Abstract

In this paper the entire triple sequence space are the generalization of the classical Maddox's paranormed sequence space have been introduced and investigated some topological properties of entire triple sequence space of binomial Poisson matrix of and

1. Introduction

A triple sequence (real or complex) can be defined as a function where and denote the set of natural numbers, real numbers and complex numbers respectively. The different types of notions of triple sequence was introduced and investigated at the initial by Sahiner et al. ^{1, 2}, Esi et al. ^{3, 4, 5, 6, 7, 8}, Dutta et al. ⁹, Subramanian et al. ¹⁰, Debnath et al. ¹¹ and many others. Throughout and denote the classes of all, entire and analytic scalar valued single sequences, respectively. We write for the set of all complex triple sequences where the set of positive integers. Then, is a linear space under the coordinate wise addition and scalar multiplication.

Let be a triple sequence of real or complex numbers. Then the series is called a triple series. The triple series is said to be convergent if and only if the triple sequence is convergent, where

A sequence is said to be triple analytic if

The vector space of all triple analytic sequences are usually denoted by . A sequence is called triple entire sequence if

The vector space of all triple entire sequences are usually denoted by The space and is a metric space with the metric

(1.1)

for all and

Consider a triple sequence The section of the sequence is defined by for all

has 1 in the position, and zero otherwise.

The Poisson matrix is defined by

Example: If and then

2. Properties of Poisson Matrix of Eigen Values and Eigen Vectors

(1) We have for

(2) The eigen vectors are orthogonal

(3) is symmetric;

(4) is positive definite.

Example: If Hence

and so on.

Now, we define the binomial poisson matrix where

where In this paper, we define the binomial Poisson triple equence spaces and as the set of all sequences whose transforms are in the spaces and respectively, that is

and

Define the triple sequence which will be frequently used as the transform of a triple sequence i.e.,

for all

Now, we may begin with the following theorem which is essential in the text.

3. Definitions and Preliminaries

3.1. Definition. is a paranorm on if

(i) with

(ii)

(iii)

(iv)

(v) If with and if with in the sense that then in the sense that

4. Main Results

4.1. Theorem. is a complete metric space paranormed by g, defined by

Proof: Let be a Cauchy sequence in Then given any there exists a positive integer depending on such that for all Hence

for all Consequently

is a Cauchy sequence in the metric space of complex numbers. Since is complete, so

as Hence there exists a positive integer such that

for all In particular, we have

Now

for each m, n, k. Thus

each m, n, k. That is Therefore, is a complete metric space.

4.2. Theorem. The entire triple sequence space is linearly isomorphic to

Proof: Now to prove that the existence of a linear bijection between thespaces and with the notation of define the transformation from and by The linearity of is trivial. Furthermore, it is obvious that

whenever

and hence is injective. Let and define the sequence

for each Then, we have

Thus, we have that and consequently is surjective. Hence, is a linear bijection. Hence the spaces and are linearly isomorphic.

5. The Basis for the Space

Let be a paranormed space. Recall that a entire triple sequence of the elements of is called a basis for if and only if, for each there exists a unique entire triple sequence of scalars such that

The series which has the sum is then called the expansion of with respect to and written as Since it is known that the poisson matrix domain of a triple sequence space has a basis if and only if has a basis we have the following, because of the isomorphism is onto, defined in the proof of the Theorem 4.2, the inverse image of the basis of the space is a basis of the new space Therefore, we have the following:

5.1. Theorem. Let for all Define the sequence of the elements of the space by

for every fixed

The sequence is a basis for the space and any has a unique representation of the form

6. The α, β and γ-duals of the Space

In this section, we state and prove the theorems determining the and duals of the space of non-absolute type.

We shall firstly give the definition of and duals of triple sequence spaces and after quoting the lemmas which are needed in proving the theorems given in this section. The set defined by

(6.1)

is called the multiplier space of the triple sequence space and One can easily observe for a triple sequence space with that the inclusions and hold.

The and duals of a triple sequence space are also referred as Köthe-Toeplitz dual, generalized Köthe-Toeplitz dual and Garling dual of a sequence space, respectively

For the give the and duals of the space of non-absolute type, we need the following Lemma.

6.1. Lemma. Let be a binomial Poisson matrix. Then the following statments hold

(1)

(6.2)

(2) (6.2) holds and

(6.3)

(3)

(6.4)

(6.5)

(6.6)

(4) (6.4), (6.5) (6.6) hold

(6.7)

(5)

(6.8)

(6)

(6.9)

(6.10)

(7)

(6.11)

(6.12)

(8) {6.11} and (6.12) hold, and

(6.13)

6.2. Theorem. Let and

for . Define the sets and as follows:

where the poisson matrix

(6.14)

Then

Proof: We choose the sequence We can easily derive that with the that

(6.15)

for all where defined by (6.14). It follows from (6.15) that whenever if and only if whenever This means that if and only if Then we observe that

Competing Interests

The authors declare that there is not any con.ict of interests regarding the publication of this manuscript.

References