On some properties of new paranormed sequence space defined by λ2 convergent sequences

R E S E A R C H

Open Access

On some properties of new paranormed

sequence space defined by

λ



-convergent

sequences

Naim L Braha

*_{Correspondence:}

[email protected]

Department of Mathematics and Computer Sciences, University of Prishtina, Avenue ‘Mother Teresa’ 5, Prishtinë, 10000, Kosovo Department of Computer Sciences and Applied Mathematics, College Vizioni per Arsim, Ferizaj, 70000, Kosovo

Abstract

In this paper, we introduce the new sequence spacel(λ2_{,p) and we will show some} topological properties like completeness, isomorphism, and some inclusion relations between this sequence spaces and some of the other sequence spaces. In addition we will compute the

α-,

β

γ

MSC: 46A45

Keywords: sequence spaces;

λ

_α-,

_{β-, and}

_γ

1 Introduction

Bywwe denote the space of all complex sequences. Ifx∈w, then we simply writex= (xk) instead ofx= (xk)∞k=. Also, we shall use the conventions that e= (, , . . .) and e(n) is the sequence whose only non-zero term is  in the nth place for each n∈N, where N={, , , . . .}. Any vector subspace ofwis called a sequence space. We shall writel, c, andcfor the sequence spaces of all bounded, convergent, and null sequences, respec-tively. Further, bylp(≤p<∞), we denote the sequence space of allp-absolutely conver-gent series, that is,lp={x= (xk)∈w:

_∞

k=|xk|p<∞}for ≤p<∞. Moreover, we write bs,cs, andcsfor the sequence spaces of all bounded, convergent, and null series, respec-tively. A sequence spaceX is called anFK space if it is a complete linear metric space with continuous coordinatespn:X→C(n∈N), whereCdenotes the complex field and pn(x) =xnfor allx= (xn)∈Xand everyn∈N. A normedFKspace is called aBK space, that is, aBK space is a Banach sequence space with continuous coordinates.

The sequence spacesl,c, andcareBKspaces with the usual sup-norm given byx∞= sup_n|x(n)|. Also, the spacelpis aBKspace with the usuallp-norm defined by

xp=

_∞

n=

|xn|p

 p

,

where ≤p<∞. A sequence (yn)∞n=in a normed spaceXis called a Schauder basis forX if for everyx∈Xthere is a unique sequence (an)∞n=of scalars such thatx=

∞

n=anyn,i.e., limnx–

n

k=anyn= . Theα-,β-, andγ-duals of a sequence spaceXare, respectively, defined by

Xα=a= (ak)∈w:ax= (akxk)∈lfor allx= (xk)∈X

,

Xβ=a= (ak)∈w:ax= (akxk)∈csfor allx= (xk)∈X

and

Xγ_{=a= (a}

k)∈w:ax= (akxk)∈bsfor allx= (xk)∈X

.

IfAis an infinite matrix with complex entriesank (n,k∈N), then we write A= (ank) instead ofA= (ank)∞n,k=. Also, we writeAnfor the sequence in thenth row ofA, that is, An= (ank)∞k=for everyn∈N. Further, ifx= (xk)∈wthen we define theA-transform ofx as the sequenceAx= (An(x))∞n=, where

An(x) =

∞

k=

an,kxk (n∈N) ()

provided the series on () is convergent for eachn∈N.

Furthermore, the sequencexis said to beA-summable toa∈CifAxconverges toa which is called theA-limit ofx. In addition, letXandYbe sequence spaces. Then we say thatAdefines a matrix mapping fromXintoYif for every sequencex∈XtheA-transform ofxexists and is inY. Moreover, we write (X,Y) for the class of all infinite matrices that mapX intoY. ThusA∈(X,Y) if and only ifAn∈Xβ for alln∈NandAx∈Y for all x∈X. For an arbitrary sequence spaceX, the matrix domain of an infinite matrixAinX is defined by

XA={x∈w:Ax∈X}, ()

which is a sequence space. The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors; see for instance [–]. In this paper, we introduce the new sequence spacel(λ,p) and we will show some topological properties as completeness, isomorphism, and some inclusion relations between this sequence spaces and some of the other sequence spaces. In addition we will compute theα-,β-, andγ-duals of these spaces.

2 Notion of the

λ

Let={λk:k= , , . . .}be a nondecreasing sequence of positive numbers tending to∞, ask→ ∞, andλn+≥·λn, for eachn∈N. From this relation it follows thatλn≥. The first difference is defined as follows:λk=λk–λk–, whereλ–=λ–= , and the second difference is defined as_(λ

k) =((λk)) =λk– λk–+λk–.

Letx= (xk) be a sequence of complex numbers, such thatx–=x–= . In [] is given the concept ofλ-convergent sequence as follows: Letx= (xk) be any given sequence of complex numbers, we will say that it convergesλ-strongly to numberxif

lim n



λn–λn– n

k=

λk(xk–x) – λk–(xk––x) +λk–(xk––x)= .

This generalizes the concept of-strong convergence in []. Let us denote

n(x) = 

λn–λn– n

k=

for alln∈N. Assume here and below that (pk), (qk) are bounded sequences of strictly positive real numbers withsup_kpk=H andmaxk(,H) =M, for  <pk<∞for allk∈N. The linear spacel(p) as defined by Madoxx [] is as follows:

l(p) = x= (xn)∈w:

∞

n=

|xn|pn<∞

, ()

which are complete spaces paranormed by

h(x) =

_∞

n=

|xn|pn

 M

. ()

3 The sequence spacel(

λ

In this section we will define the sequence spacel(λ_{,p) and prove that this sequence space} according to its paranorm is a complete linear space. We have

lλ,p= x= (xk)∈w:

∞ n= 

λn–λn– n

k=

(λk– λk–+λk–)xk

pn <∞ ()

and in case wherepn=p, for everyn∈Nwe get

l_pλ= x= (xk)∈w:

∞ n= 

λn–λn– n

k=

(λk– λk–+λk–)xk

p <∞ . ()

Letx= (xn)∈wbe any sequence; we will define then-transform of the sequencex= (xn) as follows:

_n(x) = 

λn–λn– n

k=

(λk– λk–+λk–)xk, n∈N. ()

Theorem  The sequence space l(λ,p)is the complete linear metric space with respect to the paranorm defined by

g(x) = _∞ n= 

λn–λn– n

k=

(λk– λk–+λk–)xk

pnM

. ()

Proof The linearity ofl(λ_{,p) follows from Minkowski’s inequality. In what follows we will} prove thatg(x) defines a paranorm. In fact, for anyx,y∈l(λ_{,p) we get}

_∞ n= 

λn–λn– n

k=

(λk– λk–+λk–)(xk+yk)

pnM

≤ _∞ n= 

λn–λn– n

k=

(λk– λk–+λk–)xk

pn_M

+ _∞ n= 

λn–λn– n

k=

(λk– λk–+λk–)yk

pnM

and for anyα∈R

|α|pk≤_max,|_α|M_{. ()}

It is clear thatg(θ) = ,g(x) =g(–x) for allx∈l(λ_{,p). From inequalities () and () we} find the subadditivity ofg(x) and henceg(αx)≤max{,|α|M_{}g(x). Letx}

mbe any sequence of points{xm_{} ∈l(λ}_{,p) such thatg(x}m_{–x)→ and (α}

m) also any sequence of scalars such thatαm→α. Then, since the inequality

gxm≤g(x) +gxm–x ()

holds by the subadditivity ofg, we find that{g(xm_{)}is bounded and we thus have}

gαmxm–αx

=

_∞

n=



λn–λn– n

k=

(λk– λk–+λk–)

αmxmk –αxk

pnM

≤ |αm–α| 

M_gxm₊|_α|M_gxm_{–x, ()}

which tends to zero asn→ ∞. Therefore, the scalar multiplication is continuous. Hence g is a paranorm on the spacel(λ,p). It remains to prove the completeness of the space l(λ,p). Let{xj}be any Cauchy sequence in the spacel(λ,p), wherexj={xj,x

j

, . . .}. Then, for a given > , there exists a positive integerm( ) such thatg(xj–xi) < _ for alli,j> m( ). Using the definition ofg, we obtain for each fixedn∈N

_nxj–_nxi≤

_∞

n=

_nxj–_nxipn

 M

<

 ()

for every i,j>m( ), which leads to the fact that {n(x),n(x), . . .} is a Cauchy se-quence of real numbers for every fixed n∈N. Since R is complete, it converges, say

_n(xi_)→

n(x) asi→ ∞. Using these infinitely many limits, we may write the sequence

{_n(x),_n(x), . . .}. From () asi→ ∞, we have

_nxj–_n(x)<

, j≥m( ) ()

for every fixedn∈N. By using () and boundedness of the Cauchy sequence, we have

_∞

n=

_n(x)pn

 M

≤ _∞

n=

_nxj–_n(x)pn

 M

+

_∞

n=

_nxjpn

 M

<∞. ()

Hence, we getx∈l(λ,p). Therefore, the spacel(λ,p), is complete.

Theorem  The sequence space l(λ_{,p)is a BK space.}

Proof Let us denote by= (λ_n,k)∞_n,k=the following matrix:

λn,k

=

λk–λk–+λk–

λn–λn– , ≤k≤n,

for all n,k∈N. Then the_{-transform of a sequencex∈w is the sequence}_{(x) =} (_n(x))∞_n=, where_n(x) is given by () for everyn∈N. Thus

x_l(λ_,p)=(x) l(p).

Now the proof of the theorem follows from Theorem .. given in [].

Theorem  The sequence space l(λ,p)is linearly isomorphic to the space l(p),where < pk≤H<∞.

Proof LetT:l(λ,p)→l(p) be an operator defined byx→y=(x), whereis given by (). The operatorT is linear and injective, fromT(x) =  it follows thatx= . In what follows we will prove thatTis surjective. Lety∈l(p) be any element; we definex= (xn) by

xn(λ) =

λnyn–λn–yn–

λn– λn–+λn–

; ()

then we get

_∞

n=



λn–λn– n

k=

(λk– λk–+λk–)xk

pn_M

=

_∞

n=



λn–λn– n

k=

(λkyk–λk–yk–)

pnM

=

_∞

n=

|yn|pn

 M

.

As a consequence of Theorem  and Theorem  we get the following result.

Corollary  Define the sequence e(n)_λ ∈c(λ,p)for every fixed n∈Nby

e(n)_λ =

(–)n–k λn–λn–

λk–λk–+λk– (k≤n≤k+ ),

 otherwise,

where k∈N.Then we have the following.

() The sequence(e()_λ,e () λ,e

()

λ, . . .)is a Schauder basis for the spacec(λ

_{,p)and every}

x∈c(λ,p)has a unique representation:x=

∞

n=n(x)e (n) λ.

() The sequence(e,e()_λ,e () λ,e

()

λ, . . .)is a Schauder basis for the spacec(λ

_{,p)and every}

x∈c(λ_{,p)has a unique representation:x=le+}∞

n=(n(x) –l)e (n) λ,where l=limnn(x).

Proof Since(e) =eand(e(n)_λ) =e(n)for everyn∈N, the proof of the theorem follows

from Corollary . given in [].

Theorem  The inclusion l(λ_,p)⊂c

Proof Letx∈l(λ_{,p); then it follows that}_{(x)∈l(p), from which follows that}

∞

n=

_n(x)pn<∞.

From the last relation we getlimnn(x)→, asn→ ∞, respectively,(x)∈c(p)⇒x∈ c(,p). To prove that the inclusion is strict we will show the following.

Example  Let  <pn< ,∀n∈N,xn= ( n√_n n+ –

n+√n+ n+ )·



λn–λn–+λn–, andλn=  n_{. Then}

it follows that



λn–λn– n

k=

(λk– λk–+λk–)xk=  n–

n

k=

k√_k k+  –

k+√_{k+ } k+ 

=–

√

n

n+  →, asn→ ∞.

Hencex= (xn)∈c(λ,p). On the other handx= (xn) /∈l(λ,p), for  <pn< ,∀n∈N. With

which we have proved the theorem.

Theorem  The inclusions c(λ,p)⊂c(λ,p)⊂l∞(λ,p)strictly hold.

Proof It is clear that the inclusionc(λ,p)⊂c(λ,p)⊂l∞(λ,p) holds. Further, sincec⊂ cis strict, from Lemmas  and  from [] it follows thatc(λ,p)⊂c(λ,p) is also strict. In what follows we will show that the last inclusion is strict, too. For this reason we will show the following.

Example  Let

xn= (–)n

λn–λn–

λn– λn–+λn– ,

then it follows that

_n(x) = 

λn–λn– n

k=

(λk– λk–+λk–)xk

= 

λn–λn– n

k=

(–)k(λk–λk–) = (–)n.

From the last relation it follows thatx= (xk)∈l∞(λ,p)\c(λ,p).

Theorem  The inclusion l(λ_{,p)⊂l(p)holds if and only if S(x)∈l(p)for every sequence} x∈l(λ_{,p),where≤p}

k≤H.Here S(x) ={Sn(x)}and Sn(x) =xn–n(x).

Proof The proof of the theorem is similar to Theorem . given in [].

Theorem 

() Ifpn> for alln∈N,then the inclusionlλ 

p ⊂l(λ,p)holds.

Proof () Letp= (pn) >  for alln∈Nandx∈lλ 

p . Then it follows that(x)∈l(p). Hence

lim

n

 n(x)= ,

we findn∈Nsuch that|(x)|< , for everyn>n, respectively,

_n(x)pn<_n(x) forpn> ,∀n>n.

From the last relation we getx∈l(λ,p).

() Let us suppose thatx∈l(λ,p). Then(x)∈l(p),∃n∈N, such that

∀n>n ⇒ n(x) pn

< ,

_n(x)=_n(x)pnpn _< n(x)

pn

for alln>n. Hencex∈l 

p .

4 Duals of the spacel(

λ

In this section we will give the theorems in which theα-,β-, andγ-duals are determined of the sequence spacel(λ,p). In proving the theorems we apply the technique used in []. Also we will give some matrix transformations froml(p,λ,p) intol(q) by using the matrix given in [].

Theorem  Let K={k∈N:pk≤}and K={k∈N:pk> }.Define the matrix Ma= (ma

nk)by

ma_nk= (–)

n–k_· λk

λn–λn–+λn–·an, n– ≤k≤n,

, ≤k≤n– or k>n. ()

Then

l_Ka λ,p=a= (an)∈w:Ma∈

l(p);l∞, ()

l_Ka λ,p=a= (an)∈w:Ma∈

l(p);l

. ()

Proof Letx= (xn)∈l(λ,p),a= (an)∈w, and (yn) = (n(x)). Then we have

anxn=an n

k=n–

(–)n–k λk

λn– λn–+λn–

yk=

May_n; n∈N, ()

whereMa_{is defined by (). From () it follows thatax= (a}

nxn)∈lorax= (anxn)∈l∞

wheneverx∈l(λ_{,p) if and only ifM}a_y∈l

orMay∈l∞ whenevery∈l(p). This means

thata∈lα

K(λ,p) ora∈lαK(λ,p) if and only ifMa∈(l(p),l) orMa∈(l(p),l∞).

As a direct consequence of Theorem , we get the following.

() lα

K(λ,p) ={a= (an)∈w:supKsupn∈N|

n∈K∗mank|pk<∞},

() lα

K(λ,p) =

M>{a= (an)∈w:supK∈F

K|

n∈K∗mankM–|pk<∞}, for some constant M.

In what follows we will characterize theβ- andγ-dual of the sequence spacel(λ,p).

Theorem  Let K={k∈N:pk≤},K={k∈N:pk> }.Define the sequence d= (dk), d= (d_k),and the matrix Da= (d_nka )by

d_nka =

⎧ ⎪ ⎨ ⎪ ⎩

d_k, ≤k≤n– , d_k, k=n, , k>n,

where d_k= ( ak λk–λk–+λk––

ak+

λk+–λk+λk–)λk,d  k=

akλk

λk–λk–+λk–.Then

l_Kβλ,p=lγ_Kλ,p=a= (an)∈w:Da∈

l(p),l∞, ()

l_Kβ λ,p=lγ_Kλ,p=a= (an)∈w:Da∈

l(p),c. ()

Proof Letx= (xn)∈l(λ,p). Then we obtain

n

k= akxk=

n–

k=

d_kyk+dnyn=

Day_n. ()

From () it follows thatax= (anxn)∈csorbsif and only ifDa(y)∈corl∞. This means that a= (an)∈ {l

β

K(λ,p) orl β

K(λ,p)}or a= (an)∈ {l γ

K(λ,p) orl γ

K(λ,p)}. With which

the theorem is proved.

As an immediate result of the above theorem, we get the following.

Corollary  Let(p_k)be a conjugate sequence of numbers(pk),it means that _p k +

 p_k =  and <pk<∞for all k∈N.Then

() lβ_K(λ,p) =lγ_K(λ,p) ={a= (an)∈w:dk,dk∈l∞(p)},

() lβ_K(λ,p) =l_Kγ (λ,p) =_M>{a= (an)∈w:dkM–,dkM–∈l∞(p)∩l(p)}.

5 Some matrix transformations related to sequence spacel(

λ

In this section we will show some matrix transformations between the sequence space l(λ,p) and sequence spacesl(q),c(q),c(q), andl∞(q) whereq= (qn) is a sequence of nondecreasing, bounded positive real numbers. Letx,y∈wbe connected by the relation y=_{(x). For an infinite matrixA= (a}

nk), taking into consideration Theorem , we get

m

k= ankxk=

m–

k= ankyk+

anmλn

λn– λn–+λn–

(m,n∈N), ()

where

ank=

ank

λk– λk–+λk–

– an,k+

λk+– λk+λk–

LetN, K be a finite subsets of the natural numbersNandL,T be a natural numbers. K={k∈N:pk≤}andK={k∈N:pk> }and also let (pk) be a conjugate sequence of numbers (pk). Prior to giving the theorems, let us suppose that (qn) is a nondecreasing bounded sequence of positive real numbers and consider the following conditions:

sup n

sup k∈K

n∈N ank

qn<∞, ()

∃Tsup N

k∈K

n ankT–

pk<∞, ()

∃Tsup k

n

akT –

pkqn_<∞_{, ()}

lim n |ank|

qn_{=  (}∀_k∈N_{), ()}

∀L, sup n

sup K

ankL 

qn_<∞_{, ()}

∀L, ∃Tsup n

K

akL  qn_T–pk

<∞, ()

sup n

sup K

|ank|pk<∞, ()

∃Tsup n

K

akT– pk

<∞, ()

∀L, sup n

sup K

|ak–ak|L 

qnpk_<∞_{, ()}

lim

n |ak–ak|

qn_{= ,} ∀_{k, ()}

∀L, ∃Tsup n

K

|ank–ak|L 

qn_T–pk_<∞_{, ()}

∃L, sup n

sup K

ankL –

qnpk_<∞_{, ()}

∃L, sup n

K

ankL – qnpk

<∞, ()

ankλk

λk– λk–+λk–

∞

k=

∈c(q), ∀n∈N, ()

ankλk

λk– λk–+λk–

∞

k=

∈c(q), ∀n∈N, ()

ankλk

λk– λk–+λk–

∞

k=

∈l∞(q), ∀n∈N. ()

From the above conditions we get the following.

Theorem 

A∈(l(λ,p);l(q))if and only if(), (), (),and()hold,

A∈(l(λ_,p);c

(q))if and only if(), (), (),and()hold,

Competing interests

The author declares to have no competing interests.

Received: 26 March 2014 Accepted: 16 June 2014 Published:23 Jul 2014

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10.1186/1029-242X-2014-273

Cite this article as:Braha:On some properties of new paranormed sequence space defined byλ2_-convergent