Topological properties of some sequences defined over 2-normed spaces

The paper investigates some classes of real number sequences over 2-normed spaces defined by means of Orlicz functions, a bounded sequence of strictly positive real numbers, a multiplier and a normal paranormed sequence space. Relevant properties of such classes have been investigated. Moreover, relationships among different such classes of sequences have also been studied under various parameters and conditions. Finally, the spaces are investigated for some other useful properties. The conclusion section provides many interesting facts for further research.

Then ‖.,.‖ is called a 2-norm on X and (X, ‖.,.‖) is called a linear 2-normed space (Gähler 1965). Some of the basic properties of the 2-norms includes that they are non-negative, and ‖x, y + αx‖ = ‖x, y‖ for every x, y ∊ X and any real number α.
A sequence {x n } in a linear 2-normed space (X, ‖.,.‖) is called a Cauchy sequence if lim n,m→∞ ‖x n − x m , z‖ = 0 for all z ∊ X. A sequence {x n } in a linear 2-normed space (X, ‖.,.‖) is called a convergent sequence if there is an x ∊ X such that lim n→∞ ‖x n − x, z‖ = 0 for all z ∊ X. A linear 2-normed space in which every Cauchy sequence is a convergent sequence is called a 2-Banach space.
The details about above and associated notions and results, we refer to the book by Freese and Cho (2001). Savas (2010) and Dutta (2010) can be seen for some use of the 2-norm structure in construction of sequence spaces.
Let P be a subset of the set of all scalar valued sequences w. Now we recall the following notions.
A scalar valued paranormed (Maddox 1970) sequence space (P, g P ), where g P is a paranorm on P is called monotone paranormed space if x = (x k ) ∊ P, y = (y k ) ∊ P and |x k | ≤ |y k | for all k implies g P (x) ≤ g P (y).
A sequence space P with linear topology is called a K-space provided each of the maps Let (P, g P ) be a paranormed space and (a n ) ⊂ P, where a n = a n k . If a n k → 0 as n → ∞ for each k implies g P (a n ) → 0 as n → ∞, then we say that the co-ordinate wise convergence implies convergence in g P , e.g., c 0 , ℓ 1 , ℓ ∞ , etc.
The following inequalities (Maddox 1970) will be used throughout the paper.
Proposition 1 Let (p k ) be a bounded sequence of strictly positive real numbers with 0 < p k ≤ sup p k = H, D = max(1, 2 H−1 ). Then

The new class F(‖.,.‖, M, p, s) and some other classes
In this section, we construct the new sets to be investigated and give a few descriptions of such sets along with intended aims for results concerning the sets and their possible extensions and derivatives. Let (F, g F ) be a normal paranormed sequence space with paranorm g F which satisfies the following properties: (i) g F is a monotone paranorm; (ii) coordinate wise convergence implies convergence in paranorm g F , which implies that for each (X n ) = (X n k ) ∈ F , n, k ∈ N, Let M be a Orlicz function and (N, ‖.,.‖) be a 2-normed space. We now define the new class of sequences as follows for every z ∊ N: where s ≥ 0 and {p k } is a bounded sequence of strictly +ve real numbers with inf p k > 0.
This class give rises different other classes of sequences as follows: where r is any positive integer. and so on. We define a function on F ( ., . , M, p, s) as follows which is proved to be a paranorm in the next section: The above classes of sequences of real numbers give rise to many well known sequence spaces on specifying the space F, the Orlicz function M, the bounded sequence {p k } of positive real numbers, s ≥ 0 and the base space (N, ‖.,.‖). Further, we can derive several other similar classes for study. The main results of the paper are obtained using the properties of Orlicz functions, 2-norm spaces and most importantly that are of normal paranormed spaces with monotone paranorm and coordinate wise convergence property. One may find it interesting and useful to study further the sets for several other algebraic and topological properties as well as convergence and completeness related and geometric properties. The last few results also hint for several other possible rich property of the sets.

Main results
In this section, we first examine the linearity of the sets defined above. Then the sets will be investigated for completeness under a suitably defined paranorm. Further, the sets will be examined for K-space property. The next few results will be given for the set F(‖.,.‖, M, p, s) only as for other sets the proofs can be obtained applying similar arguments.

Theorem 1 The set F(‖.,.‖, M, p, s) is linear over the set of real numbers R.
Proof M, p, s) and α, β ∈ R. Then there exist some positive numbers ρ 1 and ρ 2 such that for every z ∊ N ,.‖, M, p, s) and completes the proof.
Also, by taking α = β = 1 in the previous theorem and using the fact that g F is mono- We are only left to show that g is continuous under scalar multiplication. Let λ be any number. Then for some ρ > 0, Let ɛ > 0 be arbitrarily chosen and let K be a positive integer such that for some ρ > 0, Let 0 < |λ| < 1, using convexity of M and the property (N3) of 2-norm, for k > K we get Since M is continuous everywhere in [0, ∞) and by the definition of g F , it follows that for Let L be such that |λ n | < δ for n > L, then for n > L and k ≤ K. Hence for n > L and for all k. Hence λ n X → θ as n → ∞.  (1), where F is a K-space.
Then we have for i, j ≥ N 0 such that for every z ∊ N, Since F is a K-space, p k ≥ 0 and we can choose s suitably so that for each k and for i, j ≥ N 0 and z ∊ N. Therefore, Thus we get for each k and for i, j ≥ N 0 and for every z ∊ N. Therefore X i k becomes a Cauchy sequence in N. Since (N, ‖.,.‖) is complete, there exist X = (X k ) ∊ N such that X i k → X k as i → ∞ for each k. Since M is continuous it shows that for each k, z ∊ N and for some ρ > 0. Consequently, for each k, z ∊ N and for some ρ > 0.

Theorem 4 F(‖.,.‖, M, p, s) is a K-space if F is a K-space.
Proof Let us define a mapping by P n (X) = X n , ∀n ∈ N. To show P n is continuous.
Let (X m ) be a sequence in F(‖.,.‖, M, p, s) such that X m → g 0 as m → ∞. Then for some suitable choice of ρ > 0, Since F is a K-space, this implies that for each k and as m tending to ∞, for some ρ > 0. Since M is an Orlicz function, it follows that Consequently, X m → 0 in N. Hence the proof.

Relationship results
In this section, we shall investigate the relationship among the spaces defined in second section and their possible variants under different conditions.
In the next two results, we shall shows how the addition and composition of two different Orlicz functions effect the spaces in term of their relationship of size.
Let us define the sets for some ρ > 0.
If k ∊ N 2 , Since M 2 is non-decreasing and convex it follows that Again since M 2 satisfies Δ 2 -condition, we have So, and therefore, Hence from (2) and (3) we have for all k. Then the proof follows by the normality of F. We have the well known inclusion c 0 ⊂ c ⊂ ℓ ∞ . The following result shows that if F is replaced by these three spaces, the corresponding extended versions also preserve this inclusion.
In composite Orlicz sequence spaces, the following result gives a connection between such spaces which depend on the number of participating Orlicz functions and satisfying certain condition.
Theorem 10 Let M be an Orlicz function satisfying Δ 2 -condition and M(t)/t ≤ A for t ≥ 0, where A is a constant. If r, n ∈ N such that r > n then Proof Let r − n = ℓ > 0. Now Since M satisfying Δ 2 -condition, we have after rth step, for some constant L > 0. Therefore

Further properties
In this section, we shall investigate essentially few more properties. These properties may influence the readers to study further such spaces for several other algebraic and topological behaviours including those of dual spaces. The space F(‖.,.‖, M, p, s) is not convergence free in general. In order to establish it, it is easy to construct an example. Hence we have the following result.
Remark 1 The space F(‖.,.‖, M, p, s) is not convergence free in general.
Let us define a sequence (Y k ) as follows: Then X k = 0 implies Y k = 0, but (Y k ) ∉ F(‖.,.‖, M, p, s). However, the space F(‖.,.‖, M, p, s) is solid and symmetric in general. The following two results establish our claim with proof.
Proof Let X = (X k ) ∊ F(‖.,.‖, M, p, s), and Y = (Y m k ) be an arrangement of the sequence (X k ) such that X k = Y m k for each k ∊ N. Then For k even, X k = 1 k + 1 For k odd, Y k = 0 and for k even, Y k = k + 1 �Y k , Z� ≤ �X k , Z� for every Z ∈ N .
Hence these spaces are symmetric in general. There is a close connection between Banach spaces and 2-Banach spaces. Now we shall try to reflect this connection in our definition of the spaces as well as in the completeness result.
Consider the norm ‖.‖ defined on a linear 2-normed space (X, ‖.,.‖) by the function for any fixed y, z ∊ X and ‖y, z‖ ≠ 0. Then the function ‖.‖ is a norm on X (Freese and Cho 2001). We recall the following result and for details, we refer to (Freese and Cho 2001).
Proposition 13 If (X, ‖.,.‖) is a linear 2-normed space possessing Property (K) (Freese and Cho 2001, p. 16) and having a norm defined on it by ‖a‖ = ‖a*, a‖ + ‖b*, a‖ for a* and b* in X such that ‖a*, b*‖ ≠ 0, then X is a 2-Banach space if and only if X is a Banach space relative to this norm.
For the sake of comparison between natural norm and the norm obtain from 2-norm as described above, we shall call the later as derived 1-norm or simply derived norm.
Using this concept of derived 1-norm, we can redefine our sets over derived 1-norm instead of 2-norm and we will get the similar results of this paper. If the 2-normed space (N, ‖.,.‖) possesses the Property (K), we can modify our completeness result as follows.