# The Semi-Difference Entire Sequence Spacecs∩d1

Authors: N. Subramanian; K. Chandrasekhara Rao; K. Balasubramanian
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

---

## Abstract

LetΓ denote the space of all entire sequences. Let Λ denote the space of all analytic sequences. In this paper, we introduce a new class of sequence space, namely, the semi-difference entire sequence space cs∩d1. It is shown that the intersection of all semi-difference entire sequence spaces cs∩d1 is I⊂cs∩d1 and Γ(Δ)⊂I.


---

## Body

## 1. Introduction

A complex sequence, whosekth term is xk, is denoted by {xk} or simply x. Let w be the set of all sequences and ϕ be the set of all finite sequences. Let ℓ∞,c,c0 be the classes of bounded, convergent, and null sequence, respectively. A sequence x={xk} is said to be analytic if supk|xk|1/k<∞. The vector space of all analytic sequences will be denoted by Λ. A sequence x is called entire sequence if limk→∞|xk|1/k=0. The vector space of all entire sequences will be denoted by Γ.Given a sequencex={xk}, its nth section is the sequence x(n)={x1,x2,…,xn,0,0,…}. Let δ(n)=(0,0,…,1,0,0,…), 1 in the nth place and zeros elsewhere, s(k)=(0,0,…,1,-1,0,…), 1 in the nth place, −1 in the (n+1)th place and zeros elsewhere. An FK-space (Fréchet coordinate space) is a Fréchet space which is made up of numerical sequences and has the property that the coordinate functionals pk(x)=xk(k=1,2,3,…) are continuous.We recall the following definitions (one may refer to Wilansky [1]).An FK-space is a locally convex Fréchet space which is made up of sequences and has the property that coordinate projections are continuous. A metric space(X,d) is said to have AK (or sectional convergence) if and only if d(x(n),x)→x as n→∞, (see [1]). The space is said to have AD (or) be an AD-space if ϕ is dense in X, where ϕ denotes the set of all finitely nonzero sequences. We note that AK implies AD (one may refer to Brown [2]).IfX is a sequence space, we define (i)
X′= the continuous dual of X;(ii)
Xα={a=(ak):∑k=1∞|akxk|<∞,foreachx∈X};(iii)
Xβ={a=(ak):∑k=1∞akxkisconvergent,foreachx∈X};(iv)
Xγ={a=(ak):supn|∑k=1nakxk|<∞,foreachx∈X};(v)
letX be an FK-space ⊃ϕ. Then, Xf={f(δ(n)):f∈X′}.Xα,Xβ,Xγ are called the α (or Köthe-Töeplitz) dual of X, β—(or generalized Köthe-Töeplitz) dual of X, γ dual of X. Note that Xα⊂Xβ⊂Xγ. If X⊂Y, then Yμ⊂Xμ, for μ=α, β, or γ.Letp=(pk) be a sequence of positive real numbers with supkpk=G and D=max{1,2G-1}. Then, it is well known that for all ak,bk∈C, the field of complex numbers, for all k∈N,(1.1)|ak+bk|pk≤D(|ak|pk+|bk|pk).Lemma 1.1 (Wilansky [1, Theorem 7.2.7]).
LetX be an FK-space ⊃ϕ. Then, (i)
Xγ⊂Xf;(ii)
ifX has AK, Xβ=Xf;(iii)
ifX has AD, Xβ=Xγ.


## 2. Definitions and Preliminaries

LetΔ:w→w be the difference operator defined by Δx=(xk-xk+1)k=1∞. Let (2.1)Γ={x∈w:lim⁡k→∞⁡(|xk|1/k)=0},Λ={x∈w:sup⁡k⁡(|xk|1/k)<∞}.
Define the sets Γ(Δ)={x∈w:Δx∈Γ} and Λ(Δ)={x∈w:Δx∈Λ}.The spacesΓ(Δ) and Λ(Δ) are the metric spaces with the metric(2.2)d(x,y)=inf⁡{ρ>0:sup⁡k⁡(|Δxk-Δyk|1/k)≤1}.Because of the historical roots of summability in convergence, conservative space and matrices play a special role in its theory. However, the results seem mainly to depend on a weaker assumption, that the spaces be semi-conservative. (See Wilansky [1]).Snyder and Wilansky [3] introduced the concept of semi-conservative spaces. Snyder [4] studied the properties of semi-conservative spaces. Later on, in the year 1996 the semi replete spaces were introduced by Rao and Srinivasalu [5].In a similar way, in this paper, we define semi-difference entire sequence spacecs∩d1, and show that semi-difference entire sequence space cs∩d1 is I⊂cs∩d1 and Γ(Δ)⊂I.


## 3. Main Results

Proposition 3.1.
Γ⊂Γ(Δ) and the inclusion is strict.Proof.
Letx∈Γ. Then, we have
(3.1)|xk|1/k⟶0,ask⟶∞,(3.2)|Δxk|1/k2≤12(|xk|1/k)+12(|xk+1|1/k),by(1.1)⟶0,ask⟶∞by(3.1).
Letx∈Γ. Then, we have(3.3)(|xk|1/k)⟶0ask⟶∞.
Then,(xk)∈Γ(Δ) follows from the inequality (1.1) and (3.3).
Consider the sequencee=(1,1,…). Then, e∈Γ(Δ) but e∉Γ. Hence, the inclusion Γ⊂Γ(Δ) is strict.Lemma 3.2.
A∈(Γ,c) if and only if
(3.4)lim⁡n→∞⁡ankexistsforeachk∈N,(3.5)sup⁡n,k⁡|∑i=0kani|<∞.Proposition 3.3.
Define the setd1={a=(ak)∈w:supn,k∈N|∑j=0k(∑i=jnai)|<∞}. Then, [Γ(Δ)]β=cs∩d1.Proof.
Consider the equation(3.6)∑k=0nakxk=∑k=0nak(∑j=0kyj)=∑k=0n(∑j=knaj)yk=(Cy)n,
where C=(Cnk) is defined by
(3.7)Cnk={∑j=knaj,if0≤k≤n,0,ifk>n;n,k∈N.
Thus, we deduce from Lemma3.2 with (3.6) that ax=(akxk)∈cs whenever x=(xk)∈Γ(Δ) if and only if Cy∈c whenever y=(yk)∈Γ, that is C∈(Γ,c). Thus, (ak)∈cs and (ak)∈d1 by Lemma 3.2 and (3.5) and (3.6), respectively. This completes the proof.Proposition 3.4.
Γ(Δ) has AK.Proof.
Letx={xk}∈Γ(Δ). Then, (|Δxk|1/k)∈Γ. Hence,
(3.8)sup⁡k≥n+1(|Δxk|1/k)⟶0,ask⟶∞,(3.9)d(x,x[n])=inf⁡{ρ>0:sup⁡k≥n+1⁡(|Δxk|1/k)≤1}⟶0,asn⟶∞,byusing(3.8)⟹x[n]⟶xasn⟶∞,⟹Γ(Δ)hasAK.
This  completes the proof.Proposition 3.5.
Γ(Δ) is not solid.To prove Proposition3.5, consider (xk)=(1)∈Γ(Δ) and αk={(-1)k}. Then (αkxk)∉Γ(Δ). Hence, Γ(Δ) is not solid.Proposition 3.6.
(Γ(Δ))μ=cs∩d1 for μ=α,β,γ,f.Proof.
Step 1.
Γ(Δ) has AK by Proposition 3.4. Hence, by Lemma 1.1(ii), we get (Γ(Δ))β=(Γ(Δ))f. But (Γ(Δ))β=cs∩d1. Hence,
(3.10)(Γ(Δ))f=cs∩d1.Step 2.
SinceAK⇒AD. Hence, by Lemma 1.1(iii), we get (Γ(Δ))β=(Γ(Δ))γ. Therefore,
(3.11)(Γ(Δ))γ=cs∩d1.Step 3.
Γ(Δ) is not normal by Proposition 3.5. Hence by Proposition 2.7 of Kamthan and Gupta [6], we get
(3.12)(Γ(Δ))α≠(Γ(Δ))γ≠cs∩d1.
From (3.10) and (3.11), we have(3.13)(Γ(Δ))β=(Γ(Δ))γ=(Γ(Δ))f=cs∩d1.Lemma 3.7 (Wilansky [1, Theorem 8.6.1]).
Y⊃X⇔Yf⊂Xf where X is an AD-space and Y an FK-space.Proposition 3.8.
LetY be any FK-space ⊃ϕ. Then, Y⊃Γ(Δ) if and only if the sequence δ(k) is weakly converges in cs∩d1.Proof.
The following implications establish the result.Y⊃Γ(Δ)⇔Yf⊂(Γ(Δ))f, since Γ(Δ) has AD by Lemma 3.7.⇔Yf⊂cs∩d1, since (Γ(Δ))f=cs∩d1.⇔ for each f∈Y′, the topological dual of Y.⇔f(δ(k))∈cs∩d1.⇔δ(k) is weakly converges in cs∩d1.
This completes the proof.


## 4. Properties of Semi-Difference Entire Sequence Spacecs∩d1

Definition 4.1.
An FK-spaceΔX is called “semi-difference entire sequence space cs∩d1” if its dual (ΔX)f⊂cs∩d1.In other wordsΔX is semi-difference entire sequence space cs∩d1 if f(δ(k))∈cs∩d1 for all f∈(ΔX)′ for each fixed k.Example 4.2.
Γ(Δ) is semi-difference entire sequence space cs∩d1. Indeed, if Γ(Δ) is the space of all difference of entire sequences, then by Lemma 4.3, (Γ(Δ))f=cs∩d1.Lemma 4.3 (Wilansky [1, Theorem 4.3.7]).
Letz be a sequence. Then (zβ,P) is an AK space with P=(Pk:k=0,1,2,…), where P0(x)=supm|∑k=1mzkxk|, and Pn(x)=|xn|. For any k such that zk≠0,Pk may be omitted. If z∈ϕ,P0 may be omitted.Proposition 4.4.
Letz be a sequence. zβ is a semi-difference entire sequence space cs∩d1 if and only if z is in cs∩d1.Proof.
Suppose thatzβ is a semi-difference entire sequence space cs∩d1. zβ has AK by Lemma 4.3. Therefore zββ=(zβ)f by Lemma 1 [1]. So zβ is semi-difference entire sequence space cs∩d1 if and only if zββ⊂cs∩d1. But then z∈zββ⊂cs∩d1. Hence, z is in cs∩d1.
Conversely, suppose thatz is in cs∩d1. Then zβ⊃{cs∩d1}β and zββ⊂{cs∩d1}ββ=cs∩d1. But(zβ)f=zββ. Hence, (zβ)f⊂cs∩d1. Therefore zβ is semi-difference entire sequence space cs∩d1. This completes the proof.Proposition 4.5.
Every semi-difference entire sequence spacecs∩d1 contains Γ.Proof.
LetΔX be any semi-difference entire sequence space cs∩d1. Hence, (ΔX)f⊂cs∩d1. Therefore f(δ(k))∈cs∩d1 for all f∈(ΔX)′. So, {δ(k)} is weakly converges in cs∩d1 with respect to ΔX. Hence, ΔX⊃Γ(Δ) by Proposition 3.8. But Γ(Δ)⊃Γ. Hence, ΔX⊃Γ. This completes the proof.Proposition 4.6.
ΔX is semi-difference entire sequence space cs∩d1.Proof.
LetΔX=⋂n=1∞ΔXn. Then ΔX is an FK-space which contains ϕ. Also every f∈(ΔX)′ can be written as f=g1+g2+…+gm, where gk∈(ΔXn)′ for some n and for 1≤k≤m. But then f(δk)=g1(δk)+g2(δk)+⋯+gm(δk). Since ΔXn(n=1,2,…) are semi-difference entire sequence space cs∩d1, it follows that gi(δk)∈cs∩d1 for all i=1,2,…m. Therefore f(δk)∈cs∩d1 for all k and for all f. Hence, ΔX is semi-difference entire sequence space cs∩d1. This completes the proof.Proposition 4.7.
The intersection of all semi-difference entire sequence spacecs∩d1 is I⊂(cs∩d1)β and Γ(Δ)⊂I.Proof.
LetI be the intersection of all semi-difference entire sequence space cs∩d1. By Proposition 4.4, we see that the intersection
(4.1)I⊂∩{zβ:z∈cs∩d1}={cs∩d1}β.
By Proposition4.6 it follows that I is semi-difference entire sequence space cs∩d1. By Proposition 4.5, consequently(4.2)ΓM=Γ(Δ)⊂I.
From (4.1) and (4.2), we get I⊂{cs∩d1}β and Γ(Δ)⊂I. This completes the proof.Corollary 4.8.
The smallest semi-difference entire sequence spacecs∩d1 is I⊂(cs∩d1)β and Γ(Δ)⊂I.



---

*Source: 143974-2011-05-14.xml*