On some topological properties of generalized difference sequence spaces

© Hindawi Publishing Corp.

ON SOME TOPOLOGICAL PROPERTIES OF GENERALIZED

DIFFERENCE SEQUENCE SPACES

MIKAIL ET

(Received 22 October 1998 and in revised form 12 June 2000)

Abstract.We obtain some topological results of the sequence spaces ∆m_{(X), where} ∆m_{(X)= {x=(xk):(∆}m_{xk)∈X}, (m∈N), and X is any sequence space. We} com-pute thepα-,pβ-, andpγ-duals ofl∞,c, andc0and we investigate theN-(or null) dual of the sequence spaces∆m_(l∞),∆m_{(c), and∆}m_{(c0). Also we show that any matrix map from} ∆m_{(l∞)into aBK-space which does not contain any subspace isomorphic to∆}m_(l∞)is compact.

Keywords and phrases. Difference sequence spaces, statistical convergence, theN-dual, thepα-,pβ-, andpγ-dual.

2000 Mathematics Subject Classification. Primary 40A05, 40C05, 46A45.

1. Introduction. wdenotes the space of all scalar sequences and any subspace of

w is called a sequence space. The following sequence spaces will be used in what follows:

l∞, the space of all bounded scalar sequences; c, the space of all convergent scalar sequences;

c0, the space of all null scalar sequences;

l1, the space of all absolutely 1-summable scalar sequences; s, the space of all real sequences;

s0, the space of all statistically convergent sequences of real numbers; ∆m_(l

∞), the space of all∆m-bounded scalar sequences; ∆m_{(c), the space of all∆}m_{-convergent scalar sequences;} ∆m_{(c0), the space of all∆}m_{-null scalar sequences;} ∆m_(s

0), the space of all∆m-statistically convergent sequences of real numbers.

It is known thatl∞,c, andc0areB-spaces (Banach spaces) with their usual norm x∞= supk|xk|, where k∈ N= {1,2,...}. The sequence spaces l∞(∆m), c(∆m), c0(∆m)have been introduced by Et and Çolak [1]. These sequence spaces are BK

-spaces (Banach coordinate -spaces) with norm

x∆= m

i=1

_xi+∆m_x

∞, (1.1)

wherem∈N,∆◦_{x=(xk),∆x=(xk−xk}

+1),∆mx=(∆mxk)=(∆m−1xk−∆m−1xk+1),

and so

∆m_xk= m ν=0

(−1)v

m ν

For convenience we denote these spaces∆m_(l

∞), ∆m(c), and ∆m(c0) and call∆m

-bounded,∆m_{-convergent, and∆}m_{-null sequences, respectively. The operators}

∆(m)_, (m)_{:w →w (1.3)}

are defined by

∆(1)_xk=xk−xk −1,

(1) xk=

k

j=0

xj, (k=0,1,...),

∆(m)_=∆(1)

0∆(m−1), (m)

=(1)₀(m−1), (m≥2),

(1.4)

and

(m)

0∆(m)=∆(m)₀ (m)

=id, (1.5)

the identity onw(see [4]). For any subsetXofwlet

∆m_(X)=x=xk:∆m_{xk∈X . (1.6)}

Now we define

∆(m)_xk=m ν=0

(−1)ν

m ν

xk−ν. (1.7)

It is trivial that(∆m_{xk)∈X if and only if(∆}(m)_{xk)∈X, forX=l∞,c orc0. In [4],}

Malkowsky and Parashar also showed that the sequence spaces∆m_(l

∞),∆m(c), and ∆m_(c

0)are alsoBK-spaces with norm x∆1=sup

k

_∆(m)_{xk. (1.8)}

It is trivial that the norms (1.1) and (1.8) are equivalent. Obviously

∆(m)_:∆(m)_{(X)→X, ∆}(m)_x=y=∆(m)_xk, (m)

:X →∆(m)_(X), (m)_x=y=

(m) xk

_(1.9)

are isometric isomorphism, forX=l∞, corc0.

Hence ∆m_(l

∞), ∆m(c), and ∆m(c0)are isometrically isomorphic to l∞,c, and c0,

respectively. Thusl1is continuous dual of∆m(c)and∆m(c0).

Throughout the paper, we writekfor∞k=1and limnfor limn→∞.

LetA=(ank)be an infinite matrix of complex numbers. LetEandF beBK-spaces. We write Ax =(An(x)) if An(x)= kankxk converges for each n∈N. If Ax = (An(x))∈E for eachx=(xk)∈F, then we say thatA defines a matrix map from

FintoEand we denote it byA:F→E. By(F,E)we mean the class of matricesAsuch thatA:F→E. We denote the set{x∈w:Axexists andAx∈E}byEA. Note thatA

InB-spaceE, the following statements are equivalent (see [5]). (i) nxnis unconditionally convergent.

(ii) nxnis weakly subseries convergent; that is, weak limnnj=1xkjexists for each

increasing sequence(kn)of positive integers.

(iii) nxn is subseries convergent; that is, norm limnnj=1xkj exists with (kn)

above.

(iv) nxn is bounded multiplier convergent; that is,nxntn exists for each

se-quencet=(tn)of bounded scalars.

2. Some properties of ∆m_{(X). In this section, we will give some properties of} ∆m_(X).

Theorem2.1. LetXbe a vector space and letA⊂X. IfAis a convex set, then∆m_(A)

is a convex set in∆m_(X),

Proof. Letx, y∈∆m_{(A), then∆}m_x,∆m_{y∈A. Since∆}m_{is linear, we have}

λ∆m_{x+(1−λ)∆}m_y=∆m_{λx+(1−λ)y, (0≤λ≤1). (2.1)}

Since A is convex (λ∆m_{x+(1−λ)∆}m_{y) ∈ Aand so(λx+(1−λ)y) ∈ ∆}m_(A), (0≤λ≤1).

Lemma2.2. Letmbe a positive integer. Then

(i) ∆m₍∞

n=1An)=∞n=1∆m(An),

(ii) ∆m₍∞ n=1An)=

_∞

n=1∆m(An).

The proof is clear.

Lemma2.3. LetXbe a Banach space and letA⊂X. Then

(i) IfAis nowhere dense inX, then∆m_{(A)is nowhere dense in∆}m_(X).

(ii) IfAis dense inX, then∆m_{(A)is dense in∆}m_(X).

(iii) ∆m_{(w)=w, wheremis a positive integer.}

Proof. (i) Suppose thatA¯◦= ∅, but ∆m◦_{(A)≠∅. Then ¯Acontains no}

neighbor-hood andB(a)⊂∆m_{(A), whereB(a) is a neighborhood (or open ball) of centera}

and radius r. Hencea∈B(a)⊂∆m_(A)=∆m_{(A)¯. This implies that∆}m_{(a)∈A¯. So} B(∆m_{(a))∩A≠∅. On the other hand,B(∆}m_{(a))∩A⊂A¯. This contradicts toA¯}◦_{= ∅.}

Hence∆m◦_{(A)= ∅.}

(ii) and (iii) are trivial.

Theorem2.4. (i)The set∆m_(s

0)is dense in the spaces.

(ii) The set∆m_{(s0)is a set of the first Baire category in the spaces.}

(iii) The sets-∆m_(s

0)is a set of the second Baire category in the spaces.

Proof. The proof follows from [6, Theorem 3.1], Lemmas 2.2, and 2.3, we recall that the complementMc_{of a meager (or of the first category) subsetMof a complete}

metric spaceXis nonmeager (or of the second category).

Proof. It is trivial thatl∞∩∆m(c0)⊂l∞∩∆m(c). Now letx∈l∞∩∆m(c), thenx∈ l∞and∆m−1xk−∆m−1xk₊₁→l, (k→ ∞),∆m−1xk−∆m−1xk₊₁=l+εk(εk→0, k→ ∞).

This implies that

l=n−1_∆m−1_x

1−n−1∆m−1xn+1+n−1 n

k=1

εk. (2.2)

This yieldsl=0 andx∈l∞∩∆m(c0).

3. Dual spaces. In this section, we give the N-dual (null dual) of the sequence spaces∆m_(l

∞), ∆m(c), and∆m(c0)and thepα-,pβ-, andpγ-duals of the sequence

spaces ofl∞,c, andc0.

Definition3.1. LetXbe a sequence space and define

Xα_=a=ak: k

_{akxk<∞,∀x∈X,}

Xβ_=a=ak: k

akxkis convergent,∀x∈X

,

Xγ₌

a=ak: sup

n k

akxk<∞,∀x∈X

,

XN_{=a=ak: lim}

k akxk=0,∀x∈X

,

(3.1)

thenXα_,Xβ_,Xγ_{, andX}N_{are called theα-,β-,γ-, andN-(or nul) duals ofX, respectively.}

It is known thatX⊂Y, thenYη_⊂Xη_{forη=α-,β-,γ-, andN-, andc}

0N=l∞,l∞N= cN_=c

0[2, 3].

Lemma3.2(see [4]). Letmbe a positive integer. Then there exist positive constants

M1andM2such that

M1km≤

m+k k

≤M2km ∀k=0,1,... . (3.2)

Lemma3.3. Letx∈∆m_(c

0), then(mk+k)−1|xk| →0, (k→ ∞). Proof. The proof is trivial.

Theorem3.4. Let m be a positive integer. Then (∆m_(l

∞))N = (∆m(c))N = U1

and (∆m_(c

0))N =U2,whereU1= {a =(an):(nman)∈ c0}andU2= a =(an): _n

k=0

_n+m−k−1 m−1

an∈l∞ .

Proof. The proof of the part(∆m_(l

∞))N=(∆m(c))N =U1is easy. We show that (∆m_(c

0))N =U2. It is clear thatnk=0n+mm−1−k−1

=n+m

m =n+nm. Leta∈U2 and x∈∆m_(c

0). Then

lim_n anxn=lim_n

_n

k=0

n+m−k−1

m−1

an

_n

k=0

n+m−k−1

m−1

−1

xn=0. (3.3)

Now leta∈(∆m_(c

0))N. Then limnanxn=0 for allx∈∆m(c0). On the other hand,

for eachx∈∆m_(c

0)there exists one and only oney=(yk)∈c0such that

xn= n

k=1

n+m−k−1

m−1 yk= n k=0

n+m−k−1

m−1

yk, y0=0, (3.4)

by (1.9). Hence

lim_n anxn=lim_n

n

k=0

n+m−k−1

m−1

anyk=0 ∀y∈c0. (3.5)

If we take

ank=       

n+m−k−1

m−1

an, 1≤k≤n,

0, k > n,

(3.6)

then, we get

lim_n

∞

k=0

ankyk=lim_n

n

k=0

n+m−k−1

m−1

anyk=0 ∀y∈c0. (3.7)

HenceA∈(c0,c0)and so supn∞k=0|ank| =supnnk=0

_n+m−k−1 m−1

|an|<∞. This com-pletes the proof.

Now we give a new kind of duals of sequence sets.

Definition3.5. LetXbe a sequence spaces,p >0 and define

Xpα_=a=ak: k

_akxkp_{<∞,∀x∈X,}

Xpβ_=a=ak: k

akxkpis convergent,∀x∈X

,

Xpγ₌

a=ak: sup

n n k=0

akxkp<∞,∀x∈X

,

(3.8)

thenXpα_,Xpβ_,Xpγ_{are called thepα-,pβ-, andpγ-duals ofX, respectively. It can be}

shown thatXpα_⊂Xpβ_⊂Xpγ_{. If we takep=1 in this definition, then we obtain the} α-,β-, andγ-duals ofX.

Theorem3.6. LetX stand for l∞,c, and c0 and 0< p <∞. Then Xpη=U, for η=α,βorγ, whereU=a=(ak):k|ak|p<∞ =lp.

Proof. We give the proof for the caseX=c0andη=α. Ifa∈U, then

k

_akxkp_≤sup k

_xkp k

_akp_{<∞ (3.9)}

Now suppose thata∈(c0)pαanda∈U. Then there is a strictly increasing sequence (ni)of positive integersnisuch that

k=ni+1

k=ni+1

_akp_{> i}p_{. (3.10)}

Definex∈c0byxk=sgnak/iforni< k≤ni+1andxk=0 for 1≤k≤n1. Then we

may write

k

_akxkp₌ k=n2 k=n1+1

_akxkp_+···+k=ni+1 k=ni+1

_akxkp_+···

= k=n2

k=n1+1

_akp_+···+1 ip

k=ni+1

k=ni+1

_akp_+···

>1+1+··· = k

1= ∞.

(3.11)

This contradicts toa∈(c0)pα. Hencea∈U. The proof for the casesX=l∞orcand η=βorγis similar.

The proofs of Lemmas 3.7 and 3.8 and Theorem 3.10 are easily obtained by using the same techniques of Mishra [5, Lemmas 1 and 2 and Theorem 1], therefore we give them without proofs.

Lemma3.7. LetA:∆m_(l

∞)→Edefines a matrix map. IfAis weakly compact, then

kakis unconditionally convergent inE.

Lemma3.8. Ifkakis unconditionally convergent inE, thenA:∆m(l∞)→Edefines

a matrix map, andA(α)=kakαkfor everyα=(αk)∈∆m(l∞).

Corollary3.9. Ifkakis unconditionally convergent inE, then∆m(l∞)⊆EA. Theorem3.10. IfA:∆m_(l

∞)→Eis a weakly compact matrix map, thenAis

com-pact map.

Corollary3.11. LetEbe aBK-space such that it contains no subspace isomorphic to∆m_(l

∞). IfA:∆m(l∞)→Edefines a matrix map, thenAis compact map.

Acknowledgement. The author wishes to thank Prof. Dr Rifat Çolak for his valu-able comments on the manuscript.

References

[1] M. Et and R. Çolak,On some generalized difference sequence spaces, Soochow J. Math.21 (1995), no. 4, 377–386. MR 97b:40009. Zbl 841.46006.

[2] D. J. H. Garling,Theβ- andγ-duality of sequence spaces, Proc. Cambridge Philos. Soc.63 (1967), 963–981. MR 36#1965. Zbl 161.10401.

[3] H. Kızmaz,On certain sequence spaces. II, Int. J. Math. Math. Sci.18(1995), no. 4, 721–724. MR 96j:46005. Zbl 832.40002.

[5] S. K. Mishra,Matrix maps involving certain sequence spaces, Indian J. Pure Appl. Math.24 (1993), no. 2, 125–132. MR 94e:46019. Zbl 804.47030.

[6] T. Šalát,On statistically convergent sequences of real numbers, Math. Slovaca30(1980), no. 2, 139–150. MR 81k:40002. Zbl 437.40003.