# New Inner Product Quasilinear Spaces on Interval Numbers

Authors: Hacer Bozkurt; Yılmaz Yilmaz
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

Primarily we examine the new example of quasilinear spaces, namely, “IRn interval space.” We obtain some new theorems and results related to this new quasilinear space. After giving some new notions of quasilinear dependence-independence and basis on quasilinear functional analysis, we obtain some results on IRn interval space related to these concepts. Secondly, we present Is,Ic0,Il∞, and Il2 quasilinear spaces and we research some algebraic properties of these spaces. We obtain some new results and provide an important contribution to the improvement of quasilinear functional analysis.


---

## Body

## 1. Introduction

In 1986, Aseev generalized the notion linear spaces by introducing firmly quasilinear spaces. He used the partial order relation when he defined quasilinear spaces and then he can give consistent counterpart of results in linear spaces. For more details, the reader can refer to [1]. This work has motivated a lot of authors to introduce new results on set-valued analysis [2, 3].One of the most useful examples of a quasilinear space is the setΩC(E) of all convex compact subsets of a normed space E. The investigation of this class involves a convex interval analysis. Intervals are excellent tools for handling global optimization problems and for supplementing standard techniques. This is because an interval is an infinite set and is thus a carrier of an infinite amount of information which means global information. Further, the theory of set differential equations also needs the analysis of ΩC(E) [3].Inspired and motivated by research going on in this area, we introduce an inner product quasilinear space which is defined in [4]. Generally, in [4], we give some examples of quasi-inner product properties on ΩC(E) of all convex compact subsets of a normed space E. In our present paper we examine a new type of a quasilinear space, namely, IRn. We find some important results related to the geometric structure of the IRn quasilinear space and we examine some algebraic properties of the IRn interval space. Furthermore, we explore the concepts of Is,Ic0,Il∞, and Il2 interval sequence spaces as new examples of quasilinear spaces. Moreover, we obtain some theorems and results related to these new spaces which provide us with improving the elements of the quasilinear functional analysis.


## 2. Quasilinear Spaces and Hilbert Quasilinear Spaces

Let us start this section by introducing the definition of a quasilinear space and some of its basic properties given by Aseev [1].Definition 1.
A setX is called a quasilinear space if a partial order relation “≤,” an algebraic sum operation, and an operation of multiplication by real numbers are defined in it in such way that the following conditions hold for any elements x,y,z,v∈X and any real numbers α,β∈E1: (1)
x≤x,(2)
x≤z if x≤y and y≤z,(3)
x=y if x≤y and y≤x,(4)
x+y=y+x,(5)
x+(y+z)=(x+y)+z,(6)
there exists an elementθ∈X such that x+θ=x,(7)
α·(β·x)=(α·β)·x,(8)
α·(x+y)=α·x+α·y,(9)
1·x=x,(10)
0·x=θ,(11)
(α+β)·x≤α·x+β·x,(12)
x+z≤y+v if x≤y and z≤v,(13)
α·x≤α·y if x≤y.A linear space is a quasilinear space with the partial order relation “=.” The most popular example which is not a linear space is the set of all closed intervals of real numbers with the inclusion relation “⊆,” algebraic sum operation(1)A+B=a+b:a∈A,b∈B,and the real-scalar multiplication(2)λA=λa:a∈A.We denote this set by ΩC(R). Another one is Ω(R), the set of all compact subsets of real numbers. By a slight modification of algebraic sum operation (with closure) such as(3)A+B=a+b:a∈A,b∈B¯and by the same real-scalar multiplication defined above and by the inclusion relation we get the nonlinear QLS, ΩC(E) and Ω(E), the space of all nonempty closed bounded and convex closed bounded subsets of some normed linear space E, respectively.Lemma 2.
Suppose that any elementx in a QLS X 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 [1].Suppose thatX is a QLS and Y⊆X. Then Y is called asubspace ofX whenever Y is a QLS with the same partial order and the restriction to Y of the operations on X. One can easily prove the following theorem using the condition of being a QLS. It is quite similar to its linear space analogue.Theorem 3.
Y is a subspace of a QLS X if and only if, for every x,y∈Y and α,β∈R,αx+βy∈Y [5].LetX be a QLS. An x∈X is said to be symmetric if -x=x, where -x=(-1)x, and Xd denotes the set of all such elements. θ denotes the zero’s additive unit of X and it is minimal, that is, x=θ if x≤θ. An element x′ is calledinverse of x if x+x′=θ. The inverse is unique whenever it exists and x′=-x in this case. Sometimes x′ may not exist but -x is always meaningful in QLSs. An element x possessing an inverse is calledregular; otherwise it is calledsingular. For a singular element x we should note that x-x≠0. Now, Xr and Xs stand for the sets of all regular and singular elements in X, respectively. Further, Xr,Xd, and Xs∪{0} are subspaces of X and they are called regular, symmetric, and singular subspaces of X, respectively [5]. It is easy from the definitions that Xd⊂Xs and X=Xr∪Xs.In a linear QLS that is in a linear space, there is no singular element. Further, in a QLSX, it is obvious that any element x is regular if and only if x-x=0.Proposition 4.
In a QLSX, if x∈X and y∈Xs then x+y∈Xs.Proposition 5.
In a quasilinear spaceX every regular element is minimal [5].Definition 6.
LetX be a QLS and y∈X. The set of all regular elements preceding y is called floor of y, and Fy denotes the set of all such elements. Therefore,(4)Fy=z∈Xr:z≤y.The floor of any subset M of X is the union of floors of all elements in M and is denoted by FM [6].Definition 7.
LetX be a QLS, M⊆X and x,y∈M. M is called a proper set if the following two conditions hold: (i)
Fz≠∅ for all z∈M,(ii)
Fx≠Fy for each pair of points x,y with x≠y.OtherwiseM is called an improper set. Particularly if X is a proper set, then it is called a proper quasilinear space (briefly, proper qls) [6].Definition 8.
LetX be a quasilinear space. X is called a solid-floored quasilinear space whenever (5)y=sup⁡x∈Xr:x≤yfor each y∈X. Otherwise, X is called a non-solid-floored quasilinear space [6].Definition 9.
LetX be a QLS. Consolidation of floor of X is the smallest solid-floored QLS X^ containing X, that is, if there exists another solid-floored QLS Y containing X then X^⊆Y.
Clearly,X^=X for some solid-floored QLS X. Further, ΩC(Rn)s^=ΩC(Rn). For a QLS X, the set(6)FyX^=z∈X^r:z⪯y.is the floor of X in X^.Definition 10.
LetX be a QLS. A real function ·X:X→E1 is called a norm if the following conditions hold [1]: (14)
xX>0 if x≠0,(15)
x+yX≤xX+yX,(16)
αxX=αxX,(17)
ifx≤y, then xX≤yX,(18)
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 QLS. It follows from Lemma 2 that if any x∈X has an inverse element x′∈X, then the concept of a normed QLS coincides with the concept of a real normed linear space.LetX be a normed QLS. Hausdorff or norm metric on X is defined by the equality (7)hXx,y=inf⁡r≥0:x≤y+a1r,y≤x+a2r,air≤r.Sincex≤y+(x-y) and y≤x+(y-x), the quantity hX(x,y) is well defined for any elements x,y∈X, and(8)hXx,y≤x-yX.It is not hard to see that this function hX(x,y) satisfies all of the metric axioms.Lemma 11.
The operations of algebraic sum and multiplication by real numbers are continuous with respect to the Hausdorff metric. The norm is a continuous function with respect to the Hausdorff metric [1].Example 12.
LetE be a Banach space. A norm on Ω(E) is defined by(9)AΩE=supa∈AaE.Then Ω(E) and ΩC(E) are normed quasilinear spaces. In this case the Hausdorff metric is defined as usual:(10)hΩCEA,B=inf⁡r≥0:A⊂B+Srθ,B⊂A+Srθ,where Sr(θ) denotes a closed ball of radius r about θ∈X [1].Let us give an extended definition of inner-product. This definition and some prerequisites are given by Y. Yılmaz. We can see following inner-product as (set-valued) inner product on QLSs.Definition 13.
LetX be a quasilinear space. A mapping ·,·:X×X→Ω(R) is called an inner product on X if for any x,y,z∈X and α∈R the following conditions are satisfied:(19)
ifx,y∈Xr then x,y∈ΩC(R)r≡R,(20)
x+y,z⊆x,z+y,z,(21)
α·x,y=α·x,y,(22)
x,y=y,x,(23)
x,x≥0 for x∈Xr and x,x=0⇔x=0,(24)
x,yΩ(R)=sup⁡a,bΩ(R):a∈FxX^,b∈FyX^,(25)
ifx⪯y and u⪯v then x,u⊆y,v,(26)
if for anyε>0 there exists an element xε∈X such that x⪯y+xε and xε,xε⊆Sεθ then x⪯y.A quasilinear space with an inner product is called an inner product quasilinear space, briefly IPQLS.Example 14.
One can see easilyΩC(R), the space of closed real intervals, is a IPQLS with inner-product defined by(11)A,B=a·b:a∈A,b∈B.Every IPQLSX is a normed QLS with the norm defined by(12)x=x,xΩRfor every x∈X [4].Proposition 15.
If in an IPQLSxn→x and yn→y, then xn,yn→x,y [4].AN IPQLS is calledHilbert QLS, if it is complete according to the Hausdorff (norm) metric.Example 16.
LetE be an inner product space. Then we know that Ω(E) is an IPQLS and it is complete with respect to the Hausdorff metric. So, Ω(E) is a Hilbert QLS [4].Definition 17 (orthogonality).
An elementx of an IPQLS X is said to be orthogonal to an element y∈X if(13)x,yΩR=0.We also say that x and y are orthogonal and we write x⊥y. Similarly, for subsets m,n⊆X we write x⊥m if x⊥z for all z∈m and m⊥n if a⊥b for all a∈m and b∈n [4].An orthonormal setM⊂X is an orthogonal set in X whose elements have norm 1; that is, for all x,y∈M(14)x,yΩR=0,x≠y1,x=y.Definition 18.
LetA 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,yΩ(R)=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(15)A⊥=x∈X:x,yΩR=0,y∈A.Theorem 19.
For any subsetA of an IPQLS X,A⊥ is a closed subspace of X [4].Definition 20.
LetX be a quasilinear space, xkk=1n⊂X. The element x∈X with(16)α1x1+α2x2+⋯+αnxn≤xis said to be a quasilinear combination (ql-combination) of xkk=1n corresponding to scalars αkk=1n⊂R. ql-combination of xkk=1n corresponding to the scalars αkk=1n may not be unique, from the definition, while it is well known that the linear combination of xkk=1n corresponding to these scalars is unique. The set(17)QspA=x∈X:∑k=1nαkxk≤x,x1,x2,…,xn∈A,α1,α2,…,αn∈Ris said to be quasispan of A and is denoted by QspA. One can see easily that QspA is a subspace of X [7].It is clear thatSpA⊆QspA and SpA=QspA iff X is a linear space, where SpA is the span of A in X.Definition 21.
LetX be a QLS, xkk=1n⊂X and λkk=1n⊂R. If (18)θ≤λ1x1+λ2x2+⋯+λnxnimplies λ1=λ2=⋯=λn=0, then xkk=1n is said to be quasilinear independent (ql-independent); otherwise, it is said to be ql-dependent in X [7].Theorem 22.
Any set which hasn+1 elements has to be ql-dependent in ΩC(Rn) [7].Definition 23.
LetX be a QLS and let A be a ql-independent subset of X. If QspA=X, then the set A is called a (Hamel) basis for X [7].Definition 24.
Singular dimension of QLSX is defined as the maximum number of ql-independent elements in singular subspace of X. If this number is finite then X is called finite singular dimensional; otherwise it is called infinite singular dimensional. Further the dimension of regular subspace of X is called regular dimension of X. It is denoted by s-dim⁡X and r-dim⁡X, respectively [6].On the other hand, ifs-dim⁡X=r-dim⁡X=a then a is said to be dimension of X and is written as dim⁡X=a [6].Definition 25.
LetX be a solid-floored quasilinear space. ASchauder basis is a sequence xn of elements of X such that for every element y∈X and for every z∈Fy there exists a unique sequence of αn scalars so that(19)z=∑n=1∞αnzxn,y=sup⁡x∈Xr:x≤y.This definition is identical to the following definition. xn is a Schauder basis of X if and only if (20)hz,∑n=1kαnzxn⟶0k⟶∞,y=sup⁡x∈Xr:x≤yfor every y∈X and for every z∈Fy.Example 26.
Ω ( R ) and ΩC(R) are solid-floored quasilinear space. But singular subspace of ΩC(R) is a non-solid-floored quasilinear space.Recall that the closed interval denoted bya,b is the set of real numbers given by a,b=x∈R:a≤x≤b. Although various other types of intervals (open, half-open) appear throughout mathematics, our work will center primarily on closed intervals. In this paper, the term interval will mean closed interval.


## 3. Interval Spaces

In this section, we introduce some new examples of quasilinear space. LetIR be a set of all closed intervals of real numbers. In this case, IR is the whole compact convex subset of real numbers. Now, we can give two-dimensional interval vector such that IR2=X1,X2:X1,X2∈IR. And, so we can give(21)IRn=X1,X2,…,Xn:X1,X2,…,Xn∈IR.We should state that IRn is not a vector space that is not linear space. Also, the convergence of these sets was discussed in [8].Example 27.
LetX=X1,X2,…,Xn∈IRn and Y=Y1,Y2,…,Yn∈IRn. The algebraic sum operation on IRn is defined by the expression(22)X+Y=X1+Y1,X2+Y2,…,Xn+Ynand multiplication by a real number α∈R is defined by(23)α·X=α·X1,α·X2,…,α·Xn.If we will assume that the partial order on IRn is given by(24)X≤Y⟺Xi≤Yi1≤i≤nthen IRn is a quasilinear space with the above sum operation, multiplication, and partial order relation. From here, we get that IRn is another example of quasilinear spaces.Remark 28.
The quasilinear spaceIRn and the quasilinear space ΩC(Rn) are different from each other. For example, while the set A=(x,y):x2+y2≤1 is an element of ΩC(R2), it is not an element of IR2. On the other hand, B=1,2,{2}∈IR2 but B∉ΩC(R2). So, IRn and ΩC(Rn) are two different examples of quasilinear spaces.Example 29.
LetA=1,0⊂IR2. We can find -A=-1,0 such that A+-A=0,0. So, A is a regular subset of IR2. Also, if Bi∈R for every B=B1,B2,…,Bn∈IRn, then B is a regular subset of IRn. On the other hand, if ai≠bi for 1≤i≤n and ai,bi∈R, then B=a1,b1,a2,b2,…,an,bn is a singular subset of IRn.The norm onIRn is defined by(25)X=X1,X2,…,Xn=sup1≤i≤nXiIR.Then IRn is a normed quasilinear space. From [1], we know that IR is a normed quasilinear space with X=supx∈Xx. For example, let A=1,3,2,3,-1,0∈IR3. Then, we get A=sup1,3IR,2,3IR,-1,0IR=3.The Hausdorff metric is defined as usual:(26)hIRA,B=inf⁡r≥0:A⊆B+Srθ,B⊆A+Srθ,where Srθ is the closed ball of radius r about θ∈X.Theorem 30.
I R n interval space is a Banach Ω-space with X=sup1≤i≤nXiIR.Proof.
LetXn be any Cauchy sequence in IRn. Then, given any ϵ>0, there is N such that for all m,n>N we have(27)Xn≤Xm+A1ϵ,Xm≤Xn+A2ϵ,Ajϵ≤ϵ,j=1,2.Hence, for any 1≤i≤n, we get Aijϵ≤ϵ. On the other hand, since IRn is a quasilinear space, (28)Xin≤Xim+Ai1ϵ,Xim≤Xin+Ai2ϵfor every 1≤i≤n. This shows that Xin is a Cauchy sequence in IR. By completeness of IR, there is Xi∈IR such that Xin→Xi for all i∈1,n. Furthermore, since Xi∈IR for every i∈1,n,X=X1,X2,…,Xn∈IRn. From (28), by letting m→∞, we obtain(29)Xin≤Xi+Bi1ϵ,Xi≤Xin+Bi2ϵ,BijϵIR≤ϵ,j=1,2,for all n>n0. So, we get Bjϵ≤ϵ, since Bjϵ=supBijϵIR. Also, we obtain(30)Xn≤X+B1ϵ,X≤Xn+B2ϵfor all 1≤i≤n. This means that the sequence Xn converges to X in IRn.
Now, we will show thatIRn is an Ω-space. Suppose BX=-1,1,-1,1,…,-1,1︷nunits. From the definition of Ω-space, since (31)X=sup⁡X1,X2,…,Xn≤BX=1we have X1≤1,X2≤1,…,Xn≤1⇔X1⊆-1,1,X2⊆-1,1,…,Xn⊆-1,1. This proves the theorem.Example 31.
The spaceIRn is a normed quasilinear space with norm defined by(32)X2=∑i=1nXiIR21/2.The quasilinear spaceIRn with the inner product(33)X,Y=∑i=1nXi,YiIRis an inner product quasilinear space.Theorem 32.
I R n is a Hilbert quasilinear space with ·2 norm.Proof.
The norm in Example31 is defined by X2=∑i=1nXiIR21/2 and can be obtained from the inner product quasilinear in Example 31. Let Xn be any Cauchy sequence in IRn. Then, given any ϵ>0, there is N such that for all m,n>N we have(34)Xn≤Xm+A1ϵ,Xm≤Xn+A2ϵ,Ajϵ≤ϵ,j=1,2.Hence, for any 1≤i≤n, we get Aijϵ≤ϵ. On the other hand, since IRn is a quasilinear space, (35)Xin≤Xim+Ai1ϵ,Xim≤Xin+Ai2ϵfor every 1≤i≤n. This shows that Xin is a Cauchy sequence in IR. By completeness of IR, there is Xi∈IR such that Xin→Xi for all i∈1,n. Furthermore, since Xi∈IR for every i∈1,n,X=X1,X2,…,Xn∈IRn. From the above inequality, by letting m→∞, we obtain(36)Xin≤Xi+Bi1ϵ,Xi≤Xin+Bi2ϵ,BijϵIR≤ϵı=ϵn,j=1,2,for all n>n0. So, we get(37)Xn≤X+B1ϵ,X≤Xn+B2ϵ,and Bjϵ≤ϵ for all 1≤i≤n. This means that the sequence Xn is convergent to X in IRn.Example 33.
LetX=IR2,⊆ and take the singleton A=1,2,-1,3 in X. The q-span of A is(38)QspA=X∈IR2:λ·1,2,-1,3⊆X,λ∈R.For example, 0,2,-1,4∈QspA since 1,2,-1,3⊆0,2,-1,4 whereas -1,3,1,2∉QspA. Because there is no λ∈R such that λ·1,2,-1,3⊆-1,3,1,2, clearly, QspA≠IR2. Let B=1,0,0,1 be another element of X. For any x∈X, clearly, we can write λ1·1,0+λ2·0,1⊆x for some λ1,λ2∈R. This means QspB=X.Example 34.
InIR2,⊆, let A=2,3,-1,1 and B=-1,1,-1,0. Then A is ql-independent since 0,0⊆λ·2,3,-1,1 for λ=0, where 0,0 is the zero of the quasilinear space IR2. In contrast, the singleton B is ql-dependent since 0,0⊆β·-1,1,-1,0 for β=1. This is an unusual case in linear spaces because a nonzero singleton is clearly a linearly independent set in such case. Particularly, any subset of IR2 possessing every vector of an element related to zero must be ql-dependent even if it is a singleton.Example 35.
From Example33, QspB=IR2. Also, B is ql-independent since 0,0⊆α1·1,0+α2·0,1. From here, B=1,0,0,1 is a basis for IR2.Corollary 36.
I R 2 is a normed quasilinear space with X2=∑i=12XiIR21/2. From Example 31, IR2 is an inner product quasilinear space with (33). Further, the set 1,0,0,1 is an orthonormal subset of IR2. Hence, 1,0,0,1 is an orthonormal basis for IR2.Example 37.
LetA=-1,1,0,2∈IR2. Floor of A is (39)FA=y1,y2∈IR2r:y1,y2⊆A=y1,y2∈R2:-1≤y1≤1,0≤y2≤2.Floor of X=X1,X2,…,Xn∈IRn is (40)FIRn=⋃FX=⋃y1,y2,…,yn∈IRnr:y1∈X1,y2∈X2,…,yn∈Xn.Theorem 38.
The spaceIRn is a solid-floored quasilinear space.Proof.
LetX∈IRn. We know that X=X1,X2,…,Xn and Xi∈IR for all 1≤i≤n. From the definition of the floor of an element, we have(41)FX=Y∈IRnr:Y≤X.Since IRn is a quasilinear space, if Y≤X, then we find Yi≤Xi for every 1≤i≤n. Here, Xi∈IR for every X=X1,X2,…,Xn∈IRn and 1≤i≤n. Since Y∈Rn,Yi∈R for all i∈1,n. Hence, (42)Xi=sup⁡FXi=sup⁡Yi∈R:Yi≤Xisince IR is a solid-floored quasilinear space. This shows that X=sup⁡FX for every X=X1,X2,…,Xn∈IRn.Example 39.
The setW=X=X1,X2,…,Xn:X1,X2,…,Xn∈IRd consists of all symmetric elements of IRn. W is a non-solid-floored quasilinear space since supremum of floors of every element in W is 0. For the same reason, the subspace of IRn(43)V=X=X1,X2,…,Xn:X1,X2,…,Xn∈IRodis non-solid-floored.Lemma 40.
The spaceIRn is a proper quasilinear space.Proof.
Let us take arbitrary elementsA=A1,A2,…,An,B=B1,B2,…,Bn∈IRn such that A≠B. If AB, then AiBi for at least i∈1,n. Hence, there is at least ai∈Ai such that ai∉Bi. So ai⊆Ai and aiBi. From here, we have (44)a1,a2,…,an⊆A1,A2,…,An,a1,a2,…,anB1,B2,…,Bn.If BA, then BiAi for at least i∈1,n. Hence, there is at least bi∈Bi such that bi∉Ai. So bi⊆Bi and biAi. From here, we have b1,b2,…,bn⊆B1,B2,…,Bn=B and b1,b2,…,bnA1,A2,…,An=A. If there is not a comparison between A and B, then there is not a comparison between Ai and Bi for some i∈1,n. Hence, there exist two elements ai∈Ai, ai∉Bi and bi∈Bi, bi∉Ai for some i∈1,n. Thus ai⊆Ai, aiBi and bi⊆Bi, biAi for this i∈1,n. So, we have a1,a2,…,an⊆A,a1,a2,…,anB and b1,b2,…,bn⊆B,b1,b2,…,bnA. So, IRn is a proper quasilinear space.Example 41.
The singular subspace ofIRn(45)IRns=0∪X1,X2,…,Xn:X1,X2,…,Xn∈IRsis improper, because, in this space, floors of some elements may be empty set and floors of any two different elements may be the same.Example 42.
Regular and singular dimensions of the quasilinear spacesRn,IRn,IRns, and IRnr are as follows:(46)r-dim⁡Rn=n,s-dim⁡Rn=0,r-dim⁡IRn=n,s-dim⁡IRn=n,r-dim⁡IRns=0,s-dim⁡IRns=n,r-dim⁡IRnr=n,s-dim⁡IRnr=0.Example 43.
Let us consider the subspaceA=IR2s∪t,0:t∈R of IR2 and the elements a1=1,2,0 and a2=0,2,4 of As. The set a1,a2 is ql-independent in As since there are nonzero scalars λ1 and λ2 satisfying the inclusion 0,0⊆λ1a1+λ2a2. Hence singular dimension of A must be greater than or equal to 2. Remember that A is a subspace of IR2 and Theorem 22. Then s-dim⁡A=2. Clearly, Ar is equivalent to R and so r-dim⁡A=1.


## 4. Interval Sequence Spaces

In this section, we will introduce interval sequence spaces as a different example of quasilinear spaces. We first define the following new interval spaces. The set(47)Is=X=Xn∈IR∞:Xn=X1,X2,…,Xn,…∈IR∞is called all intervals sequence space. Let X=X1,X2,…,Xn∈IRn and Y=Y1,Y2,…,Yn∈IRn. The algebraic sum operation on Is is defined by the expression(48)X+Y=X1+Y1,X2+Y2,…,Xn+Yn,…and multiplication by a real number α∈R is defined by(49)α·X=α·X1,α·X2,…,α·Xn,….If we will assume that the partial order on Is is given by(50)X≤Y⟺Xi≤Yi1≤i≤nthen Is is a quasilinear space with the above sum operation, multiplication, and partial order relation. Is is a metric space with (51)dX,Y=∑i=1∞12ihXi,Yi1+hXi,Yifor every X,Y∈Is. Here, h is the known Hausdorff metric on IR.Now, we give a new interval sequence space, denoted byIl∞, (52)Il∞=X=Xn∈IR∞:Xnbounded,namely, the space of all bounded interval sequences Xn of real numbers such that(53)sup⁡XiIR≤∞⟺X∈Il∞.Here, the boundedness of interval sequences will be considered according to the Hausdorff metric. Il∞ is a quasilinear space with (48), (49), and (50). The sum operation and scalar multiplication are well defined since (54)sup⁡Xi+YiIR≤sup⁡XiIR+YiIR≤sup⁡XiIR+sup⁡YiIR≤∞,sup⁡λXiIR≤sup⁡λXiIR≤λsup⁡XiIR≤∞.I l ∞ is a normed quasilinear space with norm function defined by (55)X=sup⁡XnIR:n∈N.Indeed, since it is easy to verify conditions (14)–(17) we only prove condition (18). Let X≤Y+Xϵ and Xϵ≤ϵ for every X,Y∈Il∞. Then we have XϵiIR≤ϵ for all 1≤i≤∞. On the other hand, we find Xi≤Yi+Xϵi for all 1≤i≤∞ when X≤Y+Xϵ. Since IR is a normed quasilinear space, we get Xi≤Yi for every i∈1,∞. Thus, we obtain X≤Y.Denote(56)Ic0=X=Xn∈IR∞:Xn⟶0;that is, the space Ic0 consists of all convergent sequences Xn whose limit is zero such that(57)limn→∞XnIR=0⟺X∈Ic0.Ic0 is a quasilinear space the same as Il∞ with (48), (49), and (50).The spaceIc0 with the norm(58)X=supi∈NXiIRis a normed quasilinear space.A new example of the nonlinear quasilinear space is defined by(59)Il2=X=Xn∈IR∞:∑i=1∞XnIR2bounded.Il2 is a not linear space.Example 44.
I l 2 is a metric space with (60)dX,Y=inf⁡r≥0:X⊆Y+Srθ,Y⊆X+Srθ,where Sr(θ)=X∈Il2:X≤r.Proposition 45.
I l 2 with the norm defined by (61)X=∑i=1∞XiIR21/2is a normed quasilinear space but Il2 is not a Banach quasilinear space with (61).Proof.
First we show thatIl2 is a quasilinear space, then that (61) is a norm in Il2, and finally that the space is not complete. The fact that Il2 is a quasilinear space with (48), (49), and (50) is obvious. Here, sum operation and scaler multiplication are well defined since (62)∑n=1∞Xn+YnIR21/2≤∑n=1∞XnIR+YnIR21/2≤∑n=1∞XnIR21/2+∑n=1∞YnIR21/2<∞,λX=∑n=1∞λXnIR21/2=λ∑n=1∞XnIR21/2<∞for every X,Y∈Il2 and λ∈R. Also, (61) provides the (14)–(18) conditions and Il2 is a normed quasilinear space with (61).
Now, we assume thatXn is a Cauchy sequence in Il2. If(63)Xn=X1n,X2n,X3n,…then given any ϵ>0, there exists a number n0 such that(64)dXin,Xim≤ϵ.Namely, for every ϵ>0, there exists a number n0 such that(65)Xin≤Xim+Ai1ϵ,Xim≤Xin+Ai2ϵ.Note that this implies that for every fixed i∈N and for every ϵ>0 there exists a number n0 such that(66)AijϵIR≤ϵ.But this means that, for every i, the sequence Xin is a Cauchy sequence in IR and thus convergent. Denote (67)Xin⟶Xii=1,2,…,X=X1,X2,….We are not going to prove that X is an element of Il2 and that the sequence Xn does not converge to X. Indeed, from (65), by letting m→∞, we obtain(68)Xin≤Xi+Bi1ϵ,Xi≤Xin+Bi2ϵ,BijϵIR≤ϵıfor every n≥n0. Here, for every i, we get (69)Xn≤X+B1ϵ,X≤Xn+B2ϵbut(70)Bjϵ≰ϵ.So, we have that Xn is not convergent to X whether or not ∑i=1∞Xi2≤∞.Corollary 46.
I l 2 interval sequence space is not a Banach quasilinear space with (61) since(71)Xn⟶X⟹Xin⟶Xibut(72)Xin⟶Xi↛Xn⟶Xfor every Xn=X1n,X2n,….Example 47.
I l 2 , I c 0, and Il∞ interval sequence spaces are solid-floored quasilinear space. Definition of the Il2 interval sequence space X=Xn∈Il2⇔∑n=1∞XnIR2<∞. We find definition of solid-flooredness in which (73)FX=Y∈Il2r:Y≤X.Since Il2 is a quasilinear space, we find Yi∈R and Xi∈IR such that(74)Yi≤Xifor every 1≤i<∞. Here, we see clearly that Yi∈FXi. Otherwise, since IR is a solid-floored quasilinear space, we have (75)Xi=sup⁡FXi=Yi∈R:Yi≤Xifor every i. Hence, we obtain X=sup⁡FX since Il2 is a quasilinear space with “≤” relation.Example 48.
Singular subspace ofIc0 is a non-solid-floored quasilinear space. For example, we have sup⁡FB=0,0,…=0 for B=-1,1,0,0,…∈Ic0s.Example 49.
Let us recall thatIc0 is a quasilinear space with the partial order relation “⊆.” If we take X=Ic0s∪∅ where ∅=0,0,…∈c0, then r-dim⁡X=0 and s-dim⁡X=∞. The quantity of ql-independent elements in X is not finite. Indeed, the family (76)1,2,0,0,…,0,1,2,0,0,…,…,0,0,…,1,2,0,…,…is ql-independent. Let us show that any finite subset of this family is ql-independent. This also implies that this family is ql-independent: assume that(77)∅=0,0,…⊆λ11,2,0,0,…+λ20,1,2,0,0,…+⋯+λk0,0,…,1,2,0,…=λ11,2,λ21,2,…,λk1,2,0,0,…⟺λ1=λ2=⋯=λk=0.Example 50.
For the QLSX=Ic0,r-dim⁡X=∞ and s-dim⁡X=∞.Example 51.
We can say thatr-dim⁡X=2 and s-dim⁡X=∞, for the QLS(78)X=Il∞s∪0,0,…,0,k,0,0,…,0,l,0,0,…:k,l∈R.Theorem 52.
LetE1=1,0,0,0,…,E2=0,1,0,0,…,…. The sequence En is a Schauder basis for Ic0 and Ilp for (p≤1).Proof.
Now, we show thatEn is a Schauder basis for Ic0; the other proof is similar to Ic0. Let Y=Ykk=1∞∈Ic0 and Z=X1,X2,…,Xn,…∈Ic0r for every Z∈FY. We choose α1=X1,α2=X2,…,αn=Xn,…. From here, we get(79)Z-∑k=1nαkEkIc0=Z-∑k=1nXkEkIc0=X1,X2,…,Xn,…-X1,X2,…,Xn,0,0,…=0,0,…,Xn+1,Xn+2,…=supk≥n+1Xk⟶0⁡n⟶∞.Since(80)hZ,∑k=1nαkEk≤Z-∑k=1nαkEkIc0we have hZ,∑k=1nαkEk→0 for (n→∞). Now we show that this representation is unique.
To prove uniqueness, we assume that(81)Z=∑k=1∞αkEk,Z=∑k=1∞βkEkfor αk≠βk, 1≤k<∞. Since Z∈Ic0r, we get (82)0=Z-Z=∑k=1∞αkEk-∑k=1∞βkEk=α1-β1,α2-β2,…,αn-βn,…=sup1≤k<∞αk-βk.Hence, we find αk=βk for every 1≤k<∞. This proves the theorem.Example 53.
E n sequence given in Theorem 52 is an orthonormal basis for Ic0.



---

*Source: 2619271-2016-07-26.xml*