On Locally Convex Topological Vector Space Valued Null Function Space c 0 ( S , T , Φ , ξ , u ) Defined by Semi Norm and Orlicz Function

The aim of this paper is to introduce and study a new class c0 (S, T, Φ, ξ, u) of locally convex space Tvalued functions using Orlicz function Φ as a generalization of some of the well known sequence spaces and function spaces. Besides the investigation pertaining to the structures of the class c0 (S, T, Φ, ξ, u), our primarily interest is to explore some of the containment relations of the class c0 (S, T, Φ, ξ, u) in terms of different ξ and u so that such a class of functions is contained in or equal to another class of similar nature.


INTRODUCTION
We begin with recalling some notations and basic definitions that are used in this paper.Definition 1: A sequence space S is said to be solid It is noted that an Orlicz function is always unbounded and an Orlicz function satisfies the inequality Φ(γs)  γ Φ (s), 0 < γ <1 (Krasnosel'skiî & Rutickiî, 1961).
Definition 3: Lindenstrauss and Tzafriri (1971) used the notion of Orlicz function to construct the sequence for some r > 0. They proved that the space l Φ equipped with the norm defined by || ξ This becomes a Banach space, which is called an Orlicz sequence space.The space l Φ (S) is closely related to the space l p which is an Orlicz sequence space with Subsequently, Kamthan and Gupta (1981), Rao and Ren (1991), Parashar and Choudhary (1994), Chen (1996), Ghosh and Srivastava (1999), Rao and Subremanina (2004), Savas and Patterson (2005), Bhardwaj and Bala (2007), Khan (2008), Basariv and Altundag (2009), Kolk (2011), Pahari (2013), Srivastava andPahari (2011 &2013) and many others have been introduced and studied the algebraic and topological properties of various sequence spaces using Orlicz function as a generalization of several well known sequence spaces.
Definition 4: A topological linear space S is a vector space S over a topological field K (most often the complex numbers C with their standard topologies) which is endowed with a topology ℑ such that if s 1 , s 2 ∈ S, α ∈ K; the mappings: (i) vector addition S×SS such that (s 1 , s 2 ) → s 1 + s 2 and (ii) scalar multiplication K×S S such that (α, s) →α s are continuous.
This topology ℑ is called a vector topology or a linear topology on S. If ℑ is given by some metric then the topological vector space is called a linear metric space.All normed spaces or inner product spaces endowed with the topology defined by its norm or inner product are well-known examples of topological vector spaces.
A local base of topological vector space S is a collection B of neighbourhood θ such that every neighbourhood of θ contains the member of B.
A set S in a topological vector space S is said to be absorbing if for every s ∈ S there exists an α > 0 such that s ∈ ν S for all ν ∈ C such that |ν| ≥ α ; and balanced if ν S ⊂ S for every ν ∈ C such that |ν| ≤ 1.
It is called convex in S if for every α ≥ 0, we have α S + (1-α) S ⊂ S; and absolutely convex in S if it is both balanced and convex.
Definition 5: The gauge or Minkowski functional of a set A in a vector space X is a map x → q A (x) from X into the extended set R + ∪ {∞} of non-negative real numbers defined as follows: Definition 6: A seminorm (pseudonorm) on a linear space S over the scalar C with zero element θ is a subadditive function p : S → R + satisfying p(αs) = |α| p(s), for all α∈C and s∈S.
Clearly, if p(s) = 0 implies s = θ,then p is a nom.
If S is a vector space equipped with a family {p i : i ∈ I} of seminorms then there exists a unique locally convex topology ℑ on S such that each p i is ℑ -continuous (Rudin, 1991& Park, 2005).

The class c 0 (S, T, Φ Φ Φ Φ, ξ ξ ξ ξ, u) of locally convex space valued functions
Let S be an arbitrary non empty set (not necessarily countable) and F(S) be the collection of all finite subsets of S. Let (T, ℑ) be a Hausdorff locally convex topological vector space (lcTVS) over the field of complex numbers C and T * be the topological dual of T. Let U(T) denotes the fundamental system of balanced, convex and observing neighbourhoods of zero vector θ of T. p U will denote gauge or Minkowski functional of U ∈  (T).
Thus, D = {p U : U∈ (T)}is the collection of all continuous seminorms generating the topology ℑ of T. Let u and w be any functions on S → R + , the set of positive real numbers, and l ∞ (S, R + ) = { u : S → R + such that sup s u(s) < ∞}.
Further, we write ξ, η for functions on S →C -{0}, and the collection of all such functions will be denoted by s(S, C -{0}).

RESULTS
We explore the structure of the class c 0 (S, T, Φ, ξ, u) of lc TVS T -valued functions by investigating the conditions in terms of different u and ξ so that a class is contained in or equal to another similar class and thereby derive the conditions of their equality.
We shall denote the zero element of this class by θ,which we shall mean the function of θ : S → T such that θ (s) = 0, for all s∈ S.
Moreover, we shall frequently use the notations , for scalar α.But when the functions u(s) and w(s) occur, then to distinguish L, we use the notations L(u) and L(w) respectively.
Let φ ∈ c 0 (S, T, Φ, ξ, u), r 1 > 0 be associated with φ and ε > 0. Then for p U ∈ D, there exists Let us choose r such that r 1 < m r.Then for such r, using non decreasing property of Φ, we have Since p U ∈ D is arbitrary in the above discussion, therefore we easily get φ ∈ c 0 (S, T , Φ , η, u).
This proves that c 0 (S, T, Φ, ξ, u) ⊂ c 0 (S, T, Φ, η, u). Proof: Then we can find a sequence < s k > of distinct points in S such that for every k ≥ 1, We now choose t ∈ T and p V ∈ D such that p V (t) = 1 and define φ : S → T by Let r > 0 .Then for each p U ∈ D and each k ≥ 1, we have Thus for a given ε > 0, we can find a finite subset J of S satisfying This clearly shows that φ ∈ c 0 (S, T, Φ, ξ, u).
But for each k ≥ 1, in view of equations ( 1) and (2), we have This shows that φ ∉ c 0 (S, T, Φ, η, u), a contradiction.This completes the proof.
Let φ ∈ c 0 (S, T, Φ, η, u), r 1 > 0 be associated with φ and ε > 0. Then for p U ∈ D, there exists Let us choose r such that d r 1 < r.Then for such r, using non decreasing property of Φ, we have Since p U ∈ D is an arbitrary, it clearly shows that φ ∈ c 0 (S, T, Φ, ξ, u).
For any ξ, η ∈ s(S, C -{0}) such that Then there exists a sequence < s k > of distinct points in S such that for each Now, we choose t ∈ T and p V ∈ D with p V (t) = 1 and define φ : S → T by -1 k -1/u(s) t‚ for s = s k ‚ k = 1‚ 2‚… ‚ and θ‚ otherwise.
...( 4) Let r > 0 .Then for each p U ∈ D and each Thus for given ε > 0, we can find a finite subset J of S such that This shows that φ ∈ c 0 (S, T, Φ, η, u).But on the other hand for each k ≥ 1 and in view of equations ( 3) and ( 4), we have This shows that φ∉ c 0 (S, T, Φ, ξ, u), a contradiction.
This completes the proof.
Theorem 10: Proof: Assume that lim inf s w(s) u(s) > 0. Then there exists m > 0 such that w(s) > m u(s) for all but finitely many s ∈ S.
On the other hand for each k ≥ 1 and in view of equations ( 5) and ( 6), we have This shows that φ ∉ c 0 (S, T, Φ, ξ, w), a contradiction.This completes the proof.
On combining the Theorems 10 and 11, one can obtain: Proof: Assume that lim sup s w(s) u(s) < ∞.
Then there exists a constant d > 0 such that w(s) < d u(s) for all but finitely many s ∈ S.
This completes the proof.
Then there exists a sequence < s k > in S of distinct points such that w(s k ) > k u(s k ) for each k ≥ 1. .………… (7) Now, taking p V ∈ D and t ∈ T with p V (t) = 1.
The proof is now complete.

CONCLUSION
Present paper examined some conditions that characterize the linear space structures and containment relations on the locally convex topological vector space valued null functions defined by semi norm and Orlicz function.In fact, these results can be used for further generalization to investigate other properties of the function spaces using Orlicz function.