A New Paranormed Sequence Space Defined by Regular Bell Matrix

: This paper aims to construct a new paranormed sequence space by the aid of a regular matrix of Bell numbers. As well, its special duals such as  , ,   duals are presented and Schauder basis is determined. Moreover, certain matrix classes for this space are characterized.


I. Introduction
A linear subspace of the set of all real valued sequences  is known as a sequence space.The notations 0 , , c c   and p  represent the set of all convergent sequences, the set of all convergent to zero sequences, the set of all bounded sequences and the set of all sequences constituting p  absolutely convergent series, respectively.Also, these spaces are Banach spaces with regard to the following norms For brevity, here and afterwards, it means that the summation without limits runs from 0 to  .
Let H be a linear topological space over R .If there exists a subadditive function :      for all   R and , t H  then, H is known as paranormed space.
Assume here and after that   i p be a bounded sequence in R such that > 0,

S
P Then, we have from Maddox (Maddox, 1967) that for any   R and In what follows, suppose that Then, Maddox (Maddox 1968(Maddox , 1988) (consult also Nakano, 1951 andSimons, 1965) defined the linear space For each , i  N let i U be the sequence in the i th row of an infinite matrix The sequence Ut will be used in the sequel as U  transform of a sequence   = j t t   and its i th ( ) i  N entry is given by  on the condition that the above series is convergent for each .j  N U is called as a matrix mapping from a sequence space H to a sequence space G if the sequence Ut exists and Ut G  for all .

t H 
The impression

 
, H G represents the class of all infinite matrices between the spaces H and G .
This paper deals with well known Bell numbers whose th i entry is denoted by .
i B The th i Bell number represents how many different ways a set of i elements may be divided into non-empty subsets.For instance, there are five possible divisions of the numbers {1, 2, 3}: The first few Bell numbers are 1, 2,5,15,52, 203,877, 4140,... with the initial condition 0 = 1.B They can be defined as , S i j is the Stirling numbers of the second kind, counts the set partitions of {1, 2, 3,..., } i which consists of exactly j subsets or parts.
Quite recently, Karakas (Karakas, 2023) introduced a new matrix  and observed that this matrix is regular.Also, he considered the sequence spaces as the set of all sequences whose B   transforms are in the spaces p  and ,   respectively.In other words, Also, some topological properties such as giving Schauder basis, determining the   ,   and   duals, characterizing some matrix classes on   p B   and geometric properties like uniform convexity, strict convexity, super reflexivity for the resulting spaces are given.
We will use the sequence i y as the B   transform of a sequence holds for all .j  N The main purpose of this paper is to define new paranormed space . In other words, we generalize some conclusions, obtained in Karakas (Karakas, 2023).We examine a few topological features, including completeness, the Schauder basis, the   ,   and   duals.On these spaces, several matrix mappings are also classified.

II. New paranormed sequence space
In this section, we give a new paranormed space by means of the Bell matrix, then, present the Schauder basis of the resulting paranormed space and prove its completeness.
Define the sequence space Let us begin with the completeness of this new space.
it follows from Maddox (Maddox, 1988) that is linear with respect to scalar multiplication and the coordinatewise addition.Also, it is obvious that for any   R in view of (1.1) and (2.1).
  , and the fact that shows the continuity of scalar multiplication.Then, In order to demonstrate the being a Cauchy sequence of real numbers for each as n   for every fixed .i  N Taking into consideration these infinitely many limits for all fixed m  N and   0 , .
n k i   If we take the limit m   and k   in (2.3), then we have and if we apply the Minkowski's inequality, the we have


In order to provide a Schauder basis for our paranormed sequence space, we are in a position to do so.
where i  N is fixed.Then, the sequence   Now, there exists an integer 0 v such that for a given > 0 As a result, we can see for all which yields (2.4).For the purpose of the uniqueness of the representation (2.4), let We give some definitions and lemmas before our results.
The following set   , S H G is the multiplier space of H and G defined by The definitions of the   ,   and   duals of a sequence space H using this notation are as follows


In this context, cs and bs represent, respectively, the spaces of sequences with convergent and bounded series.
Lemma 4. (Grosseerdmann, 1993) for some integer > 1.K ô stands for the family of all finite subsets of N in this context.
2) Let 0 < 1 Lemma 5. (Lascarides and Maddox 1970) for some integer > 1.K .We define the following sets: Proof.We notice that the equality  .Thus by using Lemma 4, we realise that    as follows: where the matrix = ( ) mn E e is defined by where the matrix = ( ) mn E e is defined as in the proof of previous theorem.
So, we obtain the expected result from (3.1) and (3.2).

IV. Matrix transformations
In the present section, our aim is to characterize some matrix classes on the space ( , ) B p   . The following theorem includes the exact conditions for 0 < < ( 1) < sup  if and only if (4.2) holds and there is an for every m  N .Thus, we have by (1.2) that for all , m k  N .In view of the hypothesis, we obtain from (4.4) that   we have for every fixed n  N .This yields the necessity of (4.6).
Hence, for all , k m  N , the inequality Let M  N and > 1 I m  N as k   .If we consider (4.5) and the following inequality for any , c d  C and > the necessity conditions for (4.2) and (4.3) are obviously seen from previous theorem and the inclusion c    .Let's now consider the sequence ( ) n s that was specified in the proof of Theorem 3 to show that the necessity condition for (4.6).Since Vz , i.e. the V -transform of every ( Now, we recall the definition of Schauder basis prior to establishing the Schauder basis for the matrix domain of our particular matrix.Let   , H  be a paranormed space.A sequence j H   is a Schauder basis for H if and only if there exists a unique sequence of scalars i