numbers forms an important algebraic structure called quasilinear space. Now, let us give the definition of quasilinear space.A setX 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:(Q1)x⪯x(Q2)x⪯z if x⪯y and y⪯z(Q3)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 x+θ=x(Q7)αβ⋅x=αβ⋅x(Q8)α⋅x+y=α⋅x+α⋅y(Q9)1⋅x=x(Q10)0⋅x=0(Q11)α+β⋅x⪯α⋅x+β⋅x(Q12)x+z⪯y+v if x⪯y and z⪯v(Q13)α⋅x⪯α⋅y if x⪯yAny linear space is a QLS with the partial order relation “=.” An element x′ is called (additive) inverse of x∈X if x+x′=θ. The 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 QLSX has an inverse element x′∈X. Then, 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. Then, 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 Xr the set of all regular elements and by Xs the sets of all singular elements and the zero element in X, respectively. Furthermore, it can be easily shown that Xr and Xs are subspaces of X. They 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.Kℝn, the set of all compact subsets of ℝn is also quasilinear space with the inclusion and KCℝn is a subspace of Kℝn. If α,β∈ℝ and A,B∈KCℝn, then α⋅A+B=α⋅A+α⋅B,αβ⋅A=αβ⋅A,1⋅A=A and if α,β≥0, then α+β⋅A=α⋅A+β⋅A. The distance between A and B is defined by the Hausdorff metric.(3)δ∞A,B=maxsupa∈Ab∈Binfa−b,supb∈Binfa∈Aa−b.It is well known thatKCℝn is complete with this metric. Furthermore, KCℝn 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]).
LetX be a QLS. A real function ⋅X:X⟶ℝ is called a norm if the following conditions hold:(NQ1)xX>0 if x≠0(NQ2)x+yX≤xX+yX(NQ3)α⋅xX=αxX(NQ4) ifx⪯y, then ∥x∥X≤∥y∥X(NQ5) if for anyε>0 there exists an element xε∈X such that x⪯y+xε and xεX≤ε then x⪯y
A quasilinear spaceX 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. The Hausdorff metric or norm metric on X is defined by the equality.(4)hXx,y=infr≥0:x⪯y+a1r,y⪯x+a2randair≤r,i=1,2.
Sincex⪯y+x−y and y⪯x+y−x, the quantity hXx,y is well-defined for any elements x,y∈X, and it is not hard to see that the function hX satisfies all the metric axioms [7]. Also, we should note that hXx,y may not equal to x−yX if X is not a linear space. However, hXx,y≤x−yX for every x,y∈X.
We should note the following useful properties of the Hausdorff metric which are given by Aseev [7].
The operations of algebraic sum and multiplication by real numbers are continuous with respect to the Hausdorff metric. The norm is continuous and the following properties are satisfied:(a)
Suppose thatxn⟶x0 and yn⟶y0, and that xn⪯yn for any positive integer n. Then, x0⪯y0.(b)
Suppose thatxn⟶x0 and zn⟶x0. If xn⪯yn⪯zn for any n, then yn⟶x0.(c)
Suppose thatxn+yn⟶x0 and yn⟶θ. Then, xn⟶x0.
Furthermore, for eachα∈ℝ and for every x,y,u,v∈X,(5)hXα⋅x,α⋅y=α⋅hXx,y,hXx+y,z+v≤hXx,z+hXy,v.
IfA∈KCℝn or Kℝn, where Kℝn is the family of all nonempty compact subsets of ℝn, then A=supa∈Aa defines a norm on these quasilinear spaces. In this case the Hausdorff metric is defined by(6)δ∞A,B=maxsupa∈Ainfb∈Ba−b,supb∈Binfa∈Aa−b=infr≥0:A⊆B+Srθ,B⊆A+Srθ,where Srx=y∈ℝn:x−y≤r.
Letu,v∈Fℝn and α∈0,1. The algebraic sum and scalar multiplication by a real number λ∈ℝ are defined by(7)u+vx=supminuy,vx−yλux=uxλ,λ≠0,χ0x,λ=0.
And then we haveu+vα=uα+vα and λuα=λuα where for any subset A of ℝn, χA denotes the characteristic function of A. Furthermore, from [6, 15], the relation is obtained.(8)u⪯v⇔uα⊆vα,∀α∈0,1,
is a partial order onFℝn and hence it is a quasilinear space [6] with the zero element χ0.
Now let us determine two important subspace ofFℝn. Consider χa, the characteristic function of a singleton a for a∈ℝn. Obviously, χa∈Fℝn and −χa=−1⋅χa is the additive inverse of χa since the following is obtained:(9)χa−χax=sup miny∈Xχay,−χax−y=sup miny∈Xχay,χay−x=χ0.
Let us warn again that some elements may have no additive inverse; for example, consider the characteristic functionχ1,3 of 1,3⊂ℝ. Although, −χ1,3=−1⋅χ1,3=χ−3,−1∈Fℝ, it is not an additive inverse of χ1,3 in Fℝ since χ1,3−χ1,3≠χ0.Theorem 1.
All regular elements inFℝn must be in the form χa, for some a∈ℝn.Proof.
Suppose thatu≠χa and u−u=χ0. This means u−ux=χ0x for any u∈Fℝn, and so,(10)u−ux=1,x=0,0,x≠0.
Hence, forx≠0,u−ux=supy∈Xminuy,uy−x=0meaning that minuy,uy−x=0. We know that y−x≠y for x≠0. Therefore, for c∈0,1 and y∈ℝn we can consider the fuzzy set u defined by the following:(11)uy=c,y=a,0,y≠a.
Moreover, forx=0,u−u0=supy∈ℝminuy,uy=supy∈ℝuy=c. By the definition of χ0 for x=0,u−u0=1 and so c must be 1. Thus, there exists an inverse of u∈Fℝn if and only if u must be defined by(12)uy=1,y=a0,y≠a=1,y∈a0,y∉a=χay.
This contradicts with the assumptionu≠χa.
Hence, the regular subspaceFℝn is Fℝnr=χa:a∈ℝn. Let us give examples of singular elements in Fℝ.Example 1.
Letv:ℝ⟶0,1 be defined by(13)vx=1−x2,x∈−1,−12,0,x∉−1,−12.
Clearly,v∈Fℝ. Because, its α-level is(14)να=−1−α,−12,0≤α≤34,0,34<α≤1.
Andvα is compact convex for all α∈0,1 in ℝ. The function v is given in [1]. Let us try to find the inverse of v. The element v−v is defined as v−vx=supy∈Xminvy,vy−x and if x=0 then(15)v−v0=supy∈ℝminvy,vy=supy∈ℝvy=supy∈ℝ1−y2=34≠1.
So,v−v≠χ0 and this means there is no additive inverse of v. That is, v∈Fℝs.
Furthermore, for anyV∈KCℝn,χV is a singular element in Fℝn, that is, χV∈Fℝns.
We has seen that two important subspaces ofFℝn are the regular subspace Fℝns and the singular subspace Fℝnr. They are only intersect at the zero χ0. Another important subspace of Fℝn is the symmetric subspace FRnsym. An element x in a QLS X is called symmetric whenever x=−x, and Xsym denotes the set of all symmetric elements. It is a subspace of Xs and hence of X. For example, χ−3,3∈Fℝnsym since −χ−3,3=χ−3,3.


## 3. 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 numbera,b where a and b are natural numbers [10].Letukk=1m be a subset of Fℝn where m≤n and m is a positive integer. A (linear) combination of the set ukk=1m is an element z of Fℝn in the form α1u1+α2u2+⋯+αmum=z where the coefficients α1,α2,…,αm are real scalars. On the other hand, a quasilinear combination of the set ukk=1m is an element z∈Fℝn such that α1⋅u1+α2⋅u2+⋯+αm⋅um⪯z for some real scalars α1,α2,…,αm. Hence, the quasilinear combination, briefly ql-combination, is defined by the partial order relation on Fℝn. In fact, the definition of linear combination in Fℝn 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 ukk=1m is an element z of Fℝn such that(16)α1u1+α2u2+⋯+αmum⪯zandz⪯α1u1+α2u2+⋯+αmum,where the coefficients α1,α2,…,αm are real scalars. Clearly, a linear combination of ukk=1m, is a quasilinear combination of ukk=1m, but not conversely. In quasilinear spaces, there are two kinds of combinations of the set ukk=1m, 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 ukk=1m corresponding to αkk=1m is unique but quasilinear combination is not unique. For any nonempty subset A of Fℝn, span of A is given by following known definition:(17)SpA=∑k=1mαk⋅uk:u1,u2,…,um∈A,α1,α2,…,αm∈ℝ,m∈ℕ.However, QspA, the quasispan (q-span, for short) ofA, is defined by the set of all possible quasilinear combinations of A, that is,(18)QspA=z:∑k=1mαk⋅uk⪯z,u1,u2,…,un∈A,α1,α2,…,αm∈ℝ,m∈ℕ.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 Fℝn whenever QspA=Fℝn.Let us give an example.Example 2.
TakeA=χ1,5, a singleton in Fℝ. The q-span of A is(19)QspA=u∈Fℝ:λ⋅χ1,5⪯u,λ∈ℝ=u∈Fℝ:λ⋅1,5⊆uα,λ∈ℝ.
This means, for example,χ2,13∈Q spA since 2⋅χ1,5⪯χ2,13. However, χ2,13∉SpA since there is no λ∈ℝ satisfying λ⋅χ1,5=χ2,13. Note that(20)SpA=u∈Fℝ:λ⋅χ1,5=u,λ∈ℝ=u∈Fℝ:λ⋅1,5=uα,λ∈ℝ.
Furthermore,χ2,3∉QspA since we cannot find any λ∈ℝ satisfying the condition λ⋅χ1,5⪯χ2,3. Hence, QspA≠Fℝ and we say in this case A or the element χ1,5 cannot q-span Fℝ. Let us consider another singleton of a regular element in Fℝ. For 0≠a∈ℝ, χa can q-span Fℝ. In this respect, we can say χ1 can q -span Fℝ while it cannot spanFℝ. We should emphasize that singular element cannot q-span Fℝ.
This 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: LetA=x1,x2,…,xm be a subset of Fℝn. Then, QspA is a subspace of Fℝn.Definition 2 (see [10]).
(Quasilinear independence and dependence) A setA=x1,x2,…,xm in a QLSFℝn is called quasilinear independent (ql-independent, briefly) whenever the inequality(21)χ0⪯λ1⋅x1+λ2⋅x2+⋯+λm⋅xm,
holds if and only ifλ1=λ2=…=λm=0. Otherwise, A is called quasilinear dependent (ql-dependent, briefly).
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. The setA in the previous example is ql-independent since χ0⪯λ⋅χ1,5 iff 0⊆λ⋅1,5 iff λ=0. However, the singleton B=χ−2,3 is ql-dependent since 0⊆β⋅−2,3 for β=2≠0. This is an unusual case since a nonzero singleton is obviously linearly independent in linear space. On the other hand, the set χ2,3,χ−1,2 is ql-dependent. In general, we can see from the definition that any subset including an element related to zero χ0 must be ql-dependent in Fℝn. This 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 spaceℝ2. Let v1=e1⊙−1,1=t,s:s=0,t∈−1,1 and v2=e2⊙−1,1=t,s:t=0,s∈−1,1, where e1=1,0 and e2=0,1 are unit basis vectors in two dimensional vector space ℝ2. Then, v1 and v2 are horizontal and vertical bars of equal length intersecting at zero in ℝ2. Then, χv1,χv2 is a ql-dependent subset of Fℝ2 since(22)χ0,0⪯λ1⋅χv1+λ2⋅χv2iffχ0,0α⪯λ1⋅χv1α+λ2⋅χv2αiff0,0⊆λ1⋅v1+λ2⋅v2,forα∈0,1,for λ1=λ2=1. However, χu1,χu2 is a ql-independent subset of Fℝ2 where u1=e1⊙1,2 and u2=e2⊙1,3. On the other hand, let us consider u=t,s:t2+s2≤1, the unit ball of ℝ2, then the singleton χu is a ql-dependent subset of Fℝ2 since χ0,0⪯χu.Definition 3.
A ql-independent subset A of a QLSX which q-spans X is called a basis (or Hamel basis) for X.
For anya∈ℝ, the singleton χa is a basis for Fℝ. Furthermore, the set B= χ1,0,χ0,1 is a basis for Fℝ2 and similarly we can write the basis of Fℝn. This shows that the standard basis of Fℝn is just the copy of those of ℝn.
The following example is extraordinary since it presents an example of QLS which has no basis. This is an unusual case since all linear spaces have a (Hamel) basis.Example 4.
Let us consider the subspaceΛ of Fℝ where Λ=χ0∪χa,b:a<b and a,b∈ℝ. 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 ofFℝn. 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 ql-independent subset of Fℝn.S is called maximal ql-independent subset of Fℝn providing that S is ql-independent but any superset V⊃S is ql-dependent.Definition 4.
LetX be any quasilinear space. Regular (singular) dimension of Fℝn is the cardinality of any maximal ql-independent subsets of XrXs. 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 ∞.
The above mentioned definition means thatr−dimX is just the dimension of the linear subspace Xr of X and so r-dim X=dimXr. 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 χ0 of Fℝ is a 0,0-dimensional quasilinear space. Later, we will give an example of a 0,0-dimensional quasilinear space other than χ0.Theorem 2.
Fℝn is an n,n-dimensional quasilinear space.Proof.
For the sete1,e2,…,en, the standard basis of the linear space ℝn,(23)B=χe1,χe2,…,χen,
is just the basis of the linear spaceFℝnr. This means r−dimFℝn=n. Now let us consider bars.(24)v1=e1⊙1,2=t1,t2,,,tn:t1∈1,2,tk=0fork∈2,3,…nv2=e2⊙1,2=t1,t2,,,tn:t2∈1,2,tk=0fork∈1,3,4,…n.....vn=en⊙1,2=t1,t2,,tn:tn∈1,2,tk=0fork∈1,2,3,…n−1.
Then,χv1,χv2,…,χvn is a ql-independent subset of Fℝns. This implies s−dimFℝn≥n. Let us assume that A=x1,x2,…,xn,xn+1 is a ql-independent subset of Fℝns. Then, the set of its α-levels x1α,x2α,…,xnα,xn+1α is ql-independent for each α∈0,1 in the quasilinear space KCℝns of singular subspace of all compact convex subsets of ℝn. This means if(25)0⊆λ1⋅x1α+λ2⋅x2α+⋯+λn⋅xnα+λn+1⋅xn+1α,then, λ1=λ2=…=λn+1=0. None of the set xkα can contain the zero element of the vector space ℝn. Otherwise, its ql-independence would be broken. By the abovementioned assumption, we can find a1∈x1α,a2∈x2α,…,an+1∈xn+1α such that 0=λ1a1+λ2a2+…+λn+1an+1. We conclude the set a1,a2,…,an+1 is a linearly independent subset of ℝn since λ1=λ2=…=λn+1=0. This is a contradiction to the fact that r−dimFℝn=n. So, we can write s−dimFℝn=n.Example 5.
Consider the singular subspaceKCℝs of the quasilinear space KCℝ. r−dimKCℝs=0 since KCℝsr=0. Furthermore, 1,2 is ql-independent in KCℝss and so s−dimKCℝs=1. Hence, KCℝs is 0,1− dimensional. Obviously, KCℝr≡ℝ is a 1,0− dimensional quasilinear space. In this respect, all n -dimensional vector spaces on the field real numbers are n,0 -dimensional quasilinear space.
IfX=KCℝ2s∪x,y:y=0,x∈ℝ then X is a subspace of KCℝ2, the quasilinear space of all compact convex subsets of ℝ2 with the inclusion relation, and r−dimX=1 since Xr=x,y:y=0,x∈ℝ. Furthermore, the set v1,v2 in Example 3 is ql-independent and there is no 3-elements subset which is ql-independent in Xs. This proves s−dimX=2. Hence, X is a 1,2 -dimensional QLS.
Consider theQLSX=KCc0, the quasilinear space of all bounded closed convex subsets of c0, the set of all sequences convergent to zero. Xr is equivalent to c0 and so r−dimX=∞. Let us define the set(26)t,0,0,…:1≤t≤4,0,t,0,…:1≤t≤4,….
On the other hand, the set1,4⊙e1,1,4⊙e2,…(27)1,4⊙ek=0,…0,s,k⋅term,0…:s∈1,4,
is ql-independent inXs, where ek 's are coordinate vectors of c0,k=1,2,… Therefore, s−dimX=∞ and so X=KCc0 is an ∞:∞-dimensional QLS. In general, an infinite-dimensional linear space E is an ∞,0− dimensional QLS.Definition 5.
LetX be a QLS and y∈X. The set of all regular elements preceding from y is called floor of y, and ℱy denotes the set of all such elements. Therefore, ℱy=x∈Xr:x⪯y. The floor of any subset M of X is the union of floors of all elements in M and is denoted by ℱM.
Hence,ℱM=∪y∈MFy and it is clear that ℱx=x for some x∈Xr. This means that ℱXr=Xr, and so, the notion of floor is redundant in linear spaces. Furthermore, ℱXs is a subspace of the linear space Xr in the QLS X.Theorem 3.
Suppose thats−dimX≠0 in a QLSX. Then, r−dimX≤s−dimX.Example 6.
ConsiderKCℝsym=−a,a:a∈ℝ, the symmetric subspace of KCℝ. It is interesting that KCℝsym is a 0,0 -dimensional QLS, just similar to the trivial linear space.


## 4. Norm and Inner Product onFℝn

In [6], it was shown that fuzzy quasilinear spaces have a normed QLS structure with various norms. In this study, we will say that Fℝn 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(28)Dqu,v=∫01δ∞uα,vαqdα1/q,defines a metric onFℝn and it is a complete metric space by this metric. Moreover, this metric comes from the norm.(29)uq=∫01supx∈uαxqdα1/q.Let us give and prove the following theorem.Theorem 4.
For1≤q<∞,Fℝn,uq is a complete normed (Banach) quasilinear space.Proof.
Consider the functiong:0,1⟶ℝ,gα=supx∈uαx. Then, from the compactness of each α -level set in ℝn,supx∈uαx exists, and from the properties of fuzzy sets the required integral also exists. Hence, the norm is well-defined.
Now, let us verify the norm axioms:(1)
Clearly, ifu≠θ, then uq>0(2)
Ifu,v∈Fℝn, then(30)u+vq=∫01supx+y∈uα+vαx+yqdα1/q≤∫01supx+y∈uα+vαx+yqdα1/q≤∫01supx∈uαxqdα1/q+∫01supy∈vαyqdα1/q=uq+vq.(3)
Ifu∈Fℝn and λ∈ℝ, then(31)λ⋅uq=∫01supλx∈λ⋅uαλxqdα1/q=λ∫01supx∈uαxqdα1/q.(4)
Suppose thatu⪯v for u,v∈Fℝn. Then, uα⊆vα. So, we obtain sup x∈uαx≤supy∈vαy since uα,vα are a nonempty compact convex subset of ℝn. Thus, for each 1≤q<∞, we get ∫01supx∈uαxqdα1/q≤∫01supy∈vαyqdα1/q(5)
Suppose that there exists an elementwε∈Fℝn such that u⪯v+wε and wε≤ϵ for any ε>0. Then, there exists an element wεα∈KCℝn for each α∈0,1 such that uα⊆vα+wεα and wεα⊆Sεθ. Since wεα⊆KCℝn and KCℝn is a quasilinear space, we obtain uα⊆vα for any α∈0,1. This means that u⪯v.
Therefore,Fℝn is a normed quasilinear space with norm uq.
Fℝn,⋅q is a complete with uq=Dqu,θ since KCℝn is complete. Hausdorff metric for this norm can be computed by the following formula:(32)Dqu,v=infr≥0:u⪯v+a1r,v⪯u+a2randair≤r,i=1,2=∫01δ∞uα,vαqdα1/q.
We should note here that the norm metric (Hausdorff metric) cannot be obtained by the wayu−vq=Dqu,v. Only we can write Dqu,v≤u−vq.
After this stage, we will show that the normuq is an inner product norm for q=2. Now, in order to describe this inner product, let us present some of the concepts about normed quasilinear spaces that we have obtained earlier [8, 9, 16].Definition 6 (see [8]).
X is called consolidate (solid-floored) QLS whenever y=supx∈Xrx⪯y for each y∈X. Otherwise, X is called a nonconsolidate QLS, briefly, nc-QLS.
The supremum in this definition is taken on the order relation “⪯” in the definition of a QLS. Abovementioned definition assumes supx∈Xr:x⪯y exists for each y∈X. Implicitly, we say that X is consolidate if and only if y=supℱy, for each y∈X.
Let us first note that any linear space is a consolidateQLS. Indeed, Xr=X for any linear space X. So,(33)y=supx∈Xr:x⪯y=supx∈Xr:x=y=supy=y,
for any elementy∈X.Theorem 5.
Fℝn is a consolidate QLS, that is, for each u∈Fℝn,u=supv∈Fℝnr:v⪯u.Proof.
Assume thatb∈Fℝn is upper bound for v∈Fℝnr:v⪯u. We should prove that u⪯b. Suppose that u⋠b. Then, we say that uα⊈bα for a α∈0,1. This means one can find a tα∈uα such that tα∉bα. Now, let us consider the characteristic function χtα of the singleton tα. We should show that χtα⪯u. For any α∈0,1, χtαα=tα⊆uα since tα∈uα and χtαα=tα. Hence, we have χtα⪯u from the definition of partial order relation. Moreover, χtα⋠b. Because, if it had been χtα⪯b, then we had χtαα⊆bα, for any α∈0,1, i.e., tα⊆bα. However, this contradict with tα∉bα. Also, we know that χtα is a regular element of Fℝn. Consequently, we have found the regular element χtα such that χtα⪯u and χtα⋠b. But, this conflict with the admission that b is an upper bound. Thus, we obtain u⪯b. This gives Fℝn is a consolidate QLS.Example 7.
Quasilinear spacesFℝs and Fℝsym are not consolidate. Let us show that Fℝsym is not consolidate. Let us take y=χ−1,1∈Fℝsym. Then, we find(34)ℱy=x∈Fℝsymr:x⪯y=x∈χ0:x⪯y=χ0.
This showsy≠supℱy=χ0.
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. The 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 spacesX,≤ 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.Definition 8.
LetX be a QLS. Consolidation of X is the smallest consolidate QLSX^ which compatibly contains X, that is, if there exists another consolidate QLSY which compatibly contains X then X^⊆Y.
Clearly, for some consolidateQLSX,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 KCℝ^s=KCℝ.Theorem 6.
Consolidation ofKCℝs is KCℝ.Proof.
Obviously,KCℝ compatible contains KCℝs. Suppose that Z is another consolidate QLS which contains KCℝs. For an arbitrary element x of KCℝ we will show that x∈Z. If x∈KCℝs, then the proof is clear. If x∉KCℝs, then x∈KCℝr. Hence, x=a for some a∈ℝ. Assume that a∉Z. We get a−ε,a+ε∈ΩCℝs for some ε>0. So, we find a−ε,a+ε∈Z. Since Z is consolidate,(35)a−ε,a+ε=supy⊆a−ε,a+ε:y∈Zr,for any ε>0. This means there exists an element uε∈Zr such that uε⊆a−ε,a+ε in Z. Therefore, we have a∈Zr. 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 KCℝs. Thus, the assumption a∉Z is incorrect.
Similarly, one can see thatKCℝsym^=KCℝ. We can also prove that Fℝns^=Fℝn and Fℝnsym^=Fℝn with a slightly more difficult proof technique.
For any elementy of a QLS X, the set FyX^=z∈X^r:z⪯y denotes the floor of y in X^ and sometimes FyX^ 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.Example 8.
Let us take two elements−4,−1 and −2,5 in nonconsolidate QLSX=KCℝs. Then, ℱ−4,−1 is empty and ℱ−2,5=0. But ℱF−4,−1X^=t:∈−4,−1 and ℱF−2,5X^=t:t∈−2,5.
Now, let us give the definition of inner product that we previously define on quasilinear spaces [8, 17].Definition 9.
LetX be a quasilinear space having a consolidation X^. A mapping,:X×X⟶Kℝ is called an inner product on X if for any x,y,z∈X and α∈ℝ the following conditions are satisfied:(IPQ1) Ifx,y∈Xr then x,y∈Kℝr≡ℝ(IPQ2)x+y,z⊆x,z+y,z(IPQ3)α⋅x,y=α⋅x,y(IPQ4)x,y=y,x(IPQ5)x,x≥0 for x∈Xr and x,x=0⇔x=0(IPQ6)x,yKℝ=supa,bKℝ:a∈ℱxX^,b∈ℱyX^(IPQ7) ifx⪯y and u⪯v then x,u⊆y,v(IPQ8) if for anyε>0 there exists an element xε∈X such that x⪯y+xε and xε,xε⊆Sεθ then x⪯y
whereℱxX^ 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,xKℝ for every x∈X. This norm is called inner product norm. The inner-product metric is obtained by the following formula:(36)hx,y=infr≥0:x⪯y+a1r,y⪯x+a2randair2≤r2,i=1,2=infr≥0:x⪯y+a1r,y⪯x+a2randair,aiKℝrKℝ≤r2,i=1,2.
ForA,B∈KCℝ or Kℝ,A,B=ab:a∈A,b∈B is an inner product and they are Hilbert quasilinear space by this inner product norm. Furthermore, if xn⟶x and yn⟶y in a IPQLS, then xn,yn⟶x,y [9].Theorem 7.
For someA,B∈Kℝn,A,BKℝn=a,bℝn:a∈A,b∈B defines an inner product on Kℝn. Furthermore, Kℝn is a Hilbert quasilinear space by this inner product norm [9].Definition 10 (see [8]).
An elementx of an IPQLS X is said to be orthogonal to an element y∈X if x,yKℝ=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 spaceX. An element x∈X is said to be orthogonal to A, denoted by x⊥A, if x,yKR=0 for every y∈A. The set of all elements of X orthogonal to A, denoted by A⊥, is called the orthogonal complement of A and is indicated by A⊥=x∈X:x,yKℝ=0,y∈A.Theorem 8.
LetX be a quasilinear space having a consolidation X^. Then, the Schwarz inequality holds, that is, x,yKℝ≤xy for any x,y∈X.
You can see the proof of this theorem in cites [8, 9, 16].


## 5. The Inner Product onFℝn

Now, let us give some basic concepts which are used to define an inner-product onFℝn. Let Bℝk and BKCℝn denote the σ− algebras of Borel subsets of ℝk and KCℝn, respectively, where k,n are positive integers. Let F:T⟶KCℝn be a function and T⊆ℝk where we use again the symbol big F for a set-valued function. If for any B∈BKCℝn,t∈T:Ft⊂B∈Bℝk, then we say that F is measurable [15] and we write F−1A=t∈T:Ft∩A≠∅ for any subset A of ℝn.Proposition 1.
[15] The following results are equivalent:(i)
The functionF:T⟶KCℝn is measurable(ii)
F−1B∈Bℝk for every B∈Bℝk(iii)
F−1O∈Bℝk for every open subset O of ℝn(iv)
F−1C∈Bℝk for every closed subset C of ℝn(v)
The functiondx,F⋅:T⟶ℝ is measurable for every x∈ℝn(vi)
The functionF⋅:T⟶ℝ is measurable(vii)
The functionsx,F⋅:T⟶ℝ is measurable for every x∈ℝn.The statements (v)–(vii) explain the measurability of the single-valued mapping defined fromT to ℝ regarding to the Borel σ - algebras Bℝk and Bℝ. 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⟶KCℝn is measurable if it is upper semicontinuous or lower semicontinuos and hence if it is continuous. Let Fi:T⟶KCℝn be a sequence of measurable functions for i=1,2,… and suppose that limt⟶∞DFit,Ft=0 for every t∈T. Then, the limit function F:T⟶KCℝn is also measurable. A selector of a set-valued mapping F:T⟶KCℝn is a single-valued mapping f:T⟶ℝn such that ft∈FT for every t∈T, [15]. If the function F is measurable, then it’s all selector functions are also measurable. The following theorem, known as the “Castaing Representation Theorem” gives an additive characterization of the measurability of a set-valued mapping.Theorem 9 (see [15]).
The functionF:T⟶KCℝn is measurable if and only if there exist a sequence fi of measurable function F such that for each t∈TfT=∪fit:i=1,2,…¯.For any lower semicontinuous functionF:T⟶KCℝn there exist a continuous selector f of F such that ft=x for every x∈Ft 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:0,1⟶KCℝn and SF is the set of all integrable selectors of F which is defined on 0,1. The Aumann integral of F on 0,1 is defined as ∫01Ftdt=∫01ftdt:f∈SF. If SF≠∅, then the Aumann integral is exist and F is said to be Aumann integrable.The functionF is integrably bounded on 0,1 if there exists an integrable function g:0,1⟶ℝ such that Ft≤gt for almost all t∈0,1.Theorem 10 (see [15]).
IfF:0,1⟶KCℝn is measurable and integrally bounded, then it is Aumann integrable over each a,s⊂0,1 with ∫asFtdt∈KCℝn for all s∈a,1.For a set-valued mappingF as in Theorem 10, the Castaing Representation Theorem 9 applies and provides a sequence fi of integrable selectors which are pointwise dence in F. Moreover,(37)∫01Ftdt=∫01fitdt:i=1,2,…¯,and so we need only consider these selectors to evaluate∫01Ftdt.Theorem 11 (see [15]).
The Aumann integral satisfies the following properties for all Aumann integrable functionsF,G:0,1⟶KCℝn:(38)∫01Ft+Gtdt=∫01Ftdt+∫01Gtdt,∫acFtdt=∫abFtdt+∫bcFtdt,0≤a≤b≤c≤1,∫01λFtdt=λ∫01Ftdt,λ∈ℝ,∫01Ftdt⊆∫01GtifFt⊆Gtfor allt∈0,1.In addition, the Aumann integral uniquely determines its integrand.Theorem 12 (see [15]).
IfF:0,1⟶KCℝn is a measurable and integrably bounded then it is Aumann integrable over each a,s⊂0,1 with ∫asFtdt∈KCℝn for all s∈a,1.Let us give a main result.Theorem 13.
Fℝn is an inner product quasilinear space by the function(39)u,v=∫01uα,vαKℝndα=∫01a,bℝndα:a∈uα,b∈vα.
foru,v∈Fℝn.Proof.
First, we must show that the equality (5.2) is well-defined. We know from the cites [8, 9, 16], the function ,Kℝn is an inner product on the QLS Kℝn, and we proved that an inner product is a continuous function. This means it can be Aumann integrable on 0,1. Hence, the function , is well-defined.
Now, let us verify the inner-product axioms:(1)
Ifu,v∈Fℝnr, then we have the representations u=χa and v=χb for some a,b∈ℝn. Hence,(40)u,v=χa,χb=∫01χaα,χbαKℝndα=∫01a,bKℝndα=a,bKℝn=a,bℝn∈Kℝr.(2)
From Theorem11, we write(41)u1+u2,u3=∫01u1+u2α,u3αKℝndα=∫01u1α,u3αKℝn+u2α,u3αKℝndα=∫01u1α,u3Kℝnαdα+∫01u2α,u3αKℝndα=u1,u3+u2,u3.(3)
If we use Theorem11, then(42)λ⋅u,v=∫01λuα,vαKℝndα=∫01λ⋅uα,vαKℝndα=λ⋅∫01uα,vαKℝndα=λ⋅u,v.(4)
u,v=∫01uα,vαKℝndα=∫01vα,uαKℝndα=v,u.(5)
Just the same as in axiom (1), if u∈Fℝnr, then u=χa for a∈ℝn and (43)u,u=χa,χa=∫01χaα,χaαKℝndα,=∫01a,aKℝndα=a,aℝn.Let us remember thatℝn is a linear inner product space. Therefore, we say that a,aℝn≥ 0. So, u,u≥0 since u,u∈Kℝr≡ℝ for u∈Fℝnr.(6)
We shall prove thatu,uKℝ=supχa,χbKℝ:χa∈ℱu,χb∈ℱv. Now, for a∈uα,b∈vα, let us consider the mappings fab:0,1⟶ℝ such that for each α∈0,1,(44)fabα=a,bℝn..Then,fabα=∫01a,bℝndα. Hence, it is a selector of the function, ,:FRn×FRn⟶Kℝ. This means fab∈S,,, for each a∈uα and b∈vα. Furthermore,(45)fabα=a,bℝn=∫01a,bℝndα=∫01χaα,χaαKℝndα=χa,χb.From Theorem5 and from the definition of the Aumann integral we can write(46)u,v=sup⊆χa,χbKℝ:χa∈ℱu,χb∈ℱv.where the supremum is taken over the partial order relation⊆ on Kℝ. Thus, we have(47)u,v=supχa,χb:χa∈ℱu,χb∈ℱv.(7)
Suppose thatu1≤v1 and u2≤v2. Then, u1α,u2αKℝn⊆v1α,v2αKℝn and by Theorem 11(48)u1,u2=∫01u1α,u2αKℝndα⊆∫01v1α,v2αKℝndα=v1,v2.(8)
We suppose that there exists an elementxε∈Fℝn such that u≤v+xε and xε,xε⊆Sεθ for any ε>0. Then, there exists an element xεα∈Kℝn for each α∈0,1 such that uα⊆vα+xεα and xεα,xεα⊆Sεθ. Then, we get uα⊆vα for any α∈0,1 by Theorem 7. This means that u≤v.
Now, let us determine the inner product norm. Foru∈Fℝn(49)u,uKℝ=∫01uα,uKℝnαdαKℝ=∫01a,bℝndα:a,b∈uαKℝ=sup∫01a,bℝndα:a,b∈uα=sup∫01a,bℝndα:a∈uα=sup∫01a2dα:a∈uα=∫01supa∈uαa2dα=u2.Example 9.
Let us consider the setA=χv1,χv2,…,χvn in the proof of Theorem 2. For m≠r and m,r≤n(50)χvm,χvr=∫01χvmα,χvrαKℝndα=∫01vm,vrKℝndα=vm,vrKℝn=a,bℝn:a∈vm,b∈vr=0.
From Theorem7. Hence, A is an orthogonal subset of Fℝn. It cannot quasispan Fℝn.
Consider the standard basise1,e2,…,en of ℝn. Then, B=χe1,χe2,…,χen is an orthonormal basis for Fℝn. Let us show this. For m≠r,(51)χem,χer=∫01χemα,χerαKℝndα=∫01em,erKℝndα=em,erKℝnem,erℝn=0.
Furthermore,(52)χem,χem=∫01χemα,χerαKℝndα=em,emℝn=1.
There exists a copy ofℝn in Fℝn, and this copy is just the regular subspace of Fℝn. In order for a set to be a basis of Fℝn, 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 ofFℝn is that some element pairs may not be able to satisfy the parallelogram law. Especially, in the singular subspace of Fℝn, this law may not be valid. Let us give a striking example that proves this situation.Example 10.
Letu and v:ℝ⟶0,1 be defined by(53)ux=vx=1−x,x∈0,10,x∉0,1.
Clearly,u,v∈Fℝ and their α-levels are uα=vα=0,1−α. Furthermore,(54)u2=v2=∫01supa∈0,1−αa2dα=∫011−α2dα=13.
Now,(55)u+vα=uα+vα=0,2−2αandu−vα=uα−vα=0,1−α−0,1−α=α−1,1−α.
And so,(56)u+v2=∫01supa∈0,2−2αa2dα=∫012−2α2dα=4∫011−α2dα=43andu+v2=∫01supa∈0,2−2αa2dα=∫011−α2dα=13.
This implies(57)u+v2+u−v2=53≠43=2u2+u2.
Hence, the parallelogram law is not valid inFℝn.
In this exampleu=v are singular elements of Fℝn. Let us say that the parallelogram law is valid in the regular subspace of Fℝn.Theorem 14.
The parallelogram law is satisfied on the regular subspaceFℝnr, which is a Hilbert (linear) subspace of Fℝn.Let us give another important result: the Schwarz inequality.Theorem 15.
For any two elementu,v in Fℝn,(58)u,vKℝ≤uv.The proof of this result can be derived from our similar result on Hilbert quasilinear spaces.


## 6. Classical Fuzzy Sequence Spaces

A sequenceU=ukk=1∞ of fuzzy numbers is a function U from positive integers into Fℝ. The fuzzy number uk is called the k-th term of the sequence. Let U=uk be a sequence of fuzzy numbers. The sequence U of fuzzy numbers is said to be bounded if the set ‖uk‖ is bounded in ℝ where ‖uk‖=∫01supa∈ukαa2dα1/2. Furthermore, U=ukk=1∞ is said to be convergent to some u∈Fℝ whenever D2uk,u=∫01δ∞uα,ukα2dα1/2⟶0 as k⟶∞ where D2 is the inner-product metric which comes from the above inner-product norm .. In this case, we write uk⟶u or limuk=u in Fℝ Note that ‖uk−u‖⟶0 implies uk⟶u but not conversely, in general. For example, the constant sequence uk=χ0,1, for each k∈ℕ, is convergent to u=χ0,1. But(59)‖uk−u‖=‖χ0,1−χ0,1‖=‖χ−1,1‖↛0ask⟶0.More details about sequence spaces of fuzzy numbers are in [18–23] and references therein. Now, ℓ∞F and c0F denote the set of all bounded and convergent (to zero) sequences of fuzzy numbers, respectively. Hence,(60)ℓ∞F=U=ukk=1∞:uk∈Fℝandsupk‖uk‖<∞,c0F=U=ukk=1∞:uk∈Fℝand‖uk‖⟶0ask⟶∞,=U=ukk=1∞:uk∈FℝandD2uk,χ0⟶0ask⟶∞.Furthermore, for1≤p<∞,(61)ℓpF=U=ukk=1∞:uk∈Fℝand∑k=1∞‖uk‖p<∞,is the set of allpth order absolute summable sequences of fuzzy numbers. Furthermore, the set of all fuzzy numbers are denoted by wF and so wF=U=ukk=1∞:uk∈Fℝ.Theorem 16.
By the coordinate wise operations and by the coordinate wise relation of fuzzy numbers,wF is a consolidate quasilinear spaces.Proof.
We must verify the quasilinear axioms. For everyU,V,W,Z∈wF and α,β∈ℝ,(62)U+V=ukk=1∞+vkk=1∞anduk+vk=ukα+vkαfor everyk=1,2,…,α⋅U=α⋅ukk=1∞andα⋅uk=α⋅ukαfor everyk=1,2,…,U≤V⇔uk≤vk⇔ukα⊆vkαfor everyk=1,2,….(1)
U≤U since ukα⊆ukα for uk≤uk,k=1,2,….(2)
IfU≤V and V≤W, then we get uk≤vk and vk≤wk for k=1,2,…. From here, we have ukα⊆vkα and vkα⊆wkα for k=1,2,…. Since ukα,vkα,wkα∈KCℝ and KCℝ is a quasilinear space, we obtain ukα⊆wkα. This gives us uk≤wk for k=1,2,…. So, U≤W.(3)
IfU≤V and V≤U, then uk≤vk and vk≤uk for k=1,2,…. Also, we have ukα⊆vkα¯ and vkα⊆ukα for k=1,2,…. This give us ukα=vkα for k=1,2,…. Thus, U=V.(4)
Clearly,U+V=ukk=1∞+vkk=1∞=vkk=1∞+ukk=1∞=V+U.(5)
U+V+W=ukk=1∞+vkk=1∞+wkk=1∞=ukk=1∞+vkk=1∞+wkk=1∞=U+V+W.(6)
There exists an elementθ∈wF such that θ+U=U.(7)
α⋅β⋅U=α⋅β⋅ukk=1∞=αβ⋅ukk=1∞=αβ⋅U.(8)
α⋅U+V=α⋅ukk=1∞+vkk=1∞. On the otherside, we get α⋅uk+vk=α⋅ukα+vkα for every k=1,2,…,. Since KCℝ is a quasilinear space, we have α⋅ukα+vkα=α⋅ukα+α⋅vkα and α⋅uk+vk=α⋅uk+α⋅vk for every k=1,2,…. Thus, we have α⋅U+V=α⋅U+α⋅V.(9)
Since1⋅uk=1⋅ukα=ukα for k=1,2,…, we find 1⋅U=1⋅ukk=1∞=ukk=1∞=U.(10)
Since0⋅uk=0⋅ukα=0 for k=1,2,…, we find 0⋅U=0⋅ukk=1∞=0.(11)
We getα+β⋅U=α+β⋅ukk=1∞≤α⋅ukk=1∞+β⋅ukk=1∞=α⋅U+β⋅U since α+β⋅uk=α+β⋅ukα≤α⋅ukα+β⋅ukα=α⋅uk+β⋅uk for every k=1,2,….(12)
IfU≤V and W≤Z, then we get uk≤vk and wk≤zk for k=1,2,… From here, we have ukα⊆vkα and wkα⊆zkα for k=1,2,… Since ukα,vkα,wkα,zkα∈KCℝ and KCℝ is a quasilinear space, we obtain ukα+wkα⊆vkα+zkα. This gives us uk+wk≤vk+zk for k=1,2,… So, U+W≤V+Z.(13)
IfU≤V, then uk≤vk for k=1,2,… From here, we have ukα⊆vkα and α⋅ukα⊆α⋅vkα for k=1,2,… This gives us α⋅uk≤α⋅wk for k=1,2,… So, α⋅U≤α⋅W. Therefore, wF satisfies all quasilinear axioms and we get wF is a quasilinear spaces.
wF is consolidate since Fℝ is consolidate and since by the fact that(63)supz∈Fℝr:z≤ukk=1∞=ukk=1∞=U
For anyU∈wF.Theorem 17.
ℓ∞F,c0F and ℓpF for 1≤p<∞ are subspaces of wF and each of them is a quasilinear space by the operations and the relation of wF.Theorem 18.
ℓpF is a Banach quasilinear space with the norm Up=∑k=1∞‖uk‖p1/p where 1≤p<∞ and U=ukk=1∞∈ℓpF.The proof of former two theorems is not so hard. So, we rest the proof to the reader. We will try to showℓ2F is a Hilbert quasilinear space. To accomplish this, we will define an inner product on ℓ2F and show that the norm arising from this inner product corresponds to the abovementioned norm for p=2.Theorem 19.
ℓ2F is a Hilbert quasilinear space.Proof.
Let us write the equality(64)U,V=∑k=1∞uk,vk=∑k=1∞∫01ukα,vkαKℝdα,for U=ukk=1∞,V=vkk=1∞∈ℓ2F where uk,vk∈Fℝ. Obviously, is defined from ℓ2F to Kℝ, and it is well-defined since.(65)U,V≤∑k=1∞uk,vkKℝ≤∑k=1∞ukvk≤∑k=1∞uk21/2∑k=1∞vk21/2=U2V2.where we used Theorem 8. The verification of the inner product axioms are routine task since it is similar to Theorem 13.


## 7. Conclusions

In this paper, we improved the algebraic structure of the class of fuzzy setsFℝn, and we researched the some of algebraic properties such as quasilinear independence, dimension, and the algebraic basis in these spaces. Also, we proved that Fℝn is a complete normed space with special norm uq=∫01supx∈uαxqdα1/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 Fℝn. 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.



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*Source: 2466817-2022-07-20.xml*