Characterization of triple χ sequence spaces via Orlicz functions

In this paper we study of the characterization and general properties of triple gai sequence via Orlicz space of χM of χ establishing some inclusion relations.


Introduction
Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w 3 for the set of all complex sequences (x mnk ), where m, n, k ∈ N, the set of positive integers. Then, w 3 is a linear space under the coordinate wise addition and scalar multiplication.
We can represent triple sequences by matrix. In case of double sequences we write in the form of a square. In the case of a triple sequence it will be in the form of a box in three dimensional case.
Let (x mnk ) be a triple sequence of real or complex numbers. Then the series ∞ m,n,k=1 x mnk is called a triple series. The triple series ∞ m,n,k=1 x mnk is said to be convergent if and only if the triple sequence (S mnk ) is convergent, where x ijq (m, n, k = 1, 2, 3, . . . ).
The vector space of all triple analytic sequences are usually denoted by Λ 3 .
The vector space of all triple entire sequences are usually denoted by Γ 3 . The space Λ 3 and Γ 2 is a metric space with the metric with 1 in the (m, n, k) th position and zero other wise.
The triple gai sequences will be denoted by χ 3 . Consider a triple sequence x = (x mnk ). The (m, n, k) th section x [m,n,k] of the sequence is defined by x [m,n,k] = m,n,k i,j,q=0 x ijq ijq for all m, n, k ∈ N ; where ijq denotes the triple sequence whose only non zero term is a An FK-space(or a metric space)X is said to have AK property if ( mnk ) is a Schauder basis for X, or equivalently x [m,n,k] → x.
An FDK-space is a triple sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings are continuous.
with 1 in the (m, n, k) th position and zero other wise.
where ijq denotes the triple sequence whose only non zero term is a An FK-space(or a metric space)X is said to have AK property if ( mnk ) is a Schauder basis for X, or equivalently x [m,n,k] → x.
An FDK-space is a triple sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings are continuous.
If X is a sequence space, we give the following definitions: x ∈ X ; X α , X β , X γ are called α -(or Köthe-Toeplitz) dual of X, β-(or generalized-Köthe-Toeplitz) dual of X, γ-dual of X, δ-dual of X respectively. X α is defined by Gupta and Kamptan [10]. It is clear that X α ⊂ X β and X α ⊂ X γ , but X α ⊂ X γ does not hold.

Definitions and Preliminaries
A sequence x = (x mnk )is said to be triple analytic if sup mnk |x mnk |  x ∈ w 3 . Define the sets as m, n, k → ∞ for some ρ > 0 and This paper is a study of the characterization and general properties of gai sequences via triple Orlicz space of χ 3 M of χ 3 establishing some inclusion relations.

Main Results
Proposition 3.1. If M is a Orlicz function, then χ 3 M is a linear set over the set of complex numbers C.
Thus, (y mnk ) is a bounded sequence and hence an triple analytic sequence. In other words, y ∈ Λ 3 M . Therefore (χ 3 M ) * = Λ 3 M . This completes the proof.