Research | Open | Published:
Some geometric properties of the metric space
Journal of Inequalities and Applicationsvolume 2013, Article number: 28 (2013)
In this study, we consider the space with an invariant metric. Then, we examine some geometric properties of the linear metric space such as property (β), property (H) and k-NUC property.
MSC:40A05, 46A45, 46B20.
Let X be a vector space over the scalar field of real numbers and d be an invariant metric on X. We denote and as follows:
Let be a linear metric space and (resp., ) be a closed unit ball (resp., the unit sphere) of X. A linear metric space has property (β) if and only if for each and , there exists such that for each element and each sequence in with , there is an index k for which , where . If for each and , implies , a linear metric space is said to have property (H). Let be an integer. A linear metric space is said to be k-nearly uniform convex (k-NUC) if for every and , there exists such that for any sequence with , there are such that . These properties have been studied by Mongkolkeha and Pumam , Sanhan and Suantai , Cui et al.  and Cui and Hudzik .
Ahuja et al.  introduced the notions of strict convexity and U.C.I (uniform convexity) in linear metric spaces which are generalizations of the corresponding concepts in linear normed spaces. Later, Sastry and Naidu  introduced the notions of U.C.II and U.C.III in linear metric spaces and showed that these three forms are not always equivalent. Further, Junde et al. [9, 10] showed that if a linear metric space is complete and U.C.I, then it is reflexive.
In summability theory, de la Vallée-Poussin mean was first used to define the -summability by Leindler . -summable sequences have been studied by many authors including Et et al. [12, 13], Savas [14–18], Savas and Malkowsky  and Şimsek et al. [20, 21]. Let be a nondecreasing sequence of positive real numbers tending to infinity and let and . The generalized de la Vallée-Poussin mean is defined by , where for . A sequence is said to be -summable to a number ℓ if as . If , then -summability is reduced to Cesàro summability.
Let w be the space of all real sequences. Let be a bounded sequence of positive real numbers. Şimşek et al.  defined the space as follows:
where and . If for all , the notation is used in place of and the norm on is as follows:
, is a modular on and the Luxemburg norm on is defined by for all . The Amemiya norm on the space can be similarly introduced as follows:
2 Main results
In this part of the paper, our main purpose is to define a metric on and show that possesses property (β), property (H) and k-NUC property. Let be a bounded sequence of real numbers with for all . The mapping is a metric on the space , where and since the function is convex for . First, we will show that the space has property (β) under the above metric. To do this, we need the following two lemmas. To prove these lemmas, we use the technique given in Sanhan and Mongkolkeha .
Lemma 2.1 Let . If , then
Proof Let and . Then
Lemma 2.2 Let . Then for any and , there exists such that
where and .
Proof Let and . For , we take . From Lemma 2.1, we have
From (2.1) and (2.2), we obtain that . □
Theorem 2.3 The space has property (β).
Proof Let and such that and . We take . By using the diagonal method, we can find a subsequence of for each such that converges for each with , since is bounded for each . Therefore, there is for each such that . So, there is a sequence of positive integers with such that for all . Then there exists such that for all ,
By Lemma 2.2, there exists such that
where and . There exists such that if and . Let us take and . Hence, we have
From (2.3), (2.4), (2.5) and by using the convexity of the function for all , we obtain that
Therefore, we have whenever . Consequently, the space possesses property (β). □
Now, we will show that the space has k-NUC property.
Theorem 2.4 The space is k-NUC for any integer .
Proof Let and with . For each , let
Since the sequence is bounded for each , by using the diagonal method, we can find a subsequence of such that converges for each . Therefore, there is an increasing sequence with . Hence, there exists a sequence of positive integers with such that for all . Then there is such that
Let such that . Let for . From Lemma 2.2, there is a such that
where and . Then there exist positive integers () with such that . Now, define . Then we have for all . For , let and . By using (2.6), (2.7), (2.8) and the convexity of the function (), we obtain
Thus, we have for . Hence, is k-NUC. □
Since k-NUC implies NUC and NUC implies property (H), by using the previous theorem, we can give the following result.
Corollary 2.5 The space has property (H).
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The authors declare that they have no competing interests.
MC, MK and ME have contributed to all parts of the article. All authors read and approved the final manuscript.