R E S E A R C H
Open Access
On certain generalized paranormed spaces
Kuldip Raj
1and Adem Kılıçman
2**Correspondence:
2Department of Mathematics and
Institute for Mathematical Research, University Putra Malaysia (UPM), Serdang, Selangor 43400, Malaysia Full list of author information is available at the end of the article
Abstract
In the present paper we introduce and study some generalized paranormed sequence spaces defined by Musielak-Orlicz functions as well as by a sequence of modulus functions. We also study some topological properties and prove some inclusion relations between these spaces.
MSC: 40A05; 46A45; 46E30
Keywords: Orlicz function; Musielak-Orlicz function; modulus function; sequence space
1 Introduction and preliminaries
An Orlicz functionM: [,∞)→[,∞) is convex and continuous such thatM() = , M(x) > forx> . Letwbe the space of all real or complex sequencesx= (xk).
Linden-strauss and Tzafriri [] used the idea of the Orlicz function to define the following sequence space:
M=
x∈w: ∞
k= M
|x k|
ρ
<∞, for someρ>
,
which is called an Orlicz sequence space. The spaceMis a Banach space with the norm
x=inf
ρ> :
∞
k= M
|xk|
ρ
≤
.
It is shown in [] that every Orlicz sequence spaceMcontains a subspace isomorphic to
p(p≥). An Orlicz functionMsatisfies the-condition if and only if for any constant
L> there exists a constantK(L) such thatM(Lu)≤K(L)M(u) for all values ofu≥. A sequence M= (Mk) of Orlicz functions is called a Musielak-Orlicz function (see
[–]). A sequenceN= (Nk) is defined by
Nk(v) =sup
|v|u–Mk(u) :u≥
, k= , , . . .
is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz functionM, the Musielak-Orlicz sequence spacetMand its subspace hMare defined as follows:
tM=x∈w:IM(cx) <∞for somec> ,
hM=x∈w:IM(cx) <∞for allc> ,
whereIMis a convex modular defined by
IM(x) = ∞
k=
Mk(xk), x= (xk)∈tM.
We considertMequipped with the Luxemburg norm
x=inf k> :IM
x k
≤
or equipped with the Orlicz norm
x=inf k
+IM(kx):k>
.
A Musielak-Orlicz function (Mk) is said to satisfy the-condition if there exist constants
a,K> and a sequencec= (ck)∞k=∈+(the positive cone of) such that the inequality
Mk(u)≤KMk(u) +ck
holds for allk∈Nandu∈R+wheneverMk(u)≤a.
A modulus function is a functionf: [,∞)→[,∞) such that () f(x) = if and only ifx= ,
() f(x+y)≤f(x) +f(y)for allx≥,y≥, () f is increasing,
() f is continuous from right at.
It follows thatf must be continuous everywhere on [,∞). The modulus function may be bounded or unbounded. For example, if we takef(x) = xx+, thenf(x) is bounded. If f(x) =xp, <p< , then the modulusf(x) is unbounded. Subsequently, modulus function
has been discussed in [, –] and references therein.
Letl∞,c, andcdenote the spaces of all bounded, convergent, and null sequencesx= (xk)
with complex terms, respectively. The zero sequence (, , . . .) is denoted byθ.
The notion of difference sequence spaces was introduced by Kızmaz [], who studied the difference sequence spacesl∞(),c(), andc(). The notion was further generalized by Et and Çolak [] by introducing the spacesl∞(n),c(n), andc(n). Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [] who studied the spacesl∞(mn),c(mn), andc(mn).
Letm,nbe non-negative integers, then forZa given sequence space, we have
Znm=x= (xk)∈w:
nmxk
∈Z
forZ =c,candl∞ where mnx= (nmxk) = (mn–xk–nm–xk+m) and mxk=xk for all
k∈N, which is equivalent to the following binomial representation:
nmxk= n
v= (–)v
n v
Takingm= , we get the spacesl∞(n),c(n), andc(n) studied by Et and Çolak []. Takingm=n= , we get the spacesl∞(),c(), andc() introduced and studied by Kızmaz []. For more details as regards sequence spaces, see [, –] and references therein.
LetM= (Mk) be a Musielak-Orlicz function,p= (pk) be any bounded sequence of
posi-tive real numbers andu= (uk) be a sequence of strictly positive real numbers. Let (X,q) be
a space seminormed byq. In the present paper we define the following sequence spaces:
wM,nm,p,q,u=
x= (xk) :
n
n
k=
Mk
q(uknmxk)
ρ
pk
→ asn→ ∞,
for someρ>
,
wM,nm,p,q,u=
x= (xk) :
n
n
k=
Mk
q(uknmxk–L)
ρ
pk
→ asn→ ∞,
for someL∈X,ρ>
,
and
w∞M,nm,p,q,u
=
x= (xk) :sup n
n
n
k=
Mk
q(uknmxk)
ρ
pk
<∞, for someρ>
.
If we takeM(x) =x, we get wnm,p,q,u
=
x= (xk) :
n
n
k=
q(uknmxk)
ρ
pk
→ asn→ ∞, for someρ>
,
wnm,p,q,u
=
x= (xk) :
n
n
k=
q(uknmxk–L)
ρ
pk
→ asn→ ∞, for someL∈X,ρ>
,
and
w∞nm,p,q,u
=
x= (xk) :sup n
n
n
k=
q(uknmxk)
ρ
pk
<∞, for someρ>
.
If we takep= (pk) = ,∀k, we get
wM,nm,q,u
=
x= (xk) :
n
n
k=
Mk
q(uknmxk)
ρ
→ asn→ ∞, for someρ>
wM,nm,q,u=
x= (xk) :
n
n
k=
Mk
q(uknmxk–L)
ρ
→ asn→ ∞,
for someL∈X,ρ>
,
and
w∞M,nm,q,u=
x= (xk) :sup n
n
n
k=
Mk
q(uknmxk)
ρ
<∞, for someρ>
.
If we takeu= (uk) = ,∀k, we get
wM,nm,p,q=
x= (xk) :
n
n
k=
Mk
q(nmxk)
ρ
pk
→ asn→ ∞,
for someρ>
,
wM,nm,p,q=
x= (xk) :
n
n
k=
Mk
q(nmxk–L)
ρ
pk
→ asn→ ∞,
for someL∈X,ρ>
,
and
w∞M,nm,p,q
=
x= (xk) :sup n
n
n
k=
Mk
q(nmxk)
ρ
pk
<∞, for someρ>
.
The following inequality will be used throughout the paper. If ≤pk≤suppk=K,D=
max(, K–) then
|ak+bk|pk≤D
|ak|pk+|bk|pk
(.)
for allkandak,bk∈C. Also|a|pk≤max(,|a|K) for alla∈C.
The aim of this paper is to study some topological and algebraic properties of the above sequence spaces.
2 Main results
Theorem . SupposeM= (Mk)be a Musielak-Orlicz function,p= (pk)be any bounded
sequence of positive real numbers and u= (uk)be a sequence of strictly positive real
num-bers.Then the spaces w(M,n
m,p,q,u),w(M,nm,p,q,u)and w∞(M,nm,p,q,u)are
lin-ear spaces over the complex fieldC.
Proof Letx= (xk),y= (yk)∈w∞(M,nm,p,q,u) andα,β∈C. Then there exist positive
real numbersρandρsuch that
sup
n
n
n
k=
Mk
q(uknmxk)
ρ
pk
and
sup
n
n
n
k=
Mk
q(uknmyk)
ρ
pk
<∞.
Define ρ=max(|α|ρ, |β|ρ). Since (Mk) is non-decreasing, convex and so by using
inequality (.), we have
sup
n
n
n
k=
Mk
q(αuknmxk+βuknmyk)
ρ
pk
≤sup
n
n
n
k=
Mk
q(αuknmxk)
ρ
+q(βuk
n myk)
ρ
pk
≤sup
n
n
n
k= pk
Mk
q(uknmxk)
ρ
pk
+sup
n
n
n
k= pk
Mk
q(uknmyk)
ρ
pk
≤Dsup
n
n
n
k=
Mk
q(uknmxk)
ρ
pk
+Dsup
n
n
n
k=
Mk
q(uknmyk)
ρ
pk
<∞.
Thus,αx+βy∈w∞(M,nm,p,q,u). Hencew∞(M,nm,p,q,u) is a linear space. Similarly, we can provew(M,nm,p,q,u) andw(M,nm,p,q,u) are linear spaces over the field of complex numbers.
Theorem . SupposeM= (Mk)be a Musielak-Orlicz function,p= (pk)be any bounded
sequence of positive real numbers and u= (uk)be a sequence of strictly positive real
num-bers.Then the space w∞(M,nm,p,q,u)is a paranormed space with the paranorm defined
by
g(x) =inf
ρpkH :sup n
n
n
k=
Mk
q
uknmxk
ρ
pk H
≤,ρ>
,
where H=max(,supkpk).
Proof (i) Clearly, g(x)≥ for x= (xk)∈w∞(M,nm,p,q,u). Since Mk() = , we get
g(θ) = .
(ii)g(–x) =g(x).
(iii) Letx= (xk),y= (yk)∈w∞(M,nm,p,q,u) then there existρ,ρ> such that
n
n
k=
Mk
q
uknmxk
ρ
pk
≤
and
n
n
k=
Mk
q
uknmyk
ρ
pk
Letρ=ρ+ρ, then by Minkowski’s inequality, we have n n k= Mk q
uknmxk
ρ
pk
≤
ρ
ρ+ρ
n n k= Mk q
uknmxk
ρ
pk
+
ρ
ρ+ρ
n n k= Mk q
uknmyk
ρ
pk
and thus
g(x+y)
=inf
(ρ+ρ)
pk H :sup
n n n k= Mk q
uknmxk+uknmyk
ρ
pk H
≤,ρ>
≤g(x) +g(y).
(iv) Finally we prove that scalar multiplication is continuous. Letλbe any complex num-ber by definition
g(λx) =inf
(ρ)pkH :sup n n n k= Mk q
uknmλxk
ρ
pk H
≤,ρ>
=inf
|λ|rpkH :sup n n n k= Mk q
uknmxk
r
pk H
≤,ρ>
,
wherer=|ρλ|. Hence,w∞(M,nm,p,q,u) is a paranormed space.
Theorem . If <pk≤rk<∞for each k,then Z(M,mn,p,q,u)⊆Z(M,nm,r,q,u)for
Z=w,w,w∞.
Proof Letx= (xk)∈w(M,nm,p,q,u). Then there exist someρ> andL∈Xsuch that
n n k= Mk q
uknmxk–L
ρ
pk
→ asn→ ∞.
This implies that
n n k= Mk q
uknmxk–L
ρ
pk
< ( < < )
for sufficiently largek. Hence we get
n n k= Mk q
uknmxk–L
ρ rk ≤ n n k= Mk q
uknmxk–L
ρ
pk
→ asn→ ∞.
This implies thatx= (xk)∈w(M,nm,r,q,u). This completes the proof. Similarly, we can
Theorem . SupposeM= (Mk)andM= (Mk)are Musielak-Orlicz functions satisfy-ing the-condition,then we have the following results:
(i) Ifp= (pk)is a bounded sequence of positive real numbers then
Z(M,nm,p,q,u)⊆Z(M◦M,nm,p,q,u)forZ=w,w,andw∞. (ii) Z(M,n
m,p,q,u)∩Z(M,nm,p,q,u)⊆Z(M+M,nm,p,q,u)for
Z=w,w,andw∞.
Proof (i) Ifx= (xk)∈w(M,nm,p,q,u), then there exists someρ> such that
n
n
k=
Mk
q
uknmxk
ρ
pk
→ asn→ ∞.
Suppose
yk=Mk
q
uknmxk
ρ
for allk∈N. Choose <δ< , then foryk≥δwe haveyk<yδk < + yk
δ . Now (Mk) satisfies
the-condition so that there existsJ≥ such that
Mk(yk) <
Jyk
δM
k() +
Jyk
δM
k() =
Jyk
δ M
k().
We obtain
n
n
k=
Mk◦Mk
q
uknmxk
ρ
pk
= n
n
k=
Mk Mk
q
uknmxk
ρ
pk
= n
n
k=
Mk(yk) pk
→ asn→ ∞.
Similarly we can prove the other cases.
(ii) Supposex= (xk)∈w(Mk,nm,p,q,u)∩w(Mk,nm,p,q,u), then there existρ,ρ> such that
n
n
k=
Mk
q
uknmxk
ρ
pk
→, asn→ ∞.
and
n
n
k=
Mk
q
uknmxk
ρ
pk
→, asn→ ∞.
Letρ=max{ρ,ρ}. The remaining proof follows from the inequality
n
n
k=
Mk+Mkq
uknmxk
ρ
pk
≤D
n
n
k=
Mk
q
uknmxk
ρ
pk
+ n
n
k=
Mk
q
uknmxk
ρ
pk
Hence,w(Mk,n
m,p,q,u)∩w(Mk,nm,p,q,u)⊆w(Mk +Mk,nm,p,q,u). Similarly we
can prove the other cases.
Theorem . (i)If <infpk≤pk< ,then
w∞M,nm,p,q,u
⊂w∞M,nm,q,u
.
(ii)If≤pk≤suppk<∞,then
w∞M,nm,q,u⊂w∞M,nm,p,q,u.
Proof (i) Letx= (xk)∈w∞(M,nm,p,q,u). Since <infpk≤, we have
sup
n
n
n
k=
Mk
q
uknmxk
ρ
≤sup
n
n
n
k=
Mk
q
uknmxk
ρ
pk
and hencex= (xk)∈w∞(M,nm,q,u).
(ii) Letpk≥ for eachkandsupkpk<∞. Letx= (xk)∈w∞(M,nm,q,u), then for each
> such that < < , there exists a positive integern∈Nsuch that
sup
n
n
n
k=
Mk
q
uknmxk
ρ
≤ < .
This implies that
sup
n
n
n
k=
Mk
q
uknmxk
ρ
pk
≤sup
n
n
n
k=
Mk
q
uknmxk
ρ
.
Thus,x= (xk)∈w∞(M,nm,p,q,u) and this completes the proof. Theorem . The sequence space w∞(M,nm,p,q,u)is solid.
Proof Letx= (xk)∈w∞(M,nm,p,q,u),i.e.
sup
n
n
n
k=
Mk
q
uknmxk
ρ
pk
<∞.
Let (αk) be a sequence of scalars such that|αk| ≤ for allk∈N. Thus we have
sup
n
n
n
k=
Mk
q
αkuknmxk
ρ
pk
≤sup
n
n
n
k=
Mk
q
uknmxk
ρ
pk
<∞.
This shows that (αkxk)∈w∞(M,nm,p,q,u) for all sequences of scalars (αk) with|αk| ≤
for allk∈N, whenever (xk)∈w∞(M,nm,p,q,u). Hence the spacew∞(M,nm,p,q,u) is a
solid sequence space.
Proof The proof of the theorem is obvious and so we omit it.
LetF= (fk) be a sequence of modulus functions,p= (pk) be any bounded sequence of
positive real numbers andu= (uk) be a sequence of strictly positive real numbers. Let
(X,q) be a space seminormed byq. Now, we define the following sequence spaces:
wF,nm,p,q,u=
x= (xk) :
n
n
k=
fk
q(uknmxk)
ρ
pk
→ asn→ ∞,
for someρ>
,
wF,nm,p,q,u=
x= (xk) :
n
n
k=
fk
q(uknmxk–L)
ρ
pk
→ asn→ ∞,
for someρ> andL∈X
,
and
w∞F,nm,p,q,u=
x= (xk) :sup n
n
n
k=
fk
q(uknmxk)
ρ
pk
<∞, for someρ>
.
Theorem . Let F= (fk)be a sequence of modulus functions,p= (pk)be any bounded
se-quence of positive real numbers and u= (uk)be a sequence of strictly positive real numbers.
Then the spaces w(F,nm,p,q,u),w(F,nm,p,q,u),and w∞(F,nm,p,q,u)are linear spaces
over the complex fieldC.
Proof The proof of Theorem . holds along the same lines for this theorem and so we
omit it.
Theorem . Let F= (fk)be a sequence of modulus function,p= (pk)be any bounded
se-quence of positive real numbers and u= (uk)be a sequence of strictly positive real numbers.
Then w∞(F,nm,p,q,u)is a paranormed space with the paranorm defined by
g(x) =inf
ρpkH :sup n
n
n
k=
fk
q
uknmxk
ρ
pk H
≤,ρ>
, (.)
where H=max(,supkpk).
Proof The proof follows from Theorem . and so we omit it.
Theorem . Let F= (fk)be a sequence of modulus functions,p= (pk)be any bounded
se-quence of positive real numbers and u= (uk)be a sequence of strictly positive real numbers.
Then
wF,nm,p,q,u⊂wF,mn,p,q,u⊂w∞F,nm,p,q,u
,
Proof The proof is obvious.
Theorem . Let F= (fk)and G= (gk)be any two sequences of modulus functions.For any
bounded sequences p= (pk)of positive real numbers and for any two seminorms q and r.
Then
(i) wZ(F,nm,q,u)⊂wZ(F◦G,nm,q,u),
(ii) wZ(F,nm,p,q,u)∩wZ(F,mn,p,r,u)⊂wZ(F,nm,p,q+r,u),
(iii) wZ(F,nm,p,q,u)∩wZ(G,nm,p,q,u)⊂wZ(F+G,nm,p,q,u),whereZ= , ,∞.
Proof (i) We shall prove it for the relationw(F,nm,q,u)⊂w(F◦G,nm,q,u). For > , we choose δ, <δ< , such thatfk(t) < for ≤t≤δ and all k∈N. We write yk=
gk(q( mnukxk)
ρ ) and consider
n
k=
fk(yk)
=
fk(yk)
+
fk(yk)
,
where the first summation is overyk≤δand the second summation is overyk>δ. Since
Fis continuous, we have
fk(yk)
<n . (.)
By the definition ofF, we have the following relation foryk>δ:
fk(yk) < fk()
yk
δ .
Hence,
n
fk(yk)
≤δ–fk()
n
n
k=
yk. (.)
It follows from (.) and (.) thatw(F,nm,q,u)⊂w(F◦G,nm,q,u). Similarly, we can provew(F,nm,q,u)⊂w(F◦G,nm,q,u) andw∞(F,nm,q,u)⊂w∞(F◦G,nm,q,u).
The proof of (ii) and (iii) follows from (i).
Corollary . Let f be a modulus function.Then
wZ
nm,q,u⊂wZ
f,nm,q,u, for Z= , ,∞.
Theorem . Let F= (fk)be a sequence of modulus functions,p= (pk)be any bounded
se-quence of positive real numbers and u= (uk)be a sequence of strictly positive real numbers.
Then w∞(F,nm,p,q,u)is complete and seminormed by(.).
Proof Suppose (xn) is a Cauchy sequence inw
∞(F,nm,p,q,u), wherexn= (xnk)∞k= for all n∈N. So thatg(xi–xj)→ asi,j→ ∞. Suppose > is given and letsandxbe such
that sx
> andfk(
sx
)≥supk≥(pk). Sinceg(xi–xj)→, asi,j→ ∞, which means that
there existsn∈Nsuch that
gxi–xj<
This givesg(xi–xj) <sx
and
inf ρpkH :sup k≥
fk
q(uknmxik–uknmx j k)
ρ
≤,ρ>
<
sx. (.)
It shows that (xi
) is a Cauchy sequence inX. Thus, (xi) is convergent inXbecauseXis complete. Supposelimi→∞xi=xthenlimj→∞g(xi–x
j
) <sx, we get
gxi–x< sx.
Thus, we have
fk
q(uknmxik–uknmx j k)
g(xi–xj)
≤.
This implies that
fk
q(uknmxik–uknmx j k)
g(xi–xj)
≤fk
sx
and thus
quknmxik–uknmx j k
<sx
·sx<,
which shows that (uknmxik) is a Cauchy sequence inXfor allk∈N. Therefore, (uknmxik)
converges inX. Supposelimi→∞nmxik=ykfor allk∈N. Also, we havelimi→∞uknmxi= y–x. On repeating the same procedure, we obtainlimi→∞uknmxik+=yk–xk for all k∈N. Therefore by continuity offk, we get
lim
j→∞supk≥fk
q(uknmxik–uknmx j k)
ρ
≤,
so that
sup
k≥
fk
q(uknmxik–uknmxk)
ρ
≤.
Leti≥nand taking the infimum of eachρ, we have
gxi–x< .
So (xi–x)∈w
∞(F,nm,p,q,u). Hencex=xi– (xi–x)∈w
∞(F,nm,p,q,u), sincew∞(F,nm, p,q,u) is a linear space. Hence,w∞(F,nm,p,q,u) is a complete paranormed space.
Competing interests
Authors’ contributions
Both authors contributed equally during the development of manuscript and the authors read and approved the final manuscript.
Author details
1School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir 182320, India.2Department of
Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), Serdang, Selangor 43400, Malaysia.
Acknowledgements
The authors express their sincere thanks to the referees for the careful and details reading of the manuscript and suggestions on the current literature.
Received: 19 September 2014 Accepted: 14 January 2015 References
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