**Generalized Orlicz Spaces**
Pankaj Jain and Priti Upreti
**vol. 10, iss. 2, art. 37, 2009**

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## CERTAIN PROPERTIES OF GENERALIZED ORLICZ SPACES

PANKAJ JAIN PRITI UPRETI

Department of Mathematics Department of Mathematics

Deshbandhu College (University of Delhi) Moti Lal Nehru College (University of Delhi) Kalkaji, New Delhi - 110 019, India Benito Juarez Marg, Delhi 110 021, India EMail:pankajkrjain@hotmail.com

*Received:* 20 March, 2008

*Accepted:* 09 October, 2008
*Communicated by:* L.-E. Persson

*2000 AMS Sub. Class.:* 26D10, 26D15, 46E35.

*Key words:* Banach Function spaces, generalized Orlicz class, generalized Orlicz space, Lux-
emburg norm, Young function, Young’s inequality, imbedding, convergence, sep-
arability.

*Abstract:* In the context of generalized Orlicz spacesXΦ, the concepts of inclusion, con-
vergence and separability are studied.

*Acknowledgements:* The research of the first author is partially supported by CSIR (India) through the
grant no. 25(5913)/NS/03/EMRII.

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Pankaj Jain and Priti Upreti
**vol. 10, iss. 2, art. 37, 2009**

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**Contents**

**1** **Introduction** **3**

**2** **Preliminaries** **4**

**3** **Comparison of Generalized Orlicz Spaces** **8**

**4** **Convergence** **15**

**5** **Separability** **19**

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Pankaj Jain and Priti Upreti
**vol. 10, iss. 2, art. 37, 2009**

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**1.** **Introduction**

In [4], Jain, Persson and Upreti studied the generalized Orlicz spaceX_{Φ} which is a
unification of two generalizations of the LebesgueL^{p}-spaces, namely, theX^{p}-spaces
and the usual Orlicz spacesL_{Φ}. There the authors formulated the space X_{Φ} giving
it two norms, the Orlicz type norm and the Luxemburg type norm and proved the
two norms to be equivalent as is the case in usual Orlicz spaces. It was shown that
X_{Φ} is a Banach function space if X is so and a number of basic inequalities such
as Hölder’s, Minkowski’s and Young’s were also proved in the framework of X_{Φ}
spaces.

In the present paper, we carry on this study and target some other concepts in the
context ofX_{Φ}spaces, namely, inclusion, convergence and separability.

The paper is organized as follows: In Section2, we collect certain preliminaries
which would ease the reading of the paper. The inclusion property inX_{Φ} spaces has
been studied in Section3. Also, an imbedding has been proved there. In Sections4
and5respectively, the convergence and separability properties have been discussed.

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Pankaj Jain and Priti Upreti
**vol. 10, iss. 2, art. 37, 2009**

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**2.** **Preliminaries**

Let (Ω,Σ, µ) be a complete σ-finite measure space with µ(Ω) > 0. We denote
byL^{0}(Ω), the space of all equivalence classes of measurable real valued functions
defined and finite a.e. onΩ. A real normed linear spaceX ={u∈L^{0}(Ω) :kuk_{X} <

∞}is called a Banach function space (BFS for short) if in addition to the usual norm axioms,kukX satisfies the following conditions:

P1. kuk_{X} is defined for every measurable functionuonΩandu ∈ X if and only
ifkuk_{X} <∞;kuk_{X} = 0if and only if,u= 0a.e.;

P2. 0≤u≤va.e. ⇒ kuk_{X} ≤ kvk_{X};
P3. 0< u_{n}↑ua.e. ⇒ kuk_{X} ↑ kuk_{X};
P4. µ(E)<∞ ⇒ kχ_{E}k_{X} <∞;

P5. µ(E)<∞ ⇒R

Eu(x)dx≤C_{E}kuk_{X},

where E ⊂ Ω, χ_{E} denotes the characteristic function of E and C_{E} is a constant
depending only on E. The concept of BFS was introduced by Luxemburg [9]. A
good treatment of such spaces can be found, e.g., in [1]

Examples of Banach function spaces are the classical Lebesgue spacesL^{p}, 1 ≤
p≤ ∞, the Orlicz spacesL_{Φ}, the classical Lorentz spacesL_{p,q},1 ≤ p, p ≤ ∞, the
generalized Lorentz spacesΛ_{φ}and the Marcinkiewicz spacesM_{φ}.

Let X be a BFS and−∞ < p < ∞, p 6= 0. We define the space X^{p} to be the
space of all measurable functionsf for which

kfk_{X}^{p} :=k|f|^{p}k

1 p

X <∞.

For1< p <∞,X^{p}is a BFS. Note that forX =L^{1}, the spaceX^{p}coincides withL^{p}
spaces. These spaces have been studied and used in [10], [11], [12]. Very recently

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in [2], [3], Hardy inequalities (and also geometric mean inequalities in some cases)
have been studied in the context ofX^{p}spaces. For an updated knowledge of various
standard Hardy type inequalities, one may refer to the monographs [6], [8] and the
references therein.

A functionΦ : [0,∞)→[0,∞]is called a Young function if Φ(s) =

Z s

0

φ(t)dt ,

where φ : [0,∞) → [0,∞], φ(0) = 0 is an increasing, left continuous function which is neither identically zero nor identically infinite on(0,∞). A Young function Φis continuous, convex, increasing and satisfies

Φ(0) = 0, lim

s→∞Φ(s) = ∞.

Moreover, a Young functionΦsatisfies the following useful inequalities: fors ≥0, we have

(2.1)

(Φ(αs)< αΦ(s), if 0≤α <1 Φ(αs)≥αΦ(s), if α≥1.

We call a Young function anN-function if it satisfies the limit conditions

s→∞lim Φ(s)

s =∞ and lim

s→0

Φ(s) s = 0. LetΦbe a Young function generated by the functionφ, i.e.,

Φ(s) = Z s

0

φ(t)dt .

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Then the functionΨgenerated by the functionψ, i.e., Ψ(s) =

Z s

0

ψ(t)dt , where

ψ(s) = sup

φ(t)≤s

t

is called the complementary function toΦ. It is known thatΨis a Young function and thatΦis complementary to Ψ. The pair of complementary Young functions Φ, Ψsatisfies Young’s inequality

(2.2) u·v ≤Φ(u) + Ψ(v), u, v ∈[0,∞).

Equality in (2.2) holds if and only if

(2.3) v = Φ(u) or u= Ψ(v).

A Young function Φ is said to satisfy the ∆2-condition, writtenΦ ∈ ∆2, if there existk >0andT ≥0such that

Φ(2t)≤kΦ(t) for all t≥T .

The above mentioned concepts of the Young function, complementary Young
function and∆_{2}-condition are quite standard and can be found in any standard book
on Orlicz spaces. Here we mention the celebrated monographs [5], [7].

The remainder of the concepts are some of the contents of [4] which were devel- oped and studied there and we mention them here briefly.

LetXbe a BFS andΦdenote a non-negative function on[0,∞). The generalized
Orlicz classXeΦconsists of all functionsu∈L^{0}(Ω)such that

ρX(u,Φ) =kΦ(|u|)kX <∞.

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For the caseΦ(t) = t^{p},0 < p < ∞,Xe_{Φ} coincides algebraically with the spaceX^{p}
endowed with the quasi-norm

kuk_{X}^{p} =k|u|^{p}k

1 p

X.

Let X be a BFS and Φ, Ψ be a pair of complementary Young functions. The
generalized Orlicz space, denoted byX_{Φ}, is the set of allu∈L^{0}(Ω)such that

(2.4) kuk_{Φ} := sup

v

k|u·v|k_{X},

where the supremum is taken over allv ∈Xe_{Ψ}for whichρ_{X}(v; Ψ)≤1.

It was proved that for a Young functionΦ,XeΦ ⊂XΦand thatXΦ is a BFS, with
the norm (2.4). Further, on the generalized Orlicz spaceX_{Φ}, a Luxemburg type norm
was defined in the following way

(2.5) kuk^{0}_{Φ} = inf

k >0 :ρ_{X}
|u|

k ,Φ

≤1

.

It was shown that with the norm (2.5) too, the space X_{Φ} is a BFS and that the two
norms (2.4) and (2.5) are equivalent, i.e., there exists constantsc_{1}, c_{2} >0such that
(2.6) c_{1}kuk^{0}_{Φ}≤ kuk_{Φ} ≤c_{2}kuk^{0}_{Φ}.

In fact, it was proved thatc_{2} = 2.

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**3.** **Comparison of Generalized Orlicz Spaces**

We begin with the following definition:

* Definition 3.1. A BFS is said to satisfy the*L-property if for all non-negative func-

*tions*f, g ∈X, there exists a constant0< a <1

*such that*

kf+gkX ≥a(kfkX +kgkX).

*Remark 1. It was proved in [2] that the generalized Orlicz space* X_{Φ} contains the
generalized Orlicz classXe_{Φ}. Towards the converse, we prove the following:

* Theorem 3.2. Let*Φ

*be a Young function,*X

*be a BFS satisfying the*L-property and u∈X

_{Φ}

*be such that*kuk

_{Φ}6= 0. Then

_{kuk}

^{u}

Φ ∈Xe_{Φ}*.*

*Proof. Let*u ∈X_{Φ}. Using the modified arguments used in [7, Lemma 3.7.2], it can
be shown that

(3.1) ku·vkX ≤

(kukΦ ; for ρX(v; Ψ)≤1,
kuk_{Φ}ρ_{X}(v; Ψ) ; for ρ_{X}(v; Ψ)>1.

LetE ⊂ Ωbe such thatµ(E) < ∞. First assume thatu ∈ X_{Φ}(Ω) is bounded and
thatu(x) = 0forx∈Ω\E. Put

v(x) = φ 1

kuk_{Φ}|u(x)|

.

The monotonicity ofΦandΨgives that the functionsΦ

1

kuk_{Φ}|u(x)|

andΨ(|v(x)|) are also bounded. Consequently, property (P2) ofXyields that

Φ

1

kuk_{Φ}|u(x)|

X <

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∞andkΨ(|v(x)|)k_{X} <∞which by using (2.2) gives:

u·v kukΦ

X

≤

Φ |u|

kukΦ

+ Ψ(|v|) X

≤

Φ |u|

kuk_{Φ}

X

+kΨ(|v|)k_{X}

<∞.

On the other hand, using theL-property ofX and (2.3), we get that for somea >0

u·v
kuk_{Φ}

X

=

Φ |u|

kuk_{Φ}

+ Ψ(|v|) X

≥a

Φ |u|

kukΦ

X

+kΨ(|v|)k_{X}

. (3.2)

Applying (3.1) for u

kuk_{Φ},v, we find that
max(ρ_{X}(v,Ψ),1)≥

u·v kukΦ

X

and therefore, by (3.2), we get that
max(ρ_{X}(v,Ψ),1)≥a

Φ |u|

kuk_{Φ}

X

+kΨ(|v|)k_{X}

.

Now, ifρ_{X}(v,Ψ) >1, then the above estimate gives

Φ |u|

kukΦ

X

≤ρ_{X}(v,Ψ)
1

a −1

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and ifρ_{X}(v,Ψ)≤1, then

Φ |u|

kuk_{Φ}

X

+ρ_{X}(v,Ψ)≤ 1
a.
In any case

Φ |u|

kukΦ

X

<∞

and the assertion is proved for boundedu. For generalu, we can follow the modified idea of [10, Lemma 3.7.2].

*Remark 2. In view of the above theorem, for*u ∈ X_{Φ}, there existsc > 0such that
cu ∈ Xe_{Φ}. In other words, the spaceX_{Φ} is the linear hull of the generalized Orlicz
classXe_{Φ}with the assumption onX that it satisfies theL-property.

We prove the following useful result:

* Proposition 3.3. Let*Φ

*be a Young function satisfying the*∆

_{2}

*-condition (with*T = 0

*if*µ(Ω) =∞) andX

*be a BFS satisfying the*L-property. ThenX

_{Φ}=Xe

_{Φ}

*.*

*Proof. Let*u∈X_{Φ},kuk_{Φ} 6= 0. By Theorem3.2, we have
w= 1

kukΦ

·u∈Xe_{Φ}.

SinceXe_{Φ}(Ω)is a linear set, we have

kuk_{Φ}·w=u∈Xe_{Φ},
i.e.,

XΦ ⊂XeΦ.

The reverse inclusion is obtained in view of Remark1and the assertion follows.

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Let Φ_{1} andΦ_{2} be two Young functions. We writeΦ_{2} ≺ Φ_{1} if there exists con-
stantsc >0,T ≥0such that

Φ_{2}(t)≤Φ_{1}(ct), t ≥T .
Now, we prove the following inclusion relation:

* Theorem 3.4. Let*X

*be a BFS satisfying the*L-property andΦ

_{1}

*,*Φ

_{2}

*be two Young*

*functions such that*Φ

_{2}≺Φ

_{1}

*and*µ(Ω) <∞. Then the inclusion

X_{Φ}_{1} ⊂X_{Φ}_{2}

*holds.*

*Proof. Since*Φ_{2} ≺Φ_{1}, there exists constants,c > 0,T ≥0such that

(3.3) Φ_{2}(t)≤Φ_{1}(ct), t ≥T .

Letu ∈X_{Φ}_{1}. Then in view of Theorem3.2, there existsk >0such thatku∈Xe_{Φ}_{1},
i.e.,ρ_{X}(ku; Φ_{1})<∞. Denote

Ω_{1} =

x∈Ω;|u(x)|< cT k

.

Then forx∈Ω\Ω1,|u(x)| ≥ ^{cT}_{k} , i.e.,
k

c|u(x)| ≥T

so that the inequalities (3.3) withtreplaced by ^{k}_{c}|u(x)|gives
Φ_{2}

k c|u(x)|

≤Φ_{1}(k|u(x)|)

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which implies that

Φ_{2}
k

c|u(x)|

X

=

Φ_{2}
k

c|u(x)|

χ_{Ω}_{1} + Φ_{2}
k

c|u(x)|

χΩ\Ω_{1}

X

≤

Φ_{2}
k

c|u(x)|

χ_{Ω}_{1}

X

+

Φ_{2}
k

c|u(x)|

χΩ\Ω1

X

≤Φ_{2}(T)kχ_{Ω}_{1}k_{X} +kΦ_{1}(k|u(x)|)χΩ\Ω_{1}k_{X}

= Φ2(T)kχΩ1kX +ρX(ku; Φ1)

<∞.

Consequently, ^{k}_{c}u ∈ Xe_{Φ}_{2} ⊂ X_{Φ}_{2}, i.e., ^{k}_{c}u ∈ X_{Φ}_{2}. But sinceX_{Φ}_{2} is in particular a
vector space we find thatu∈X_{Φ}_{2} and we are done.

The above theorem states that Φ_{2} ≺ Φ_{1} is a sufficient condition for the alge-
braic inclusionX_{Φ}_{1} ⊂ X_{Φ}_{2}. The next theorem proves that the condition, in fact, is
sufficient for the continuous imbeddingX_{Φ}_{1} ,→X_{Φ}_{2}.

* Theorem 3.5. Let*X

*be a BFS satisfying the*L-property andΦ

_{1}

*,*Φ

_{2}

*be two Young*

*functions such that*Φ

_{2}≺Φ

_{1}

*and*µ(Ω) <∞. Then the inequality

kuk_{Φ}_{2} ≤kkuk_{Φ}_{1}
*holds for some constant*k >0*and for all*u∈X_{Φ}_{1}*.*

*Proof. Let*Ψ1 and Ψ2 be the complementary functions respectively to Φ1 andΦ2.
ThenΦ_{2} ≺Φ_{1}implies thatΨ_{1} ≺Ψ_{2}, i.e., there exists constantsc_{1}, T_{1} >0such that

Ψ1(t)≤Ψ2(c1t) for t ≥T1

or equivalently

Ψ_{1}
t

c_{1}

≤Ψ_{2}(t) for t≥c_{1}T_{1}.

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Further, ift ≤c_{1}T_{1}, then the monotonicity ofΨgives
Ψ_{1}

t
c_{1}

≤Ψ_{1}(T_{1}).

The last two estimates give that for allt >0

(3.4) Ψ_{1}

t
c_{1}

≤Ψ_{1}(T_{1}) + Ψ_{2}(t).

By the property (P4) of BFS,kχ_{Ω}k_{X} <∞. Denoteα= (Ψ_{1}(T_{1})kχ_{Ω}k_{X} + 1)^{−1} and
k = ^{c}_{α}^{1}. Clearly0< α <1. We know that for a Young functionΦand0< β <1,

(3.5) Φ(βt)≤βΦ(t), t >0.

Now, letv ∈ XeΨ2 be such that ρX(v; Ψ2) ≤ 1. Then, using (3.5) forβ = α and
t= ^{|v(x)|}_{c}

1 and (3.4), we obtain that
ρ_{X} v

k; Ψ_{1}

=

Ψ_{1}

α|v(x)|

c1

X

≤α

Ψ_{1}

|v(x)|

c_{1}

X

≤αkΨ_{1}(T1) + Ψ2(|v(x)|)k_{X}

≤α(Ψ_{1}(T_{1})kχ_{Ω}k_{X} +kψ_{2}(|v(x)|)k_{X})

=α(Ψ_{1}(T_{1})kχ_{Ω}k_{X} +ρ_{X}(v; Ψ_{2}))

≤αα^{−1} = 1.

Thus we have shown thatρX(v; Ψ2) ≤1impliesρX v k; Ψ1

≤ 1and consequently

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using the definition of the generalized Orlicz norm, we obtain
kuk_{Φ}_{2} = sup

ρ(v;Ψ2)≤1

k(|u(x)v(x)|)k_{X}

=k sup

ρ(v;Ψ2)≤1

u(x)v(x) k

X

≤k sup

ρ(^{v}k;Ψ1)^{≤1}

u(x)v(x) k

X

=k sup

ρ(w;Ψ1)≤1

k|u(x)w(x)|k_{X}

=k· kuk_{Φ}_{1}
and the assertion is proved.

*Remark 3. If*Φ_{1}andΦ_{2}are equivalent Young functions (i.e.,Φ_{1} ≺Φ_{2}andΦ_{2} ≺Φ_{1})
then the normsk·k_{Φ}

1 andk·k_{Φ}

2 are equivalent.

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**4.** **Convergence**

Following the concepts in the Orlicz spaceL_{Φ}(Ω), we introduce the following defi-
nitions.

* Definition 4.1. A sequence*{un}

*of functions in*XΦ

*is said to converge to*u ∈ XΦ

*,*

*written*u

_{n}→u, if

n→∞lim ku_{n}−uk_{Φ} = 0.

* Definition 4.2. A sequence*{u

_{n}}

*of functions in*X

_{Φ}

*is said to converge in*Φ-mean

*to*u∈X

_{Φ}

*if*

n→∞lim ρ_{X}(u_{n}−u; Φ) = lim

n→∞kΦ(|u_{n}−u|)k_{X} = 0.

We proceed to prove that the two convergences above are equivalent. In the se- quel, the following remark will be used.

*Remark 4. Let*ΦandΨbe a pair of complementary Young functions. Then in view
of Young’s inequality (2.2), we obtain foru∈Xe_{Φ},v ∈Xe_{Ψ}

k|uv|k_{X} ≤ kΦ(|u|)k_{X} +kΨ(|v|)k_{X}

=ρ_{X}(u; Φ) +ρ_{X}(v; Ψ)
so that

kuk_{Φ} ≤ρ_{X}(u; Φ) + 1.
Now, we prove the following:

* Lemma 4.3. Let*Φ

*be a Young function satisfying the*∆

_{2}

*-condition (with*T = 0

*if*µ(Ω) =∞) andr

*be the number given by*

(4.1) r =

(2 *if* µ(Ω) =∞,

Φ(T)kχ_{Ω}k_{X} + 2 *if* µ(Ω)<∞.

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*If there exists an*m ∈N*such that*

(4.2) ρ_{X}(u; Φ)≤k^{−m},

*where*k *is the constant in the*∆_{2}*-condition, then*
kuk_{Φ} ≤2^{−m}r .

*Proof. Let*m∈Nbe fixed. Consider first the case whenµ(Ω)<∞and denote
Ω_{1} ={x∈Ω : 2^{m}|u(x)| ≤T}.

Then forx∈Ω_{1}, we get

(4.3) Φ(2^{m}|u(x)|)≤Φ(T)

and forx∈Ω\Ω_{1}, by repeated applications of the∆_{2}-condition, we obtain
(4.4) Φ(2^{m}|u(x)|)≤k^{m}Φ(|u(x)|).

Consequently, we have using (4.3) and (4.4)

kΦ(2^{m}|u(x)|)k_{X} =kΦ(2^{m}(|u(x)|))χ_{Ω}_{1} + Φ(2^{m}|u(x)|)χΩ\Ω1k_{X}

≤ kΦ(2^{m}(|u(x)|))χ_{Ω}_{1}k_{X} +kΦ(2^{m}|u(x)|)χΩ\Ω_{1}k_{X}

≤Φ(T)kχΩ1kX +k^{m}kΦ(|u(x)|)kX

≤Φ(T)kχ_{Ω}k_{X} +k^{m}ρ_{X}(u; Φ)

≤Φ(T)kχ_{Ω}k_{X} + 1

=r−1.

In the caseµ(Ω) =∞we takeΩ_{1} =φand then (4.4) directly gives
kΦ(2^{m}|u(x)|)kX ≤1≤r−1

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sincer = 2forµ(Ω) =∞. Thus in both cases we have
kΦ(2^{m}|u(x)|)k_{X} ≤r−1
which further, in view of Remark4gives

k2^{m}u(x)k_{Φ} ≤r
or

kuk_{Φ} ≤2^{−m}r
and we are done.

Let us recall the following result from [4]:

* Lemma 4.4. Let*u∈X

_{Φ}

*. Then*

ρ_{X}(u; Φ)≤ kuk^{0}_{Φ} *if* kuk^{0}_{Φ} ≤1
*and*

ρ_{X}(u; Φ)≥ kuk^{0}_{Φ} *if* kuk^{0}_{Φ} >1,

*where*kuk^{0}_{Φ} *denotes the Luxemburg type norm on the space*X_{Φ}*given by (2.5).*

Now, we are ready to prove the equivalence of the two convergence concepts defined earlier in this section.

* Theorem 4.5. Let*Φ

*be a Young function satisfying the*∆2

*-condition. Let*{un}

*be*

*a sequence of functions in*XΦ

*. Then*un

*converges to*u

*in*XΦ

*if and only if*un

*converges in*Φ-mean tou*in*X_{Φ}*.*

*Proof. First assume that* u_{n} converges in Φ-mean to u. We shall now prove that
u_{n} → u. Given ε > 0, we can choosem ∈ N such that ε > 2^{−m}r, wherer is as

**Generalized Orlicz Spaces**
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**vol. 10, iss. 2, art. 37, 2009**

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given by (4.1). Now, sinceu_{n}converges inΦ-mean tou, for thism, we can find an
M such that

ρ_{X}(u_{n}−u; Φ) ≤k^{−m} for n ≥M
which by Lemma4.3implies that

ku_{n}−uk_{Φ}≤2^{−m}r < ε for n≥M
and we get thatu_{n} →u.

Conversely, first note that the two norms k·k_{Φ} and k·k^{0}_{Φ} on the space X_{Φ} are
equivalent and assume, in particular, that the constants of equivalence arec_{1}, c_{2}, i.e.,
(2.6) holds.

Now, letun, u∈XΦso that

ku_{n}−uk_{Φ} ≤c_{1}.
Then (2.6) gives

ku_{n}−uk^{0}_{Φ} ≤1
which, in view of Lemma4.4and again (2.6), gives that

ρ_{X}(u_{n}−u; Φ)≤ ku_{n}−uk^{0}_{Φ}

≤ 1

c_{1}ku_{n}−uk_{Φ}.

The Φ-mean convergence now, immediately follows from the convergence in XΦ.
*Remark 5. The fact that the* Φ-mean convergence implies norm convergence does
not require the use of∆_{2}-conditions. It is required only in the reverse implication.

**Generalized Orlicz Spaces**
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**vol. 10, iss. 2, art. 37, 2009**

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**5.** **Separability**

*Remark 6. It is known, e.g., see [7, Theorem 3.13.1], that the Orlicz space*L_{Φ}(Ω)
is separable ifΦsatisfies the ∆_{2}-condition (withT = 0 ifµ(Ω) = 0). In order to
obtain the separability conditions for the generalized Orlicz spaceX_{Φ}, we can depict
the same proof with obvious modifications except at a point where the Lebesgue
dominated convergence theorem has been used.

In the framework of general BFS, the following version of the Lebesgue domi- nated convergence theorem is known, see e.g. [1, Proposition 3.6].

* Definition 5.1. A function*f

*in a Banach function space*X

*is said to have an abso-*

*lutely continuous norm in*X

*if*kf χ

_{E}

_{n}k

_{X}→0

*for every sequence*{E

_{n}}

^{∞}

_{n=1}

*satisfying*E

_{n}→φ µ-a.e.

* Proposition A. A function*f

*in a Banach function space*X

*has an absolutely con-*

*tinuous norm iff the following condition holds; whenever*f

_{n}{n = 1,2, . . .}

*and*g

*are*µ-measurable functions satisfying|f

_{n}| ≤ |f|

*for all*n

*and*f

_{n}→ g µ-a.e., then kf

_{n}−gk

_{X}→0.

Now, in view of Remark6and PropositionAwe have the following result.

* Theorem 5.2. Let*X

*be a BFS having an absolutely continuous norm and*Φ

*be a*

*Young function satisfying the*∆

_{2}

*-condition (with*T = 0

*if*µ(Ω) = 0). Then the

*generalized Orlicz space*X

_{Φ}

*is separable.*

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**vol. 10, iss. 2, art. 37, 2009**

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**References**

[1] C. BENNETT AND *R. SHARPLEY, Interpolation of Operators, Academic*
Press, London, 1988.

[2] P. JAIN, B. GUPTA AND D. VERMA, Mean inequalities in certain Banach
**function spaces, J. Math. Anal. Appl., 334(1) (2007), 358–367.**

[3] P. JAIN, B. GUPTA AND D. VERMA, Hardy inequalities in certain Banach function spaces, submitted.

[4] P. JAIN, L.E. PERSSONANDP. UPRETI, Inequalities and properties of some
* generalized Orlicz classes and spaces, Acta Math. Hungar., 117(1-2) (2007),*
161–174.

[5] M.A. KRASNOSEL’SKIIAND *J.B. RUTICKII, Convex Functions and Orlicz*
*Spaces, Noordhoff Ltd. (Groningen, 1961).*

[6] A.KUFNER, L.MALIGRANDAAND*L.E.PERSSON, The Hardy Inequailty -*
*About its History and Some Related Results, (Pilsen, 2007).*

[7] A. KUFNER, J. OLDRICHAND*F. SVATOPLUK, Function Spaces, Noordhoff*
Internatonal Publishing (Leydon, 1977).

[8] A. KUFNERAND*L.E. PERSSON, Weighted Inequalities of Hardy Type, World*
Scientific, 2003.

[9] W.A.J. LUXEMBURG, Banach Function Spaces, Ph.D. Thesis, Technische Hogeschoo te Delft (1955).

[10] L. MALIGRANDAANDL.E. PERSSON, Generalized duality of some Banach
**fucntion spaces, Proc. Konin Nederlands, Akad. Wet., 92 (1989), 323–338.**

**Generalized Orlicz Spaces**
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**vol. 10, iss. 2, art. 37, 2009**

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[11] L.E. PERSSON, Some elementary inequalities in connection withX^{p}-spaces,
*In: Constructive Theory of Functions, (1987), 367–376.*

*[12] L.E. PERSSON, On some generalized Orlicz classes and spaces, Research*
*Report 1988-3, Department of Mathematics, Luleå University of Technology,*
(1988).

[13] M.M. RAOAND*Z.D. REN, Theory of Orlicz spaces, Marcel Dekker Inc. (New*
York, Basel, Hong Kong, 1991).