A Study Involving the Completion of a Quasi-2-Normed Space
The fundamental aim of this paper is to introduce and investigate a new property of a quasi-2-normed space based on a question given by Park (2006) for the completion quasi-2-normed space. Finally, we also find an answer for Park's question.
1. Introduction, Definitions, and Notations
In 1928, Menger introduced the notion called n-metrics (or generalized metric) . But many mathematicians had not paid attentions to Menger’s theory about generalized metrics. But several mathematicians, A. Wald, L. M. Blumenthal, W. A. Wilson, and so forth, have developed Menger’s idea.
In 1965, Gähler limits Menger’s considerations to . Gähler’s study is more complete in view of the fact that he develops the topological properties of the spaces in question. Gähler also proves that if the space is a linear normed space, then it is possible to define 2-norm .
Since 1963, S. Gähler, Y. J. Cho, R. W. Frees, C. R. Diminnie, R. E. Ehret, K. Iséki, A. White, and many others have studied on 2-normed spaces and 2-metric spaces. It is well known that is complete but is not complete. Since is dense in , it is the said that is completion of . It is very important that an incomplete space can be completed in similar sense. Complete spaces, in other words Banach spaces, play quite important role in many branches of mathematics and its applications. Many mathematicians showed the existence of the completion of normed spaces (for more information, see [3–5]). We will also show the completion of quasi-2-normed spaces via similar sense.
Definition 1. Let be a real linear space with , and let , be a function. Then, let is called linear 2-normed spaces if and are linearly dependent, , , , for all and all .
Example 2. Let denotes the Euclidean vector three spaces. Let and define
Then, is a 2-normed space, and this space is complete (for more information, see ).
Also,(1) in addition to ,, and , if there is a constant such that , for all is called a quasi-2-normed space;(2) a -norm defined on a linear space is said to be uniformly continuous in both variables, if for any , there exists a neighbourhood of such that , whenever - and - are in , which is independent of the choice of , , , ;(3) a pseudo--norm is defined to be real-valued function having all the properties of -norm except the condition that implies the linear dependence of and (for details, see ).
Example 3. Let be a linear space with , and let be -norm on . is quasi-2-norm on , and is a quasi-2-normed space.
Solution 1. By using conditions ,, and in -normed spaces, however, using , we show that if and only if, namely,and are linearly dependent:It is easy to see for , that is, , using the property of , we readily see the following applications: for all , we are now ready to prove the property of , that is, for all , , , so, we complete the solution of Example 3.
Theorem 4. Let be a linear space with , and let be -norm on , for constant , which are and . There exists quasi--norm on defined as
Proof. It is evident to show conditions ,, and ; therefore, It is sufficient to prove the condition of as follows: For all , , since there exists a constant , namely, by substituting , we show that is a quasi--norm on .
Definition 5. Let be a quasi--normed space. (a)A sequence is a Cauchy sequence in a linear quasi--normed space if and only if , for every in .(b)A sequence in is called a convergent sequence if there is an such that , for every in .(c)A quasi--normed space in which every Cauchy sequence converges is called complete.
Definition 6. Let and be quasi--normed spaces. (a′) A mapping is said to be isometric or isometry, if for all ,, (b′) The space is said to be isometry with the space if there exists a bijective isometry of onto . The spaces and are called isometric spaces.
Theorem 7. If a sequence is a Cauchy sequence in a linear -normed space , then exists, for every in (for proof, see ).
Theorem 8. If is a linear space having a uniformly continuous -norm defined on it, then for any two Cauchy sequences and in , (for proof, see ).
Definition 9. The two Cauchy sequences and in a linear -normed space are said to be equivalent, denoted by , if for every neighbourhood of , there is an integer such that implies that , compare .
2. The Completion of a Quasi-2-Normed Space
In , Park introduced quasi--normed spaces and gave some results on -normed spaces. Also, he introduced a question which was “Construct a completion of a quasi-2-norm.” In this section, we give an answer to this question.
Theorem 10. is equivalent to in a linear -normed space if and only if for every in (for proof, see ).
Theorem 11. The relation ~ on the set of the Cauchy sequences in is an equivalence relation on .
Proof. It is clear that (reflexivity) and when .
Let and ,, Then, . So, ~ is an equivalence relation on .
Theorem 12. If and are equivalent to and in a linear 2-normed space , respectively, then is equivalent to and is equivalent to (for proof, see ).
Denote by is the set of all equivalence classes of the Cauchy sequences in . Let ,, , and so forth denote the elements of . Define an addition and a scalar multiplication on as follows:(i): the set of sequences equivalent to , where is in and in ,(ii): the set of sequences equivalent to , where is in . It is clear that these two operations are well defined, since they are independent of the choice of elements from and . So, is a linear space with operations.
Theorem 13. If is a linear space having a uniformly continuous -norm defined on it, then for pairs of the equivalent Cauchy sequences and and , then
Theorem 14. If and are Cauchy sequences in a linear 2-normed space , then is a Cauchy sequence in .
Proof. We see that we can readily see that when , is a Cauchy sequence in .
Whenever is a space having a uniformly continuous -norm defined it which is possible to define real-valued function on the space . The function is defined as follows.
For any two elements and in ,
where and .
Since the limit exists and is independent of the choice of the elements in and , the function is well defined.
Theorem 15. If is a linear space having a uniformly continuous -norm defined on it and and are the Cauchy sequences in and , respectively, then the function defined by is a pseudo-quasi -norm on .
Proof. By using definition of -normed spaces, we see the following: let it is easy to see as follows: on account of definition of -normed space, that is, ,,, , , are the Cauchy sequences, This shows that is a pseudo-quasi--norm on .
Let be the subset of consisting of those equivalence classes which contain a Cauchy sequence for which . At most one sequence of this kind can be in each equivalence class. If and are in and if the corresponding Cauchy sequences are and with and , for every , then we have
Thus, and are isometrics. This isometry will be used to show that is dense in (for details, see ).
Theorem 16. If is a linear space having a uniformly continuous quasi--norm defined on it, then (for proof, see ).
Theorem 17. If is a linear space having a uniformly continuous quasi--norm which is defined as then is complete and the pair is called the completion of a quasi--normed space.
Proof. In order to see that is complete, we have to show that every Cauchy sequence in is convergent in . Let be a Cauchy sequence in and ,.
Because of then we can write , for each ; also, we have, Last from inequality when right hand side will be equal to . Thus this shows us that is a Cauchy sequence in .
Use of and are isometric there is a Cauchy sequence in that corresponding .
On the other hand, there is such that Last from the inequality as and is dense in , So, arbitrary of a Cauchy sequence is convergent to . Then, is complete.
The authors would like the thank to referees for their suggestions and comments, which have improved the presentation of the paper.
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