A Revisit to n-Normed Spaces through Its Quotient Spaces
Harmanus Batkunde, Hendra Gunawan

TL;DR
This paper investigates properties of n-normed spaces via their quotient spaces, establishing equivalences of continuity types and proving a fixed point theorem for contractive mappings.
Contribution
It introduces a new approach to analyze n-normed spaces through quotient spaces, including fixed point results for contractive mappings.
Findings
All types of continuity are equivalent in this context
A fixed point theorem for contractive mappings is established
Properties of n-normed spaces are characterized via quotient space norms
Abstract
In this paper, we study some features of n-normed spaces with respect to norms of its quotient spaces. We define continuous functions with respect to the norms of its quotient spaces and show that all types of continuity are equivalent. We also study contractive mappings on n- normed spaces using the same approach. In particular, we prove a fixed point theorem for contractive mappings on a closed and bounded set in an n-normed space.
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A Revisit to -Normed Spaces through
Its Quotient Spaces
Harmanus Batkunde∗
Analysis and Geometry Research Group,
Faculty of Mathematics and Natural Sciences,
Bandung Institute of Technology
Bandung 40132, Indonesia
(correspondence) [email protected]
Hendra Gunawan
Analysis and Geometry Research Group,
Faculty of Mathematics and Natural Sciences,
Bandung Institute of Technology
Bandung 40132, Indonesia
(Date: March 29, 2019)
Abstract.
In this paper, we study some features of -normed spaces with respect to norms of its quotient spaces. We define continuous functions with respect to the norms of its quotient spaces and show that all types of continuity are equivalent. We also study contractive mappings on -normed spaces using the same approach. In particular, we prove a fixed point theorem for contractive mappings on a closed and bounded set in an -normed space.
Key words and phrases:
continuous mappings, fixed points, -normed spaces, quotient spaces.
1991 Mathematics Subject Classification:
Primary 46B20; Secondary 54B15
1. Introduction
We will study some features of the -normed spaces with its quotient spaces as a tool. In the 1960’s, the concept of -normed spaces was investigated by S. Gähler [8, 9, 10, 11]. Let be a nonnegative integer and be a real vector space with . The pair is called an -normed space where is an -norm on which satisfies the following conditions:
- (i)
; if and only if linearly dependent.
- (ii)
is invariant under permutation.
- (iii)
for any .
- (iv)
.
Some researchers studied further various aspects of these spaces [1, 2, 12, 13, 15, 7, 16, 17]. In particular, Brzdȩk and Ciepliński [5] proved a fixed point theorem for operators acting on some classes of functions, with values in -Banach spaces. Ekariani et al. [6] also proved a fixed point theorem of a contractive mapping in as an -normed space. They used a norm which is equivalent to the usual norm on . The norm that they used is simpler than the normed used by Brzdȩk and Ciepliński. This norm is derived from the -norm using a linearly independent set consisting of vectors.
In this paper, we provide a more general approach to prove a fixed point theorem in an -normed space. First, we define some quotient spaces of an -normed space. In each quotient space we define a norm which is derived from the -norm in a certain way. This norm is simpler than the norm that Brzdȩk and Ciepliński or Ekariani et al. used. Using these norms, we investigate continuous mappings and contractive mappings in the -normed space and prove a fixed point theorem for contractive mappings on a closed and bounded set in the -normed space. Using this approach we will also prove a fixed point theorem in as an -normed space. Compared to the approaches used by previous researchers [13, 15, 16, 5, 6], our approach provides a more general way which can be used to study properties of -normed spaces.
2. Preliminaries
Let us begin with the construction of quotient spaces of an -normed space. Let be an -normed space, and be a linearly independent set in . For , consider . We define the following subspace of :
[TABLE]
For each , the corresponding coset in is defined by
[TABLE]
Then we have if , then . We define the quotient space
[TABLE]
The addition and scalar multiplication apply in this space. Moreover, we define the following norm on :
[TABLE]
Using the above construction, we get quotient spaces. Moreover, for an , the set that contains all quotient spaces constructed above is called class- collection [4].
Observe that each term of (2.1) is a norm of quotient spaces in class- collection. Here (2.1) can be written as
[TABLE]
One can see that the norm we define in (2.1) is more simpler than the norm that Brzdȩk and Ciepliński used since we define it with respect to a linearly independent set consisting of vectors.
For an we will use the phrase ’norms of class- collection’ which means ’all norms of each quotient space in the class- collection’. Furthermore, using the norms of class- collection, we have observed some topology characteristics of an -normed space as presented in the following.
Definition 2.1**.**
[3] Let be an -normed space and . We say a sequence converges with respect to the norms of class- collection to if for any , there exists an such that for we have
[TABLE]
for every . In this case we also say
[TABLE]
for every . If does not converge, we say it diverges.
By the above definition, we have the following theorem.
Theorem 2.2**.**
[3]* Let be an -normed space and . A sequence is convergent with respect to the norms of class- collection if and only if it is convergent with respect to the norms of class- collection.*
Corollary 2.3**.**
Let be an -normed space and . Then, if a sequence in converges with respect to the norms class- collection, then it is also converges with respect to the norms of class- collection.
Definition 2.4**.**
[3] Let be an -normed space and . A sequence is called a Cauchy sequence with respect to the norms of class- collection if for any , there exists an such that, for every , we have
[TABLE]
for every . In other words
[TABLE]
for every .
Theorem 2.5**.**
[3]* Let be an -normed space and . If is convergent with respect to the norms of class- collection, then is Cauchy with respect to the norms of class- collection.*
Corollary 2.6**.**
Let be an -normed space and . Then, if a sequence in is a Cauchy sequence with respect to the norms class- collection, then it is also a Cauchy sequence with respect to the norms of class- collection.
Remark 2.7*.*
For , we find that all types of convergent sequence with respect to the norms of class- collection and to the norms of class- colection are equivalent. The equivalence also applies to all types of Cauchy sequence with respect to the norms of class- collection and to the norms of class- collection. In this regard, we may simply use the word ’converges or Cauchy’ instead of ’converges or Cauchy with respect to the norms of class- collection’. Furthermore if every Cauchy sequence in converges, then is complete. By the word ’complete’, we mean ’complete with respect to the norms of class- collection’, for some .
Now let us move to the definition of closed sets with respect to the norms of class- collection, for any .
Definition 2.8**.**
[3] Let be an -normed space and . The set is called closed if for any sequence in that converges in , its limit belongs to .
Note that by saying ’closed’ we mean ’closed with respect to class- collection’, for some . Next is the definition of bounded sets.
Definition 2.9**.**
[3] Let be an -normed space, and be a nonempty set. The set is called bounded with respect to the norms of class- collection if and only if for any there exists an such that
[TABLE]
for every .
We also have the following theorem which says that all types of bounded set are equivalent.
Theorem 2.10**.**
[3]* Let is an -normed space, and nonempty. The set is bounded with respect to the norms of class- collection if and only if it is bounded with respect to class- collection.*
Furthermore, for the completeness of the -normed space with respect to the norms of class- collections we have the following theorem.
Theorem 2.11**.**
Let be an -normed space and . Then is complete with respect to the norms of class- collection if and only if is complete with respect to the norms of class- collection.
Proof.
Suppose that is complete with respect to the norms of class- collection and . Take any Cauchy sequence with respect to the norms of class- collection in . Then by Theorems 2.2 and 2.5 we have is a convergent sequence with respect to the norms of class- collection in . This tells us that is complete with respect to the norms of class- collection. The converse is similar. ∎
Remark 2.12*.*
As in Corollary 2.6, one can see that the equivalence also applies to boundedness and completeness of a set. Then, we will use the word ’bounded’ instead of the phrase ’bounded with respect to the norms of class- collection’, for some .
Remark 2.13*.*
For a fixed , the convergence of a sequence, the closedness and the boundedness of a set with respect to the norms of class- collection may be investigated with respect to some norms we choose such that
[TABLE]
Moreover, the least number of norms that can be used to investigate these notions is . Next, by using a similar approach we will study continuous mappings, contractive mappings, and also prove a fixed point theorem of contractive mappings of a closed and bounded in an -normed space.
3. Continuous Mappings with Respect to the Norms of Class- Collection
We shall now discuss the continuity of a mapping with respect to the norms of class- collection in an -normed space.
Definition 3.1**.**
Let be an -normed space and be an -normed space and , . Suppose that .
- (i)
We say that is continuous with respect to the norms of class- collections at if and only if for any there exists a such that for with for every , we have for every . 2. (ii)
We say that is continuous with respect to the norms of class- collections on if and only if continuous with respect to the norms of class- collection at each .
Moreover, if , then we say ’ is continuous with respect to the norms of class- collections’ instead of ’continuous with respect to the norms of class- collections’. The norms , are the norms of class- collections in and respectively. Note that we define class- collections by using a linearly independent set consisting of and vectors in and respectively. Based on the definition, we have the following theorem.
Theorem 3.2**.**
Let be an -normed space and be an -normed space and , . A mapping is continuous with respect to class- collections if and only if is continuous with respect to class- collections.
Proof.
Let be a continuous mapping with respect to the norms of class- collections at , and . For any , there is such that for any with for every , we have for every . Then for any with for every , we have for every . This means is continuous with respect to the norms of class- collection.
The converse is obvious. ∎
Corollary 3.3**.**
Let and be an -normed space and an -normed space respectively, and . If a mapping is continuous with respect to class-, then continuous with respect to the norms of class-.
Remark 3.4*.*
By the above corollary, we can see that all types of continuity of a mapping with respect to the norms of class- collections are equivalent. Then from now on we will use the word ’continuous’ instead of ’continuous with respect to the norms of class- collection’.
Now we present a proposition that gives a relation between a convergent sequence and a continuous mapping with respect to the norms of class- collection.
Proposition 3.5**.**
Let and be an -normed space and an -normed space respectively, , . Suppose that is continuous. If converges to , then
[TABLE]
Proof. Let , , and . Since is continuous, there is a such that if for every , then . Since converges to with respect to norms of class-, choose such that for we have . It then follows that whenever . Therefore converges to .
4. Fixed Point Theorem for Contractive Mappings with Respect to The Norms of Class- Collection
In this section, we shall discuss contractive mappings with respect to the norms of class- collection in an -normed space and its fixed point theorem.
Definition 4.1**.**
Let be an -normed space and . A mapping is called contractive with respect to the norms of class- collection if there is a such that for any we have
[TABLE]
for every with
Following the definitions of continuous mappings and contractive mappings, we have the following proposition and theorem.
Proposition 4.2**.**
Let be an -normed space and . If is a contractive mapping with respect to the norms of class- collection, then is continuous.
Proof.
For an , let be a contractive mapping with respect to the norms of class- collection, then there is a such that for any we have
[TABLE]
for every .
For any , choose . Then, for where for every , we have
[TABLE]
for every . Therefore, is continuous in ∎
Note that, for an a contractive mapping with respect to the norms of class- collection are continuous with respect to the norms of class- collection, for any . Moreover, we have the following theorem.
Theorem 4.3**.**
Let be an -normed space, and . If is a contractive mapping with respect to the norms of class- collection, then is a contractive mapping with respect to the norms of class- collection.
Proof.
Suppose that is a contractive mapping with respect to the norms of class- collection, then there is a such that for any we have
[TABLE]
for every . Let , by (4.1) we have
[TABLE]
for every . It therefore follows from the above inequalities that
[TABLE]
for every . If we consider the norms of class- collection, then this means is a contractive mapping with respect to the norms of class- collection. ∎
We also have this following theorem which will be used later.
Theorem 4.4**.**
Let be an -normed space, and . If is a contractive mapping with respect to the norms of class- collection, then is also a contractive mapping with respect to the norm of class- collection.
Proof.
Let be a contractive mapping with respect to the norms of class- collection for an . Then there is a such that for every we have
[TABLE]
for every . One can see that (4.2) contains inequalities. From these inequalities, we have
[TABLE]
or
[TABLE]
which means that is a contractive mapping with respect to the norm of class- collection. ∎
Finally, we provide a fixed point theorem for a contractive mapping in a closed and bounded set on an -normed space, with respect to the norms of class- collection.
Theorem 4.5**.**
Let be an -normed space, and is nonempty, closed and bounded. Suppose that is complete (with respect to the norms of class- collection). If T: is a contractive mapping with respect to the norms of class- collection, then has a unique fixed point.
Proof.
Fix an . Let and be a sequence in such that
[TABLE]
Since is a contractive mapping, there is a such that for we have
[TABLE]
for every . By using induction, we have
[TABLE]
for every . Now we show that is a Cauchy sequence in . Let . Without loss of generality, we take and , with . Then we have
[TABLE]
for every . Also, since is bounded, for any there exists an such that , for every . Then we have
[TABLE]
for every . Since , we have
[TABLE]
for every , which means that is a Cauchy sequence.
Moreover, since is complete with respect to the norms of class- collection and is closed then , with . Propositions 3.5 and 4.2 imply that
[TABLE]
Therefore, has a fixed point in with respect to the norms of class- collection. Next, we want to show the uniqueness of the fixed point with respect to the norms of class- collection. Assume that is another fixed point of . Because is a contractive mapping, there is a such that
[TABLE]
for every . This is true only for , for every . This means that or has a unique fixed point. Therefore, for any , a contractive mapping has a unique fixed point. ∎
5. Concluding Remarks
Let us consider the -summable sequences (for ) containing all sequences of real numbers for which . As in [14], one may equip this space with the following -norm
[TABLE]
with
Now, one can see that if is a linearly independent set in , the norm of class- collection can be writen as
[TABLE]
This is precisely the norm that was used in [6] as a bridge to prove the fixed point theorem on .
In this paper, we provide more alternatives to study as an -normed space. Note that the usual norm on is defined by
[TABLE]
Furthermore, we have some equivalence relations between norm dan .
Theorem 5.1**.**
[6]* Let be a linearly independen set on . Then the norm defined in (5.2) is equivalent to the usual norm on . Precisely, we have*
[TABLE]
for every .
Since is a Banach space, by Theorem 5.1 we have the following corollary.
Corollary 5.2**.**
[6]* The normed space is a Banach space.*
With our approach, we have the proposition below.
Proposition 5.3**.**
Let be a linearly independent set on . Then the norm of class -collection which is defined in (2.1) is equivalent with the norm in (5.2). Precisely we have
[TABLE]
Proof.
Consider the -normed space and be a linearly independent set on . For any , recall the norm of class- collection is
[TABLE]
and
[TABLE]
Then we have
[TABLE]
From Theorem (5.1) and by using Hölder inequality we have
[TABLE]
Then we have the inequality we want. ∎
Note that, we write instead of , because the element of class- collection of an -normed space is the -normed space itself. One might notice that the formula of the norm of class- collection is simpler than the norm that Ekariani et al used and both norms are equivalent. Since the norm equivalent to the usual norm on , we have the following corollaries.
Corollary 5.4**.**
Let be a linearly independent set on . Then the norm is equivalent to the usual norm on . Precisely we have
[TABLE]
Corollary 5.5**.**
The normed space is a Banach space.
Now, we present a more general fixed point theorem on as follows.
Theorem 5.6**.**
Consider the -normed space and let be nonempty, closed and bounded. If is a contractive mapping with respect to norms of class- collection for an , then has a unique fixed point.
Proof.
Fix an and let be a contractive mapping with respect to norms of class- collection on . By Theorem 4.4, is contractive with respect to the norm of class- collection on . By Corollary (5.5) is complete with respect to the norm of class- collection. Hence, has a unique fixed point. ∎
Remark 5.7*.*
We can also prove Theorem 5.6 using Theorem 4.5, noting the fact that is complete with respect to the norms of class- collection (see Theorem 2.11).
Acknowledgment
This work is supported by ITB Research and Innovation Program 2019. The first author is also supported by LPDP Indonesia.
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