On Certain Topological Structures of Summable Paranormed Sequence Space Defined in Two-Normed Space

The aim of this paper is to introduce and study a new class  (S, –, u– ) of sequences with values in 2Banach space as a generalization of the familiar space of summable sequences . We explore some of the preliminary results that characterize the topological linear structure of  the class  (S, –, u – ) when topologized it with suitable natural paranorm.


INTRODUCTION
So far, a bulk number of works have been done on various types of paranormed spaces.The concept of paranorm is closely related to linear metric space and its studies on sequence spaces were initiated by Maddox (1969) and many others.
Before proceeding with the main results,we begin with recalling some of the notations and basic definitions that are used in this paper.
Definition 1: A paranormed space (S, P ) is a linear space S with zero element together with a function P : S R + (called a paranorm on S) which satisfies the following axioms: (i) P ( ) = 0; (ii) P (s) = P (-s), for all s S; (iii) P (s 1 + s 2 ) P (s 1 ) + P (s 2 ) , for all s 1 , s 2 S; and (iv) Scalar multiplication is continuous i.e., if < n > is a sequence of scalars with n → as n → ∞ and < s n > is a sequence of vectors with P (s n − s ) → 0 as n → ∞, then P ( n s n − s) → 0 as n → ∞.Note that the continuity of scalar multiplication is equivalent to (ii) if n 0 as n and s be any element in S, then P ( n s) 0, see Wilansky (1978).
Definition 2: Let S be a linear space of dimension > 1 over K, the field of real or complex numbers The concept of 2-normed space was initially introduced by S. GÄahler (1963) as an interesting linear generalization of a normed linear space, which was subsequently studied by Iseki (1976), White and Cho (1984), Freese et al. (1992), Freese and Cho (2001) We shall denote (C) by .Any linear subspace of is then called a sequence space.
Definition 4: A sequence space S is said to be solid if Definition 5:

The Class  (S, -, u -) of 2-Normed Space Valued Sequences
Let u -= < u k > and v -= < v k > be any sequences of strictly positive real numbers and be the sequences of non zero complex numbers.
We now introduce the following classes of 2-normed space S-valued vector sequences In fact, this class is a generalization of the familiar sequence spaces, studied in Pahari (2011Pahari ( , 2013Pahari ( & 2014)), Srivastava and Pahari (2011, 2011& 2013) , using 2-norm

RESULTS
In this section, we shall investigate some results that characterize the linear topological structure of the class  Let s for each t S.This shows that < k s k >  (S, -, u -) and hence  (S, -, u -) is normal.
Theorem 2: For any u - Proof.
Then there exist m > 0 and a positive integer K such that This clearly implies that s - (S, -, u -) and hence  (S, -, u -)  (S, -, u -).This completes the proof.
Theorem 3: For any Proof.
Suppose 0 u k v k < ∞ for all but finitely many values This shows that there exists K 1 such that || k s k , t || for all k K and for each t S.Thus || k s k , t || v k || k s k , t|| u k for all k K and for each t S and consequently Further, let according as k is odd or even integers and hence lim inf k z k > 0. Further, Hence both the conditions of Theorem 4 are satisfied.
Now for each t S, we have But on the other hand, let us choose t S such that || s, t || = 1.Then for each even integer k, we have Theorem 6:  (S, -, u -) forms a linear space over the field of complex numbers C if < u k > is bounded above. Proof.

Narayan Prasad Pahari
We prove below that  (S, -, u -) with respect to P forms a paranormed space.

Let
C and s -= < s k > , w -= < w k >  (S, -, u -).Then we can easily verify that P satisfy the following properties of paranorm.
Next if s - (S, -, u -) , then for > 0 there exists an integer K such that Further if n , we can find N such that for n N, then for each t S, we have Thus for each t S, P ( n s -) for all n N , and hence (b) follows.
We prove the completeness of  (S, -, u -) with respect to the metric d(s -, t -) = P (s -t -).
Let < s -(n) > be a Cauchy sequence in  (S, -, u -) .Then for 0 < < 1 , there exists N such that for all n, m N and for each t S, we have and so for all n, m N and k 1 and for each t S, .
This shows that for each k, < s This proves the completeness of  (S, -, u -).
Narayan Prasad Pahari -) of 2-normed space S-valued sequences by endowing it with suitable natural paranorm.Throughout the work, we denote z k = | k k -1 | u k , sup u k = M and for scalar , A [ ] = max (1, | |).But when the sequences < u k > and < v k > occur, then to distinguish M we use the notations M(u) and M(v) respectively.Theorem 1: The space  (S, -, u -) forms a solid.Proof.
If (i)  lim inf k z k > 0; and (ii) u k v k ,for all but finitely many values of k, then ( S,-, u -) ( S, -, v -) .In the following example, we conclude that ( S, -, u -) may strictly be contained in ( S, -, v -) inspite of the satisfaction of both conditions of Theorem 4. Example 5: Let ( S, ||., .|| ) be a 2-normed space and consider a sequence s -= < s k > defined by s k = 1 k 2k s, if k = 1, 2, 3,…,where s S and s .
(a) P (s -(n) ) 0 and n imply P ( n s -(n) ) 0; and (b) n 0 implies P ( n s -) 0 for each s - (S, -, u -).Now to prove (a) suppose | n | L for all n 1, then in view of (3) , we have for some L > 0 and for all n 1.Thus for every n and r, (for each t S i.e. s -(n) s in  (S,-, u -) , as n .
for each t S .It is said to be a Cauchy if there are t and w in S such that t and w are linearly Banach space if every Cauchy sequence < s n > in S is convergent to some s S.