Some classes of Cesàro type difference sequences over n normed spaces

(1)

R E S E A R C H

Open Access

Some classes of Cesàro-type difference

sequences over

n

-normed spaces

Hemen Dutta

*

*_{Correspondence:}

hemen_dutta08@rediﬀmail.com Department of Mathematics, Gauhati University, Guwahati, Assam 781014, India

Abstract

In this paper, we introduce some Cesàro-type diﬀerence sequences spaces deﬁned over a real linearn-normed space and investigate the spaces for completeness under suitablen-norm in each case. Relevant relations among the classes of sequences are examined. We also introduce the notion ofn-BK-spaces and show that the spaces can be made ann-BK-space under certain condition. Further, we compute the

Köthe-Toeplitz duals of the spaces, wherever possible within the scope of the research of this article.

MSC: 40A05; 46A20; 46D05; 46A45; 46E30

Keywords: Cesàro summability; diﬀerence sequences;n-normed spaces; completeness; Köthe-Toeplitz dual

1 Introduction

The studies of linear transformation on sequence spaces are called summability. The ear-liest idea of summability theory was perhaps contained in a letter written by Leibnitz to Wolf (), in which the sum of the oscillatory series -+-+- - - - as given by Leib-nitz was _. Studies on sequence space were further extended through summability the-ory. Summability theory, or in short summability, is the theory of the assignment of limits, which is fundamental in analysis, function theory, topology and functional analysis.

Throughout w, ∞, p, l, c and c denote the spaces of all, bounded, p-absolutely

summable, absolutely summable, convergent and null sequencesx= (xk) with complex

terms, respectively.

The zero element of a normed linear space (n.l.s.) is denoted byθ. A completen.l.s.is called a Banach space.

p( <p<∞) denotes the space of all complex sequences such that

k|xk|p<∞, called

as the space ofp-absolutely summable sequences. The spacepforp≥ is complete under

the norm deﬁned byx= (_k|xk|p)/p.

For  <p< ,pis a completep-normed space,p-normed byx=

∞

k=|xk|p.

ABK-space (introduced by Zeller []) (X, · ) is a Banach space of complex sequences x= (xk), in which the co-ordinate maps are continuous, that is,|xnk–xk| →, whenever xn_–_x_→_{ as}_n_{→ ∞}_{, where}_xn_{= (}_xn

k) for alln∈Nandx= (xk).

Let (X, · ) be a normed linear space, andλis a scalar-valued sequence space, then the vector-valued sequence space orX-valued sequence spaceλ(X) is deﬁned asλ(X) ={(xk) :

xk∈Xfor allk∈Nandxk ∈λ}.

(2)

Clearly,λ(X) is a linear space under coordinate wise addition and scalar multiplication

The Cesàro means (also called Cesàro averages) of sequence{an}are the terms of

se-quence{cn}, wherecn=

n

i=anis the arithmetic mean of the ﬁrstnelements of{an}. This

concept is named after Ernesto Cesàro. It is known that if{an}converges tol, then{an}also

converges to the same limit. This means that the operation of taking Cesàro means pre-serves convergent sequences and their limits. This is the basis of taking Cesàro means as a summability method in the theory of divergent sequences. If the sequence of the Cesàro means is convergent, the series is said to be Cesàro summable. There are certainly many examples, for which the sequence of Cesàro means converges, but the original sequence does not. For example, sequence{an}={(–)n}which is Cesàro summable to .

have been introduced and studied by Shiue [], and it was observed that p ⊂Cesp

( <p< ) is strict, although it does not hold forp= .

Nag and Lee [] deﬁned and studied the Cesàro sequence spaceXpof non-absolute-type

as follows:

Further, Orhan [] deﬁned and studied the following sequence spaces:

(3)

and

O∞() =

x= (xk) :sup n≥

 n

n

k=

|xk|<∞

.

He showed that for ≤p<∞, the inclusionsOp()⊂Xp() and Cesp⊂Op() are strict.

Mursaleenet al.[] deﬁned and studied the Cesàro diﬀerence sequence spaceXp(),

i.e.,

Xp 

=

x= (xk) :

∞

n=

 n

n

k= xk

p

<∞, ≤p<∞

,

and

X∞ =

x= (xk) :sup n≥

 n

n

k= xk

<∞

,

wherexk=xk–xk+.

For uniformity of the literature, henceforth, we shall writeCp instead ofXp andC∞

instead ofX∞.

For some useful works on Cesàro-type summable spaces, we refer to [–]. LetEandFbe two sequence spaces. Then theFdual ofEis deﬁned asEF₌_{₍_x

k)∈w:

(xkyk)∈Ffor all (yk)∈E}.

ForF=l, the dual is termed asα-dual (Köthe-Toeplitz dual) ofEand denoted byEα. If

X⊂Y, thenYα_⊂_Xα_.

For initial and useful works on the notion of Köthe-Toeplitz duals, we refer to [–]. Letn∈N andX be a real vector space of dimensiond, wheren≤d. A real-valued function·, . . . ,·onXnsatisfying the following four conditions:

(N) x,x, . . . ,xn= if and only ifx,x, . . . ,xnare linearly dependent,

(N) x,x, . . . ,xnis invariant under permutation,

(N) αx,x, . . . ,xn=|α|x,x, . . . ,xn, for anyα∈R,

(N) x+x,x, . . . ,xn ≤ x,x, . . . ,xn+x,x, . . . ,xn

is called ann-norm onX, and the pair (X,·, . . . ,·) is called ann-normed space.

A trivial example of ann-normed space isX=Rn_{equipped with the following Euclidean}

n-norm:

x,x, . . . ,xnE= abs

⎛ ⎜ ⎜ ⎝

x · · · xn

..

. . .. ... xn · · · xnn

⎞ ⎟ ⎟ ⎠,

wherexi= (xi, . . . ,xin)∈Rnfor eachi= , , . . . ,n.

(4)

Letn∈N and (X,·,·) be a real inner product space of dimensiond=n. Deﬁne the following function·, . . . ,·|·,·onX× · · · ×X(n+  factors) by

x, . . . ,xn–|y,z=

x,x · · · x,xn– x,z

..

. . .. ... ...

xn–,x · · · xn–,xn– xn–,z y,x · · · y,xn– y,z

.

Then one may check that this function satisﬁes the following ﬁve properties:

(I) x, . . . ,xn–|xn,xn= ;x, . . . ,xn–|xn,xn= if and only ifx, . . . ,xnare linearly

dependent;

(I) x, . . . ,xn–|xn,xn=xi, . . . ,xin–|xin,xinfor every permutation(i, . . . ,in)of

(, . . . ,n);

(I) x, . . . ,xn–|y,z=x, . . . ,xn–|z,y;

(I) x, . . . ,xn–|y,αz=αx, . . . ,xn–|y,z;

(I) x, . . . ,xn–|y,z+z=x, . . . ,xn–|y,z+x, . . . ,xn–|y,z.

Accordingly, we can deﬁne·, . . . ,·onX× · · · ×X(nfactors) by

x, . . . ,xn=x, . . . ,xn–|xn,xn/,

that is,

x, . . . ,xn=

x,x · · · x,xn

..

. . .. ...

xn,x · · · xn,xn

 

.

Forn= , we know that · is a norm, while forn= ,·,· deﬁnes a -norm. Note further that forn= ,xgives the length ofx, while forn= ,x,xrepresents the

area of the parallelogram spanned byxandx. Forn=  andX=R, one may observe

thatx,x,xis nothing but the volume of the parallelepiped spanned byx,xandx,

that is,x,x,x=|x.(x×x)|.

Sequence (xk) in ann-normed space (X,·, . . . ,·) is said to converge to someL∈Xin

then-norm if

lim

k→∞xk–L,u, . . . ,un=  for everyu, . . . ,un∈X.

Sequence (xk) in ann-normed space (X,·, . . . ,·) is said to be Cauchy with respect to the

n-norm if

lim

k,l→∞xk–xl,u, . . . ,un=  for everyu, . . . ,un∈X.

If every Cauchy sequence inXconverges to someL∈X, then Xis said to be complete with respect to then-norm. Any completen-normed space is said to ben-Banach space.

(5)

The notion of diﬀerence sequence space was introduced by Kizmaz [], who studied the diﬀerence sequence spaces∞(),c() andc(). The notion was further generalized

by Et and Colak [] by introducing the spaces_∞(m_),_c₍m_{) and}_c

(m), wherenis a

non-negative integer. In general, we have the following deﬁnition of diﬀerence sequence spaces.

Letmbe non-negative integers, then forZ, a given sequence space, we have

Z m=x= (xk)∈w: mxk

∈Z,

wherem_x_{= (}m_x

k) = (m–xk–m–xk+) andxk=xkfor allk∈N, which is

equiva-lent to the following binomial representation:

mxk= quence space. Letmbe a non-negative integer and ≤p<∞, then we introduce the fol-lowing sequence spaces for every non zeroz, . . . ,zn–∈X:

We procure the following result, which will help us in establishing the results of this article.

Lemma .(Tripathy, Esi and Tripathy [])

(6)

(i) The spaceCpis a Banach space,normed by

x= _∞

i=

 i

i

k=

xk

p_p

.

(ii) The spaceOpis a Banach space,normed by

x= _∞

i=

 i

i

k= |xk|p



p

.

(iii) The spacepis a Banach space,normed by

x= _∞

k= |xk|p



p

.

(b) (i) The spaceC∞is a Banach space,normed by

x=sup

i

 i

i

k=

xk

.

(ii) The spaceO∞is a Banach space,normed by

x=sup

i

 i

i

k= |xk|.

Deﬁnition . Ann-BK-space (X,·, . . . ,·) is ann-Banach space of real sequencesx= (xk) in which the co-ordinate maps are continuous.

3 Construction ofn-norms and relevant properties

In this section, we constructn-norms on the introduced spaces of previous section and investigate the spaces for completeness and some relations among them. The proof of the following result is a routine veriﬁcation.

Proposition . The classes of sequences Cp(m,·, . . . ,·), Op(m,·, . . . ,·), p(m, ·, . . . ,·),C∞(m,·, . . . ,·)and O∞(m,·, . . . ,·)for≤p<∞are linear spaces over the ﬁeld of reals.

Theorem .

(a) Let≤p<∞,and the base spaceXis ann-Banach space.Then

(i) The spaceCp(m,·, . . . ,·)is ann-Banach space,n-normed by x_,_x_{, . . . ,}_xnm

CP = ifx

_,_x_{, . . . ,}_xn_{are linearly dependent and}

=m_k_=xk,z, . . . ,zn–+ (

∞

i=i

i

k=mxk,z, . . . ,zn–p)



p _{for every}

z, . . . ,zn–∈Xifx,x, . . . ,xnare linearly independent.

(ii) The spaceOp(m,·, . . . ,·)is ann-Banach space,n-normed by x_,_x_{, . . . ,}_xnm

OP = ifx

(7)

=m_k_=xk,z, . . . ,zn–+ (

n-norm, deﬁned as above.

Here, we prove the completeness for the space C∞(m,·, . . . ,·), and for the other spaces, it will follow on applying similar arguments.

(8)

⇒(m_xs

k) is a Cauchy sequence inC∞(·, . . . ,·), which is complete (it is easy to check

thatC∞(·, . . . ,·) is complete).

Hence, (n_xs

k) converges for eachk∈N. Letlims→∞mnxsk=ykfor eachk∈N.

Letk= , we have

lim

s→∞

m_xs =_slim_→∞

m

υ=

(–)υ

m

υ

x+υ=y. (..)

We have

lim

s→∞x s

k=xk, fork=  +υ, forυ= , , . . . ,m– . (..)

Thus, from (..) and (..), we have thatlim_s_→∞xs

+m exists. Letlims→∞xs+m=x+m.

Proceeding in this way inductively, we have thatlim_s_→∞xs

k=xkexists for eachk∈N.

Now, for everyz, . . . ,zn–∈X

lim

t m

k=

xs_k–xt_k,z, . . . ,zn–= m

k=

xs_k–xk,z, . . . ,zn–<ε for alls≥n.

Again, using the continuity ofn-norm, we ﬁnd that for everyz, . . . ,zn–∈X

 i

i

k=

mxs_k–lim

t→∞ m_xt

k,z, . . . ,zn–

<ε for alls≥nandi∈N.

Hence, for everyz, . . . ,zn–∈X

sup

i

 i

i

k=

mxs_k–mxk,z, . . . ,zn–

<ε for alls≥n.

Thus, for everyu_{, . . . ,}_un_in_C

∞(m,·, . . . ,·) xs–x,u, . . . ,un_C_∞m< ε for alls≥n.

Hence, (xs_–_x₎_∈_C_∞₍m_,_·_{, . . . ,}_·_{). Since}_C_∞₍m_,_·_{, . . . ,}_·_{) is a linear space, so we have}

for alls≥n,

x=xs_– _xs_–_x_∈_C

∞ m,·, . . . ,·.

Hence,C∞(m,·, . . . ,·) is complete and as such is ann-Banach space.

Corollary . The spaces Cp(m,·, . . . ,·),Op(m,·, . . . ,·),p(m,·, . . . ,·),C∞(m, ·, . . . ,·)and O∞(m,·, . . . ,·)for≤p<∞are n-BK-spaces if the base space X is an n-Banach space.

Proof The proof is obvious in view of the previous theorem.

(9)

Proof Here, we prove the result forZ=Cp and for the other results, it will follow on

ap-Then for each positive integerr, we get

r

follows from the following example.

(10)

Theorem .

(a) Op(m,·, . . . ,·)⊂Cp(m,·, . . . ,·)⊂C∞(m,·, . . . ,·),and the inclusions are

strict.

(b) Op(m,·, . . . ,·)⊂O∞(m,·, . . . ,·)⊂C∞(m,·, . . . ,·),and the inclusions are strict.

Proof The proof is easy, so it is omitted.

Remark . p(m,·, . . . ,·)⊂Op(m,·, . . . ,·). For this, consider the following

exam-ple.

Example . Letp=  and a -norm·,·onX=R _{as in Example .. Let}_m_{=  and}

consider sequence{xk}={(, , ), (, , ), (, , ), (, , ), . . .). Thenxk= (, , ) fork=

 andxk= (, , ) for allk> . Then (xk)∈(,·,·) but (xk) /∈O(,·,·).

Theorem . If≤p<q,then

(i) Cp(m,·, . . . ,·)⊂Cq(m,·, . . . ,·).

(ii) p(m,·, . . . ,·)⊂q(m,·, . . . ,·).

(iii) Op(m,·, . . . ,·)⊂Oq(m,·, . . . ,·).

Proof The proof is easy, so it is omitted.

4 Computation of the Köthe-Toeplitz duals

In order to compute Köthe-Toeplitz dual, we ﬁrst deﬁne the following. Ann-functional is a real-valued mapping with domainA× · · · ×An, whereA, . . . ,Anare linear manifolds

of a linearn-normed space.

LetFbe ann-functional with domainA× · · · ×An.F is called a linearn-functional

whenever for all_a

,a, . . . ,an∈A,a,a, . . . ,an∈A, . . . ,na,na, . . . ,nan∈Anand

allα, . . . ,αn∈R, we have

(i) F(_a

+a+· · ·+an,a+a+· · ·+an, . . . ,na+na+· · ·+nan) =

≤i,i,...,in≤nF( _a

i,

_a i, . . . ,

n_a in)and

(ii) F(αa, . . . ,αnan) =α· · ·αnF(a, . . . ,an).

LetFbe ann-functional with domainD(F).Fis called bounded if there is a real constant K≥ such that|F(a, . . . ,an)| ≤Ka, . . . ,anfor all (a, . . . ,an)∈D(F). IfFis bounded,

we deﬁne the norm ofF,Fby

F=glbK:F(a, . . . ,an)≤Ka, . . . ,anfor all (a, . . . ,an)∈D(F)

.

IfFis not bounded, we deﬁneF= +∞.

It is easy to check the following two results. In this context, one may refer to George [].

Proposition . A linear n-functional F is continuous if and only if it is bounded.

Proposition . Let B∗be the set of bounded linear n-functionals with domain B× · · · ×

(11)

For anyn(> )-normed spaceE, we denote byE∗the continuous dual ofE. There is a need to explore in detail on this notion of continuous duality forn-normed spaces.

We have the deﬁnition of Köthe-Toeplitz dual of sequence spaces with base space an n-normed space as follows.

LetEbe ann-normed linear space, normed by·, . . . ,·E. Then we deﬁne the

Köthe-Toeplitz dual of the sequence spaceZ(E) whose base space isEas

Z(E)α=(yk) :yk∈E∗,k∈Nand xk,u, . . . ,unEyk,v, . . . ,vnE∗

∈

for everyv, . . . ,vn∈E∗,u, . . . ,un∈E, (xk)∈Z(E)

.

It is easy to check thatφ⊂Xα_{. If}_X_⊂_Y_{, then}_Yα_⊂_Xα_.

Let us consider

SCp m,·, . . . ,·

=x= (xk) :x∈Cp m,·, . . . ,·

,x=· · ·=xm= 

.

Then SCp(m,·, . . . ,·) is a subspace ofCp(m,·, . . . ,·) for ≤p<∞. We can have

similar subspaces for other spaces as well.

Now, we procure the following results which will be helpful in establishing our result.

Lemma .(M. Et []) x∈SC∞(m)implies thatsup_kk–m_|_x k|<∞.

Lemma . x∈SC∞(m,·, . . . ,·) implies that sup_kk–m_x

k,u, . . . ,un<∞ for every

u, . . . ,un∈X.

Proof The proof follows using similar techniques as applied in the proof of Lemma .. Let us set

U=

a= (ak) :

∞

k=

kmak,z, . . . ,znX∗<∞for everyz, . . . ,zn∈X∗

.

Theorem . The Köthe-Toeplitz dual of the space SCp(m,·, . . . ,·)is U,i.e., [SC∞(m, ·, . . . ,·)]α₌_U_.

Proof Ifa∈U, then

∞

k=

ak,z, . . . ,znX∗xk,u, . . . ,unX

=

∞

k=

kmak,z, . . . ,znX∗ k–mxk,u, . . . ,unX

<∞,

for eachx∈SC∞(m,·, . . . ,·) (by Lemma .). Hence,x∈[SC∞(m,·, . . . ,·)]α_.

Next, leta∈[SC∞(m,·, . . . ,·)]α_{. Then}∞

k=ak,z, . . . ,znX∗xk,u, . . . ,unX<∞for

eachx∈SC∞(m,·, . . . ,·). We deﬁne sequencex= (xk) by

xk=

(12)

and chooseu, . . . ,un∈Xsuch that

km,u, . . . ,un_X=km,u, . . . ,unX=

, k≤m, km_, _k_>_m_.

We may write for everyz, . . . ,zn∈X∗,

∞

k=

kmak,z, . . . ,znX∗

=

∞

k=

km,u, . . . ,un_Xak,z, . . . ,znX∗

=

m

k=

km,u, . . . ,un_Xak,z, . . . ,znX∗+

∞

k=

km,u, . . . ,un_Xak,z, . . . ,znX∗

<∞.

This implies thata∈U.

Theorem . [SC∞(m_,_·_{, . . . ,}_·_)]α_{= [}_C

∞(m_,_·_{, . . . ,}_·_)]α_.

Proof SinceSC∞(m,·, . . . ,·)⊂C∞(m,·, . . . ,·), we have

C∞ m,·, . . . ,·α⊂SC∞ m,·, . . . ,·α.

Leta∈[SC∞(m,·, . . . ,·)]α _and_x_∈_C

∞(m,·, . . . ,·). If we take sequencex= (xk) as

follows

xk=

xk, k≤m,

x_k, k>m,

wherex = (x_k)∈SC∞(m,·, . . . ,·). Then we may write ∞

k=

ak,z, . . . ,znX∗xk,u, . . . ,unX

=

m

k=

ak,z, . . . ,znX∗xk,u, . . . ,unX+

∞

k=

ak,z, . . . ,znX∗xk,u, . . . ,un_X

<∞.

This implies thata∈[C∞(m,·, . . . ,·)]α.

Theorem . [O∞(m,·, . . . ,·)]α_{= [}_C

∞(m,·, . . . ,·)]α_.

(13)

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The research work of this article is supported by the University Grant Commission, New Delhi-110002, India as a minor research project under F. No. 39-935/2010 (SR).

Received: 1 August 2013 Accepted: 12 September 2013 Published:07 Nov 2013

References

1. Zeller, K: Theorie der Limitierungsverfahren. Springer, Berlin (1958) 2. Shiue, JS: On the Cesàro sequence spaces. Tamkang J. Math.1, 19-25 (1970) 3. Siddiqi, AH: 2-normed spaces. Aligarh Bull. Math.7, 53-70 (1980)

4. Orhan, C: Cesàro diﬀerence sequence spaces and related matrix transformations. Commun. Fac. Sci. Univ. Ankara, Sér. A32, 55-63 (1983)

5. Mursaleen, M, Khatib, MA, Qamaruddin: On diﬀerence Cesàro sequence spaces of non-absolute type. Bull. Calcutta Math. Soc.89, 337-342 (1997)

6. Ng, PN: Matrix transformations on Cesàro sequence spaces of nonabsolute type. Tamkang J. Math.10, 215-221 (1979) 7. Ng, PN, Lee, PY: Cesàro sequence spaces of nonabsolute type. Comment. Math.20, 429-433 (1978)

8. Freedman, AR, Sember, JJ, Raphael, M: Some Cesàro-type summability spaces. Proc. Lond. Math. Soc.37, 508-520 (1978)

9. Köthe, G, Toeplitz, O: Lineare Räume mit unendlich vielen koordinaten und Ringe unenlicher Matrizen. J. Reine Angew. Math.171, 193-226 (1934)

10. Lascarides, CG: A study of certain sequence spaces of Maddox and a generalization of a theorem of Iyer. Pac. J. Math.,

38(2), 487-500 (1971)

11. Maddox, IJ: Continuous and Köthe-Toeplitz duals of certain sequence spaces. Proc. Camb. Philol. Soc.65, 431-435 (1969)

12. Mursaleen, M, Gaur, AK, Saiﬁ, AH: Some new sequence spaces their duals and matrix transformations. Bull. Calcutta Math. Soc.88, 207-212 (1996)

13. Gunawan, H: Onn-inner product,n-norms, and the Cauchy-Schwarz inequality. Sci. Math. Jpn. (Online)5, 47-54 (2001)

14. Dutta, H: An application of lacunary summability method ton-norm. Int. J. Appl. Math. Stat.,15(D09), 89-97 (2009) 15. Gähler, S: Über die Uniformisierbarkeit 2-metrischer Räume. Math. Nachr.28, 235-244 (1964)

16. Gähler, S: Über 2-Banach Räume. Math. Nachr.42, 335-347 (1969)

17. Gähler, S: Untersuchungen über verallgemeinertem-metrische Räume, II. Math. Nachr.40, 229-264 (1969) 18. Gähler, S, Siddiqi, AH, Gupta, SC: Contributions to non-Archimedean functional analysis. Math. Nachr.69, 162-171

(1975)

19. Gunawan, H: The space ofp-summable sequences and its naturaln-norm. Bull. Aust. Math. Soc.,64(1), 137-147 (2001) 20. Gunawan, H, Mashadi, M: On ﬁnite dimensional 2-normed spaces. Soochow J. Math.,27(3), 631-639 (2001) 21. Gunawan, H, Mashadi, M: Onn-normed spaces. Int. J. Math. Math. Sci27(10), 631-639 (2001)

22. Dutta, H: Onn-normed linear space-valued null, convergent and bounded sequences. Bull. Pure Appl. Math.4(1), 103-109 (2010)

23. Kim, SS, Cho, YJ: Strict convexity in linearn-normed spaces. Demonstratio Math29, 739-744 (1996) 24. Misiak, A:n-inner product spaces. Math. Nachr.140, 299-319 (1989)

25. Misiak, A: Orthogonality and orthonormality inn-inner product spaces. Math. Nachr.143, 249-261 (1989) 26. Malèeski, A: Strongn-convexn-normed spaces. Mat. Bilt.21, 81-102 (1997)

27. Diminnie, C, Gähler, S, White, A: 2-inner product spaces. Demonstrario Math.6, 525-536 (1973) 28. Gähler, S: 2-metrische Räume und ihre topologische Struktur. Math. Nachr.26, 115-148 (1963) 29. Gähler, S: 2-normed spaces. Math. Nachr.28, 1-43 (1964)

30. Gähler, S: Lineare 2-normierte Räume. Math. Nachr.28, 1-43 (1964) 31. Kizmaz, H: On certain sequence spaces. Can. Math. Bull.24(2), 169-176 (1981)

32. Et, M, Colak, R: On generalized diﬀerence sequence spaces. Soochow J. Math.,21(4), 377-386 (1995)

33. Tripathy, BC, Esi, A, Tripathy, BK: On a new type of generalized diﬀerence Cesàro Sequence spaces. Soochow J. Math.,

31(3), 333-340 (2005)

34. George, WA Jr.: 2-Banach spaces. Math. Nachr.42, 43-60 (1969)

35. Et, M: On some generalized Cesàro diﬀerence sequence spaces. Istanbul Univ. fen fak. Mat. Dergisi,55-56, 221-229 (1996-1997)

10.1186/1687-1847-2013-286