Research Article | Open Access
New Inner Product Quasilinear Spaces on Interval Numbers
Primarily we examine the new example of quasilinear spaces, namely, “ 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 interval space related to these concepts. Secondly, we present , and 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.
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 . 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 set of all convex compact subsets of a normed space . 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 .
Inspired and motivated by research going on in this area, we introduce an inner product quasilinear space which is defined in . Generally, in , we give some examples of quasi-inner product properties on of all convex compact subsets of a normed space . In our present paper we examine a new type of a quasilinear space, namely, . We find some important results related to the geometric structure of the quasilinear space and we examine some algebraic properties of the interval space. Furthermore, we explore the concepts of , and 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 .
Definition 1. A set 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 and any real numbers : (1),(2) if and ,(3) if and ,(4),(5),(6)there exists an element such that ,(7),(8),(9),(10),(11),(12) if and ,(13) if .
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 operationand the real-scalar multiplicationWe denote this set by . Another one is , the set of all compact subsets of real numbers. By a slight modification of algebraic sum operation (with closure) such asand by the same real-scalar multiplication defined above and by the inclusion relation we get the nonlinear QLS, and , the space of all nonempty closed bounded and convex closed bounded subsets of some normed linear space , respectively.
Lemma 2. Suppose that any element in a QLS has an inverse element . Then the partial order in is determined by equality, the distributivity conditions hold, and, consequently, is a linear space .
Suppose that is a QLS and . Then is called a subspace of whenever is a QLS with the same partial order and the restriction to of the operations on . 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. is a subspace of a QLS if and only if, for every and .
Let be a QLS. An is said to be symmetric if , where , and denotes the set of all such elements. denotes the zero’s additive unit of and it is minimal, that is, if . An element is called inverse of if . The inverse is unique whenever it exists and in this case. Sometimes may not exist but is always meaningful in QLSs. An element possessing an inverse is called regular; otherwise it is called singular. For a singular element we should note that Now, and stand for the sets of all regular and singular elements in , respectively. Further, , and are subspaces of and they are called regular, symmetric, and singular subspaces of , respectively . It is easy from the definitions that and .
In a linear QLS that is in a linear space, there is no singular element. Further, in a QLS , it is obvious that any element is regular if and only if .
Proposition 4. In a QLS , if and then .
Proposition 5. In a quasilinear space every regular element is minimal .
Definition 6. Let be a QLS and . The set of all regular elements preceding is called floor of , and denotes the set of all such elements. Therefore,The floor of any subset of is the union of floors of all elements in and is denoted by .
Definition 7. Let be a QLS, and . is called a proper set if the following two conditions hold: (i) for all ,(ii) for each pair of points with .
Otherwise is called an improper set. Particularly if is a proper set, then it is called a proper quasilinear space (briefly, proper qls) .
Definition 8. Let be a quasilinear space. is called a solid-floored quasilinear space whenever for each . Otherwise, is called a non-solid-floored quasilinear space .
Definition 9. Let be a QLS. Consolidation of floor of is the smallest solid-floored QLS containing , that is, if there exists another solid-floored QLS containing then .
Clearly, for some solid-floored QLS . Further, . For a QLS , the setis the floor of in
Definition 10. Let be a QLS. A real function is called a norm if the following conditions hold : (14) if ,(15),(16),(17)if , then ,(18)if for any there exists an element such that and then .
A quasilinear space with a norm defined on it is called normed QLS. It follows from Lemma 2 that if any has an inverse element , then the concept of a normed QLS coincides with the concept of a real normed linear space.
Let be a normed QLS. Hausdorff or norm metric on is defined by the equality
Since and , the quantity is well defined for any elements , andIt is not hard to see that this function 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 .
Example 12. Let be a Banach space. A norm on is defined byThen and are normed quasilinear spaces. In this case the Hausdorff metric is defined as usual:where denotes a closed ball of radius about .
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. Let be a quasilinear space. A mapping is called an inner product on if for any and the following conditions are satisfied:(19)if then ,(20),(21),(22),(23) for and ,(24),(25)if and then ,(26)if for any there exists an element such that and then .
A quasilinear space with an inner product is called an inner product quasilinear space, briefly IPQLS.
Example 14. One can see easily , the space of closed real intervals, is a IPQLS with inner-product defined by
Every IPQLS is a normed QLS with the norm defined byfor every .
Proposition 15. If in an IPQLS and , then .
AN IPQLS is called Hilbert QLS, if it is complete according to the Hausdorff (norm) metric.
Example 16. Let be an inner product space. Then we know that is an IPQLS and it is complete with respect to the Hausdorff metric. So, is a Hilbert QLS .
Definition 17 (orthogonality). An element of an IPQLS is said to be orthogonal to an element ifWe also say that and are orthogonal and we write . Similarly, for subsets we write if for all and if for all and .
An orthonormal set is an orthogonal set in whose elements have norm 1; that is, for all
Definition 18. Let be a nonempty subset of an inner product quasilinear space . An element is said to be orthogonal to , denoted by , if for every . The set of all elements of orthogonal to , denoted by , is called the orthogonal complement of and is indicated by
Theorem 19. For any subset of an IPQLS is a closed subspace of .
Definition 20. Let be a quasilinear space, . The element withis said to be a quasilinear combination (ql-combination) of corresponding to scalars . ql-combination of corresponding to the scalars may not be unique, from the definition, while it is well known that the linear combination of corresponding to these scalars is unique. The setis said to be quasispan of and is denoted by . One can see easily that is a subspace of .
It is clear that and iff is a linear space, where is the span of in .
Definition 21. Let be a QLS, and . If implies , then is said to be quasilinear independent (ql-independent); otherwise, it is said to be ql-dependent in .
Theorem 22. Any set which has elements has to be ql-dependent in .
Definition 23. Let be a QLS and let be a ql-independent subset of . If , then the set is called a (Hamel) basis for .
Definition 24. Singular dimension of QLS is defined as the maximum number of ql-independent elements in singular subspace of . If this number is finite then is called finite singular dimensional; otherwise it is called infinite singular dimensional. Further the dimension of regular subspace of is called regular dimension of . It is denoted by and , respectively .
On the other hand, if then is said to be dimension of and is written as .
Definition 25. Let be a solid-floored quasilinear space. A Schauder basis is a sequence of elements of such that for every element and for every there exists a unique sequence of scalars so thatThis definition is identical to the following definition. is a Schauder basis of if and only if for every and for every .
Example 26. and are solid-floored quasilinear space. But singular subspace of is a non-solid-floored quasilinear space.
Recall that the closed interval denoted by is the set of real numbers given by . 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. Let be a set of all closed intervals of real numbers. In this case, is the whole compact convex subset of real numbers. Now, we can give two-dimensional interval vector such that . And, so we can giveWe should state that is not a vector space that is not linear space. Also, the convergence of these sets was discussed in .
Example 27. Let and . The algebraic sum operation on is defined by the expressionand multiplication by a real number is defined byIf we will assume that the partial order on is given bythen is a quasilinear space with the above sum operation, multiplication, and partial order relation. From here, we get that is another example of quasilinear spaces.
Remark 28. The quasilinear space and the quasilinear space are different from each other. For example, while the set is an element of , it is not an element of . On the other hand, but . So, and are two different examples of quasilinear spaces.
Example 29. Let . We can find such that . So, is a regular subset of . Also, if for every , then is a regular subset of . On the other hand, if for and , then is a singular subset of .
The norm on is defined byThen is a normed quasilinear space. From , we know that is a normed quasilinear space with . For example, let . Then, we get .
The Hausdorff metric is defined as usual:where is the closed ball of radius about .
Theorem 30. interval space is a Banach -space with .
Proof. Let be any Cauchy sequence in . Then, given any , there is such that for all we haveHence, for any , we get . On the other hand, since is a quasilinear space, for every . This shows that is a Cauchy sequence in . By completeness of , there is such that for all . Furthermore, since for every . From (28), by letting , we obtainfor all . So, we get , since . Also, we obtainfor all . This means that the sequence converges to in .
Now, we will show that is an -space. Suppose . From the definition of -space, since we have . This proves the theorem.
Example 31. The space is a normed quasilinear space with norm defined by
The quasilinear space with the inner productis an inner product quasilinear space.
Theorem 32. is a Hilbert quasilinear space with norm.
Proof. The norm in Example 31 is defined by and can be obtained from the inner product quasilinear in Example 31. Let be any Cauchy sequence in . Then, given any , there is such that for all we haveHence, for any , we get . On the other hand, since is a quasilinear space, for every . This shows that is a Cauchy sequence in . By completeness of , there is such that for all . Furthermore, since for every . From the above inequality, by letting , we obtainfor all . So, we getand for all . This means that the sequence is convergent to in .
Example 33. Let and take the singleton in . The q-span of isFor example, since whereas . Because there is no such that , clearly, . Let be another element of . For any , clearly, we can write for some . This means .
Example 34. In , let and . Then is ql-independent since for , where is the zero of the quasilinear space . In contrast, the singleton is ql-dependent since for . 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 possessing every vector of an element related to zero must be ql-dependent even if it is a singleton.
Example 35. From Example 33, . Also, is ql-independent since . From here, is a basis for .
Example 37. Let Floor of is Floor of is
Theorem 38. The space is a solid-floored quasilinear space.
Proof. Let . We know that and for all . From the definition of the floor of an element, we haveSince is a quasilinear space, if , then we find for every . Here, for every and . Since for all . Hence, since is a solid-floored quasilinear space. This shows that for every .
Example 39. The set consists of all symmetric elements of . is a non-solid-floored quasilinear space since supremum of floors of every element in is . For the same reason, the subspace of is non-solid-floored.
Lemma 40. The space is a proper quasilinear space.
Proof. Let us take arbitrary elements such that . If , then for at least . Hence, there is at least such that . So and . From here, we have If , then for at least . Hence, there is at least such that . So and . From here, we have and . If there is not a comparison between and , then there is not a comparison between and for some Hence, there exist two elements , and , for some . Thus , and , for this . So, we have and . So, is a proper quasilinear space.
Example 41. The singular subspace of is 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 spaces , and are as follows:
Example 43. Let us consider the subspace of and the elements and of . The set is ql-independent in since there are nonzero scalars and satisfying the inclusion . Hence singular dimension of must be greater than or equal to . Remember that is a subspace of and Theorem 22. Then Clearly, is equivalent to and so .
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 setis called all intervals sequence space. Let and . The algebraic sum operation on is defined by the expressionand multiplication by a real number is defined byIf we will assume that the partial order on is given bythen is a quasilinear space with the above sum operation, multiplication, and partial order relation. is a metric space with for every . Here, is the known Hausdorff metric on .
Now, we give a new interval sequence space, denoted by , namely, the space of all bounded interval sequences of real numbers such thatHere, the boundedness of interval sequences will be considered according to the Hausdorff metric. is a quasilinear space with (48), (49), and (50). The sum operation and scalar multiplication are well defined since
is a normed quasilinear space with norm function defined by Indeed, since it is easy to verify conditions (14)–(17) we only prove condition (18). Let and for every . Then we have for all . On the other hand, we find for all when . Since is a normed quasilinear space, we get for every . Thus, we obtain .
The space with the normis a normed quasilinear space.
A new example of the nonlinear quasilinear space is defined by is a not linear space.
Example 44. is a metric space with where .
Proposition 45. with the norm defined by is a normed quasilinear space but is not a Banach quasilinear space with (61).
Proof. First we show that is a quasilinear space, then that (61) is a norm in , and finally that the space is not complete. The fact that is a quasilinear space with (48), (49), and (50) is obvious. Here, sum operation and scaler multiplication are well defined since for every and . Also, (61) provides the (14)–(18) conditions and is a normed quasilinear space with (61).
Now, we assume that is a Cauchy sequence in . Ifthen given any , there exists a number such thatNamely, for every , there exists a number such thatNote that this implies that for every fixed and for every there exists a number such thatBut this means that, for every , the sequence is a Cauchy sequence in and thus convergent. Denote We are not going to prove that is an element of and that the sequence does not converge to . Indeed, from (65), by letting , we obtainfor every . Here, for every , we get butSo, we have that is not convergent to whether or not .
Corollary 46. interval sequence space is not a Banach quasilinear space with (61) sincebutfor every .
Example 47. , and interval sequence spaces are solid-floored quasilinear space. Definition of the interval sequence space . We find definition of solid-flooredness in which Since is a quasilinear space, we find and such thatfor every . Here, we see clearly that . Otherwise, since is a solid-floored quasilinear space, we have for every . Hence, we obtain since is a quasilinear space with “” relation.
Example 48. Singular subspace of is a non-solid-floored quasilinear space. For example, we have for .
Example 49. Let us recall that is a quasilinear space with the partial order relation “.” If we take where , then and . The quantity of ql-independent elements in is not finite. Indeed, the family 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
Example 50. For the QLS and .
Example 51. We can say that and , for the QLS