On Some New Sequence Spaces in 2-Normed Spaces Using Ideal Convergence and an Orlicz Function
© E. Savaş. 2010
Received: 25 July 2010
Accepted: 17 August 2010
Published: 8 September 2010
The purpose of this paper is to introduce certain new sequence spaces using ideal convergence and an Orlicz function in 2-normed spaces and examine some of their properties.
The notion of ideal convergence was introduced first by Kostyrko et al.  as a generalization of statistical convergence which was further studied in topological spaces . More applications of ideals can be seen in [3, 4].
The concept of 2-normed space was initially introduced by Gähler  as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [6, 7]). Recently, a lot of activities have started to study summability, sequence spaces and related topics in these nonlinear spaces (see, [8–10]).
Recall in  that an Orlicz function is continuous, convex, nondecreasing function such that and for , and as .
Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary  and others.
Let be a real vector space of dimension where A 2-norm on is a function which satisfies (i) if and only if and are linearly dependent, (ii) , (iii) , and (iv) . The pair is then called a -normed space .
Quite recently Savaş  defined some sequence spaces by using Orlicz function and ideals in 2-normed spaces.
In this paper, we continue to study certain new sequence spaces by using Orlicz function and ideals in 2-normed spaces. In this context it should be noted that though sequence spaces have been studied before they have not been studied in nonlinear structures like -normed spaces and their ideals were not used.
2. Main Results
(iv) Finally using the same technique of Theorem of Savaş  it can be easily seen that scalar multiplication is continuous. This completes the proof.
gives us the result.
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