Abstract and Applied Analysis

Volume 2012 (2012), Article ID 951374, 10 pages

http://dx.doi.org/10.1155/2012/951374

## Topological Quasilinear Spaces

Department of Mathematics, Inonu University, 44280 Malatya, Turkey

Received 14 March 2012; Accepted 22 July 2012

Academic Editor: Jing Ping Wang

Copyright © 2012 Yılmaz Yılmaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce, in this work, the notion of topological quasilinear spaces as a generalization of the notion of normed quasilinear spaces defined by Aseev (1986). He introduced a kind of the concept of a quasilinear spaces both including a classical linear spaces and also nonlinear spaces of subsets and multivalued mappings. Further, Aseev presented some basic quasilinear counterpart of linear functional analysis by introducing the notions of norm and bounded quasilinear operators and functionals. Our investigations show that translation may destroy the property of being a neighborhood of a set in topological quasilinear spaces in contrast to the situation in topological vector spaces. Thus, we prove that any topological quasilinear space may not satisfy the localization principle of topological vector spaces.

#### 1. Introduction

In [1], Aseev introduced the concept of quasilinear spaces both including classical linear spaces and modelling nonlinear spaces of subsets and multivalued mappings. Then, he proceeds a similar way to linear functional analysis on quasilinear spaces by introducing notions of the norm and quasilinear operators and functionals. Further, 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 operators, and set-valued analysis [2–4].

One of the most useful example of a quasilinear space is the set of all convex compact subsets of a normed space . The investigation of this class involves convex and 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. We refer the reader to [5] for detailed information about global optimization related to interval analysis. Further, the theory of set differential equations also needs the analysis of [3].

There are various ways introducing and handling quasilinear spaces. Another important treatment is those of Markow's approach (see [6, 7]). However, we think that Aseev's treatment provides the most suitable base and necessary tools to proceed a similar analysis on quasilinear spaces to those of classical linear functional analysis. Further, it reflects more aspects of set-valued algebra and analysis by the advantages of the ordering relation. After the introduction of normed quasilinear spaces and bounded quasilinear operators in [1], we think that the investigation of quasilinear topologies on a quasilinear spaces and the introduction of some new results may provide important contributions to the improvement of the quasilinear functional analysis.

#### 2. Preliminaries and Some New Results on Quasilinear Spaces

Let us start this section by giving some notation and preliminary results. We mainly follow the terminology of [1, 8]. For some topological space , the notation stands for the family of all neighborhoods of an . Let be a topological vector space (TVS, for short), and . Then if and only if and . This is the localization principle of TVSs.

A set is called a *quasilinear space *(QLS, for short), [1], if a partial ordering 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 scalars :

A linear space is a QLS with the partial ordering relation “ if and only if ”.

Perhaps the most popular example of nonlinear QLSs is the set of all closed intervals of real numbers with the inclusion relation “” , algebraic sum operation and the real-scalar multiplication We denote this set by . Another one is , the set of all compact subsets of real numbers. In general, and stand for the space of all nonempty closed bounded and nonempty convex and closed bounded subsets of any normed linear space , respectively. Both are QLSs with the inclusion relation and with a slight modification of addition as follows: and with the real-scalar multiplication above.

Hence, convex}.

Lemma 2.1 (see [1]). *In a QLS the element is minimal, that is, if . *

*Definition 2.2. *An element is called an inverse of an if . If an inverse element exists, then it is unique. An element having an inverse is called regular; otherwise, it is called singular.

We show later that the minimality is not only a property of but also is shared by the other regular elements.

Lemma 2.3 (see [1]). *Suppose that each element in the QLS has an inverse element . Then the partial ordering in is determined by equality, the distributivity conditions hold, and, consequently, is a linear space. *

Corollary 2.4 (see [1]). *In a real linear space, equality is the only way to define a partial ordering such that conditions (2.1) hold. *

It will be assumed in what follows that . An element in a QLS is regular if and only if if and only if .

*Definition 2.5. *Suppose that is a QLS and . is called a subspace of whenever is a quasilinear space with the same partial ordering and the same operations on .

Theorem 2.6. * is a subspace of a QLS if and only if for every and , . *

Proof of this theorem is quite similar to its classical linear algebraic counterpart.

Let be a QLS and be a subspace of . Suppose that each element in has an inverse element ; then by Lemma 2.3 the partial ordering on is determined by the equality. In this case the distributivity conditions hold on , and is a linear subspace of .

*Definition 2.7. *Let be a QLS. An element is said to be symmetric provided that , and denotes the set of all such elements. Further, and stand for the sets of all regular and singular elements in , respectively.

Theorem 2.8. *,, and are subspaces of .*

*Proof. * is a subspace since the element is the inverse of .

is a subspace of . Let and . The assertion is clear for . Let and suppose that , that is, for some . Then and so . This implies that . Analogously we obtain if . This contradiction shows that .

The proof for is similar.

, , and are called *regular*, *symmetric*, and *singular subspaces* of , respectively.

*Example 2.9. *Let and and . is the singular subspace of . However, the set of all singletons constitutes and is a linear subspace of . In fact, for any normed linear space , each singleton , is identified with , and hence is considered as the regular subspace of both and .

Proposition 2.10. *In a quasilinear space every regular element is minimal. *

*Proof. * We must show that implies that for each . Consider
Hence by the minimality of . This implies that by the uniqueness of the inverse element.

*Example 2.11. *Consider again the subspace of in the former example. is the only minimal element in , and there is no else minimal element in .

Let be a quasilinear space. A real function is called a *norm*, [1], if the The following conditions are satisfied:

A quasilinear space with a norm defined on it is called* normed quasilinear* space. It follows from Lemma 2.3 that if any has an inverse, then the concept of a normed quasilinear space coincides with the concept of a real normed linear space.

Hausdorff metric or norm metric on a normed QLS is defined by the following equality:

Since and , the quantity is well defined for any elements , further [1]. It is not hard to see that satisfies all of the metric axioms.

Lemma 2.12 (see [1]). *The operations of algebraic operations of addition and scalar multiplication are continuous with respect to the Hausdorff metric. The norm is continuous function with respect to the Hausdorff metric. *

Lemma 2.13 2.13 (see [1]). *(a) Suppose that and , and that for any positive integer . Then . (b) Suppose that and . If for any , then . (c) Suppose that and ; then .*

*Example 2.14 2.14 (see [1]). *Let be a real complete normed linear space (a real Banach space). Then is a complete normed quasilinear space with partial ordering given by equality. Conversely, if is complete normed quasilinear space and any has an inverse element , then is a real Banach space, and the partial ordering on is the equality. In this case . Note that for nonlinear QLSs, in general.

*Example 2.15 2.15 (see [1]). *For example, if is a Banach space, then a norm on is defined by . Then and are normed quasilinear spaces. In this case the Hausdorff metric is defined as usual:
where stands for -centered closed ball with radius in .

#### 3. Topological Quasilinear Spaces

*Definition 3.1. *A topological quasilinear space (TQLS, for short) is a topological space and a quasilinear space such that the algebraic operation of addition and scalar multiplication are continuous, and, following conditions are satisfied for any :

Any topology , which makes be a topological quasilinear space, will be called a *quasilinear topology*. The conditions (3.1) and (3.2) provide necessary harmony of the topology with the ordering structure on .

*Example 3.2. *Let be a TVS. Then, for any and for any , if and only if there exists a neighborhood of satisfying such that for some or for some . In fact, this is true by the localization principle of TVSs since and are neighborhood of . So we can obtain desired by taking or . This provides the condition (3.2). Further, the condition (3.1) obviously holds. Hence, is a TQLS.

We later show that some TQLSs may *not* satisfy the localization principle.

*Remark 3.3. *In condition (3.2), for some , we may find a satisfying such that both and for some . This comfortable situation depends on the selection of . However, we may not find such a suitable for some even in TVSs.

*Example 3.4. *Consider real numbers with usual metric. Take , , and . Then any satisfying must be a subset of . Further and gives , , and hence can only include .

*Remark 3.5. *In a semimetrizable TQLS the condition (3.1) and the condition (3.2) can be reformulated by balls as follows:
equivalently,

A TQLS with a (semi)metrizable quasilinear topology will be called a *(semi)metric QLS*.

*Example 3.6. *Any normed QLS is a Hausdorff TQLS. By the definition and for some , whence the condition (3.2) holds.

Proposition 3.7. * Let X be a Hausdorff TQLS and . If for any there exists some such that , then .*

*Proof. *Suppose that there exists some satisfying for every , but is not true. Then there exists distinct open neighborhoods and of and , respectively. Since , this implies by the condition (3.2) that for any with the property we cannot find satisfying . This is a contradiction to the hypothesis.

*Example 3.8. * In Proposition 3.7, the condition “ is Hausdorff” is indispensable. Let us consider and the function
for some . We can construct a topology on by in such a way that if and only if for some (we later call as a seminorm on ). is a semimetrizable topology by the semimetric
Let
Then, there is not separate neighborhoods of the points and of in this topology. So, cannot be a Hausdorff topology. Now let be arbitrary and define
Then for every there exists some such that . But, .

Theorem 3.9. *Let be a TQLS. Then and are closed in . *

*Proof. * is a net in converging to an . By the continuity of algebraic operations and . This means since for each , whence . The proof is easier for .

The result of this theorem may not be true for . Let and define for each . Then .

*Definition 3.10. * Let be a quasilinear space. A paranorm on is a function satisfying the following conditions. For every ,(1),
(2),
(3),
(4),
(5)if is a sequence of scalars with and X with , then and (continuity of scalar multiplication),(6)if , then .

The pair with the function satisfying the conditions (1)–(6) is called a paranormed QLS.

It follows from Lemma 2.3 that if any has an inverse element , then the concept of paranormed quasilinear space coincides with the concept of a real paranormed linear space.

The paranorm is called *total* if, in addition, we have

The equality

defines a semimetric on a paranormed quasilinear space . is metric whenever is total.

*Example 3.11. *Let us prove the above assertion. First of all we should note that is well defined since and at least. Now and if , and so , for that implies since . Obviously is symmetric. Further, for every element and such that and , observe that . Similarly, for every and such that and . Since and , we get by the definition.

Let be total and . Then for any there exist elements , such that , and , for , . Hence the totality conditions imply that and , that is, .

Further, we have the inequality .

Note that these definitions are inspired from the definitions in [1] about normed quasilinear spaces. The proofs of some facts given here are quite similar to that of Aseev's corresponding results.

If the first condition in the definition of norm in a QLS is relaxed into the condition and if the condition (2.10) of norm is removed, we then obtain the definition of a seminorm. A quasilinear space with a seminorm is called a seminormed QLS. By the same way in linear spaces one can prove that a seminorm on a QLS is a paranorm. Thus we have the following implication chain among the kinds of QLSs:

(normed)seminormed QLS (total paranormed)paranormed QLS (metric)semimetric QLS (Hausdorff)TQLS.

*Example 3.12. *Let be a QLS. The discrete topology on is not a quasilinear topology since the continuity of the scalar multiplication is not satisfied.

*Example 3.13. *The function
in Example 3.8 is a seminorm on . Further, the function
is a paranorm on but is not a seminorm.

*Definition 3.14. *Let be a semimetric QLS and be an element of . Then, the nonnegative number
is called diameter of .

For each regular element , since . Hence this definition is redundant in linear spaces. Further it should not be confused with the classical notion of the diameter of a subset in a semimetric space for which it is defined by for any .

For example, in , and However, for the (singleton) subset of , .

following result is half of the localization principle of TVSs

Theorem 3.15. *Let be a TQLS, , and is a set containing 0. If , then .*

*Proof. *The proof is only an application of the fact that the translation operator , , is continuous by the continuity of the algebraic sum operation.

Although the converse of this theorem is true in TVSs, it may not be true in some TQLSs.

*Example 3.16. *Consider again and its closed unit ball . Now, for , we show that is not a neighborhood of . A careful observation shows that doesnot contain elements (intervals) for which the diameter is smaller than 1. However, every -centered ball with radius contains a singleton if and contains an interval such as if since
That is, contains elements with diameter smaller than . However, neither a singleton nor such an element belongs to . This implies that for every . Eventually, the set cannot contain an -centered ball.

Thus, the localization principle may not be satisfied about a singular element in . The example alludes that translation by a singular element destroys the property of being a neighborhood in a TQLS. The following theorem states that the translation by a regular element preserves neighborhoods, and so the localization principle holds for these elements.

Theorem 3.17. *Let X be a TQLS and . Then .*

*Proof. *Consider again the operator in the proof of Theorem 3.15. In this case the inverse exists and just is the continuous operator . Hence is a homeomorphism and so preserves the neighborhoods.

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