- Open Access
Some geometric properties of the metric space
© Çinar et al.; licensee Springer 2013
- Received: 18 September 2012
- Accepted: 26 December 2012
- Published: 22 January 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.
- linear metric space
- Luxemburg norm
- de la Vallée-Poussin mean
- k-NUC property
- property (H)
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.
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 .
where and .
From (2.1) and (2.2), we obtain that . □
Theorem 2.3 The space has property (β).
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 .
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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