Some Geometric and Topological Properties of a New Sequence Space Defined by De La Vallée-poussin Mean Neci̇p Şi̇mşek, Ekrem Savaş, and Vatan Karakaya

In summability theory, de la Vallée-Poussin’s mean is first used to define the (V, λ)-summability by Leindler [9]. Malkowsky and Savaş [14] introduced and studied some sequence spaces which arise from the notion of generalized de la ValléePoussin mean. Also the (V, λ)-summable sequence spaces have been studied by many authors including [6] and [20]. Recently, there has been a lot of interest in investigating geometric properties of several sequence spaces. Some of the recent work on sequence spaces and their geometrical properties is given in the sequel: Shue [21] first defined the Cesáro sequence spaces with a norm. In [11], it is shown that the Cesáro sequence spaces cesp (1 ≤ p < ∞) have Kadec-Klee and Local Uniform Rotundity(LUR) properties. Cui-Hudzik-Pluciennik [4] showed that Banach-Saks of type p property holds in these spaces. In [15], Mursaleen et al studied some geometric properties of normed Euler sequence space. Karakaya [7] defined a new sequence space involving lacunary sequence space equipped with the Luxemburg norm and studied KadecKlee(H), rotund(R) properties of this space. Quite recently, Sanhan and Suantai [19] generalized normed Cesáro sequence spaces to paranormed sequence spaces by making use of Köthe sequence spaces. They also defined and investigated modular structure and some geometrical properties of these generalized sequence spaces. In addition, some related papers on this topic can be found in [1],[2],[5],[16],[17] and [23].