Some Lacunary Sequence Spaces of Invariant Means Defined by Musielak Orlicz Functions on 2 Norm Space

(1)

http://dx.doi.org/10.4236/am.2014.516248

Some Lacunary Sequence Spaces of

Invariant Means Defined by Musielak-Orlicz

Functions on 2-Norm Space

Mohammad Aiyub

Department of Mathematics, University of Bahrain, Sakhir, Bahrain Email: maiyub2002@gmail.com

Received 11 July 2014; revised 15 August 2014; accepted 25 August 2014

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of sequences of Musielak-Orlicz functions, invariant means and lacunary convergence on 2-norm space. We establish some inclusion relations between these spaces under some conditions.

Keywords

Invariant Means, Musielak-Orlicz Functions, 2-Norm Space, Lacunary Sequence

1. Introduction

Let ω be the set of all sequences of real numbers _∞,c and c0 be respectively the Banach spaces of bounded, convergent and null sequences x=

( )

xk with

( )

xk ∈ or  the usual norm x =supk xk ,

where k∈ = 1, 2, 3,, the positive integers.

The idea of difference sequence spaces was first introduced by Kizmaz [1] and then the concept was generalized by Et and Çolak [2]. Later on Et and Esi [3] extended the difference sequence spaces to the sequence spaces:

( )

m

{

( )

:

( )

m

}

,

v k v

X ∆ = x= x ∆ x ∈X

for X =_∞, c and c0, where v=

( )

vk be any fixed sequence of non zero complex numbers and

(

) (

1 1

)

1 .

m m m

vxk v xk v xk

− −

+

∆ = ∆ − ∆

(2)

( )

0

1 , for all .

m i m

v k k i k i i

m

x v x k

i + +

=

 

∆ = − _{ } ∈

 

∑



The sequence spaces ∆mv

( )

∞ ,

( )

m v c

∆ and

( )

0

m v c

∆ are Banach spaces normed by

1

.

m

m i i v i

x_∆ v x x_∞

=

∑

+ ∆

The concept of 2-normed space was initially introduces by Gahler [4] as an interesting linear generalization of normed linear space which was subsequently studied by many others [5] [6]). Recently a lot of activities have started to study summability, sequence spaces and related topics in these linear spaces [7] [8]).

Let X be a real vector space of dimension d, where 2≤ < ∞d . A 2-norme on X is a function

.,. :X×X → which satisfies:

1) x y, =0 if and only if x and y are linearly dependent, 2) x y, = y x, ,

3) αx y, = α x y, ,α∈, 4) x y, + ≤z x y, + x z, .

The pair

(

X, .,.

)

is called a 2-normed space. As an example of a 2-normed space we may take X =2

being equiped with the 2-norm x y, = the area os paralelogram spaned by the vectors x and y, which may be given explicitly by the formula

11 22 1 2

21 22 ,

E

x x

x x abs

x x

 

=  

 .

Then clearly

(

X, .,.

)

is 2-normed space. Recall that

(

X, .,.

)

is a 2-Banach space if every cauchy sequence in X is convergent to some x∈X .

Let σ be a mapping of the positive integers into itself. A continuous linear functional φ on ∞ is said to be an invariant mean or σ-mean if and only if

1) φ

( )

x ≥0, when the sequence x=

( )

x_n has, xn≥0 for all n, 2) φ

( )

e =1,e=

(

1,1,1,

)

,

3) φ

( )

x_σ_{( )}_n =φ

( )

x for all x∈∞.

If x=

( )

xk , where Tx=

( )

Txk =

( )

xσ( )k . It can be shown that

( )

{

: lim _kn , uniformly in

}

k

V_σ = x∈_∞ t x =l n

lim ,

l= −σ x where

( )

1( ) 2( ) ( )

1

n k n

n _n

kn

x x x x

t x

k

σ σ σ

+ + + +

=

+ 

[9].

In the case σ is the translation mapping n→ +n 1, σ -mean is often called a Banach limit and V_σ the set of bounded sequences of all whose invariant means are equal is the set of almost convergent sequence [10].

By Lacunary sequence θ =

( )

kr ,r=0,1, 2, where k0=0 we mean an increasing sequence of non neg- ative integers hr=

(

kr−kr−1

)

→ ∞

(

r→ ∞

)

. The intervals determined by θ are denoted by Ir =

[

kr−1−kr

]

and the ratio 1

r r

k

k−

will be denoted by qr. The space of lacunary strongly convergent sequence Nθ was de-

fined by Freedman et al. [11] as follows:

( )

1

: lim 0 for some .

r

i k

r _{k I}

r

N x x x l

h

θ _→∞

=

 

 

= = − = 

 



∑



An Orlicz function is a function M: 0,

[

∞ →

)

[

0,∞

)

which is continuous, non-decreasing and convex with

( )

0 0, 0

( )

(3)

It is well known that if M is convex function and M

( )

0 =0 then M

( )

λx ≤λM x

( )

, for all λ with

0≤ ≤λ 1.

Lindenstrauss and Tzafriri [12] used the idea of Orlicz function and defined the sequence space which was called an Orlicz sequence space M such as

( )

1

: k , for some 0

M k

k

x

x x M ρ

ρ ∞

=

   

 

=_ = _ _< ∞ > _

   



∑





which was a Banach space with the norm

1

inf 0 : k 1

k

x

x ρ M

ρ ∞

=

   

 

=  >  ≤ 

   



∑



which was called an Orlicz sequence space. The M was closely related to the space p which was an Orlicz

sequence space with M t

( )

= tp, for 1≤ < ∞p . Later the Orlicz sequence spaces were investigated by Prashar and Choudhry [13], Maddox [14], Tripathy et al. [15]-[17] and many others.

A sequence of function M =

( )

Mk of Orlicz function is called a Musielak-Orlicz function [18] [19]. Also a

Musielak-Orlicz function Φ = Φ

( )

k is called complementary function of a Musielak-Orlicz function M if

( )

sup

{

( )

: 0 , for

}

1, 2, 3, .

k t t s Mk s s k

Φ = − ≥ = 

For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space lM and its subspaces M

are defined as follow:

( )

{

: , for some 0

}

M k M

l = x=x ∈ω I cx < ∞ c>

( )

{

: , for all 0

}

M = x=xk∈ω IM cx < ∞ c>



where IM is a convex modular defined by

( )

1

, .

M k k k M

k

I x M x x x l

∞

=

∑

= ∈

We consider lM equipped with the Luxemburg norm

inf 0 : M 1

x

x k I

k

   

= _ > _{ }≤ _

 

 

or equipped with the Orlicz norm

( )

(

)

0 1

inf 1 M : 0 .

x I kx k

k

 

=  + > 

 

The main purpose of this paper is to introduce the following sequence spaces and examine some properties of the resulting sequence spaces. Let M =

( )

Mk be a Musielak-Orlicz function,

(

X, .,.

)

is called a 2-normed

space. Let p=

( )

p_k be any sequences of positive real numbers, for all k∈ and u=

( )

u_k such that

(

)

0 1, 2, 3,

k

u ≠ k=  . Let s be any real number such that s≥0. By ω

(

2−X

)

we denote the space of all sequences defined over

(

X, .,.

)

. Then we define the following sequence spaces:

( )

(

)

(

)

, , , , , , .,.

1

2 :sup , 0, 0

k

r

m v

p m

kn v k s

k k k

r n rk I

M p u s

t x

x x X k u M z s

h

θ

σ ω

ω ρ

ρ ∞

−

∈

  ∆

 

 _ _ _∆ __ 

 _ _ __ 

=_ = ∈ − < ∞ > ≥ _

  

 _ _ __ 

 

(4)

( )

(

)

(

)

, , , , , .,.

1

2 ::lim , 0 for some , 0, 0

k

r

m v

p m

kn v k s

k k k

r _{k I}

r M p u s

t x

x x X k u M z l s

h

θ

σ ω

ω ρ

ρ

−

∈

  ∆

 

 _ _ _∆ __ 

 _ _ __ 

=_ = ∈ − = > ≥ _

  

 _ _ __ 

 

∑

( )

(

)

(

)

0 , , , , , .,.

1

2 ::lim , 0 0, 0

k

r

m v

p m

kn v k s

k k k

r r k I

M p u s

t x

x x X k u M z s

h

θ

σ ω

ω ρ

ρ −

∈

  ∆

 

 _ _ _∆ __ 

 _ _ __ 

=_ = ∈ − = > ≥ _

  

 _ _ __ 

 

∑

Definition 1. A sequence space E is said to be solid or normal if

(

αkxk

)

∈E whenever

( )

xk ∈E and for

all sequences of scalar

( )

αk with αk ≤1 [20].

Definition 2. A sequence space E is said to be monotone if it contains the canonical pre-images of all its

steps spaces, [20].

Definition 3. If X is a Banach space normed by . , then ∆m

( )

X is also Banach space normed by

( )

1

.

m

m k

k

x_∆ x f x

=

∑

+ ∆

Remark 1. The following inequality will be used throughout the paper. Let p=

( )

pk be a positive sequence

of real numbers with 0< pk ≤suppk =G,

(

)

1 max 1, 2G

D= − . Then for all a bk, k∈ for all k∈. We

have

(

)

.

k k k

p p p

k k k k

a +b ≤D a + b (1)

2. Main Results

Theorem 1. Let M =

( )

Mk be a Musielak-Orlicz function, p=

( )

pk be a bounded sequence of positive real

number and θ =

( )

kr be a lacunary sequence. Then , , , , , .,.

( )

,

m v M p u s

θ

σ

ω ∞

  ∆

  , , , , , .,.

( )

m

v

M p u s

θ

σ ω

  ∆

 

and , , , , , .,. 0

( )

m v M p u s

θ

σ ω

  ∆

  are linear spaces over the field of complex numbers.

Proof 1. Let x

( )

x_k ,y

( )

y_k θ,M p u s, , , , .,. 0

( )

m_v

σ ω

 

= = ∈_ _ ∆ and α β, ∈C . In order to prove the result we

need to find some ρ3 such that,

(

)

(

)

3 1

lim , 0, uniformly in .

k

r

p m

nk v k k

s

k k

r k I r

t x y

u k M z n

h

α β

ρ

−

→∞ _∈

  _∆ ₊ 

   ₌

 _ _

  

 

∑

Since

( ) ( )

, , , , , , .,. 0

( )

m

k k v

x y θ M p u s

σ ω

 

∈_ _ ∆ , there exist positive ρ ρ1, 2 such that

(

)

1

lim , 0 uniformly in

k

r

p m

kn v k s

k k r

k I r

t x

u k M z n

h ρ

−

→∞ _∈

  _∆ 

   ₌

  

 

 

∑

and

(

)

1

lim , 0 uniformly in .

k

r

p m

kn v k s

k k r

k I r

t x

u k M z n

h ρ

−

→∞ _∈

  _∆ 

   ₌

  

 

 

∑

(5)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

3 3 3 1 2 1 , 1 , , 1 , , , k r k r k r r p m

nk v k k s k k k I r p m m

nk v k nk v k s k k k I r p m m

nk v k nk v k s

k k k I

r

m nk v k s

k k k I

r

t x y

u k M z

h

t x t y

u k M z z

h

t x t y

u k M z z

h

t x

D

u k M z

h α β ρ α β ρ ρ ρ ρ ρ − ∈ − ∈ − ∈ − ∈   _∆ ₊              _∆ _∆     ≤ +          _∆ _∆     ≤ +         _∆  ≤ 

∑

(

)

,

0, as , uniformly in .

k k

r

p p

m nk v k s k k k I r t y D

u k M z

h r n ρ − ∈     _∆    ₊                 → → ∞

∑

So that

(

x_k

) (

y_k

)

θ,M p u s, , , , .,. 0

( )

_vm .

σ

α + β ∈_ω _ ∆ This completes the proof. Similarly, we can prove that

( )

, , , , , .,. m v

M p u s

θ

σ ω

  ∆

  and θ,M p u s, , , , .,.

( )

m_v

σ

ω ∞

  ∆

  are linear spaces.

Theorem 2. Let M=

( )

Mk be a Musielak-Orlicz function, p=

( )

pk be a bounded sequence of positive

real number and θ =

( )

kr be a lacunary sequence. Then

( )

0 , , , , , .,. m

v M p u s

θ

σ ω

  ∆

  is a topological linear

space totalparanormed by

( )

(

)

1

inf : , 1 for some , 1, 2,

k r r H p m m

nk v k

p H s

k k k

k rk I

t x

g x x u k M z r

h ρ ρ ρ − ∆ = ∈  _ _ _ __ _  ∆  _ _   _ _ __  = + _ _ _ ≤ = _     _ _ _ __ _     

∑



Proof 2. Clearly g_∆

( )

x =g_∆

( )

−x . Since M_k

( )

0 =0, for all k∈. we get g∆

( )

θ =0, for x=θ. Let

( )

k

x= x , y

( )

yk ,M p u s, , , , .,. 0

( )

mv

θ

σ ω

 

= ∈_ _ ∆ and let us choose ρ >1 0 and ρ >2 0 such that

( )

(

)

1

sup , 1 1, 2, 3,

k

r

p m

nk v k s

r k k

r k I

t x

h u k M z r

ρ − − ∈   _∆     _≤ ₌       

∑

 and

( )

(

)

1 2

sup , 1 1, 2, 3,

k

r

p m

nk v k s

r k k

r k I

t y

h u k M z r

ρ − − ∈   _∆     _≤ ₌       

∑



Let ρ ρ ρ= 1+ 2, then we have

(

)

(

)

( )

(

)

( )

(

)

1 1 1

1 2 1

1 1

1 2 2

sup ,

sup , 1.

k r k r k r p m

nk v k k

s

r k k

r k I

p m

nk v k

s

r k k

r k I

p m

nk v k

s

r k k

r k I

t x y

h u k M z

t x

h u k M z

t y

h u k M z

ρ

ρ ρ ρ

ρ

ρ ρ ρ

(6)

Since ρ>0, we have

(

)

(

)

( )

1 1 1 1 1 1 1

inf : , 1 for some 0, 1, 2,

1

inf : , 1 for some

k r r k r r H p m m

nk v k k

p H s

k k k k

k rk I

H p m

m

nk v k

p H s

k k k

k rk I

g x y

t x y

x y u k M z r

h

t x

x u k M z

h ρ ρ ρ ρ ρ ∆ − = ∈ − = ∈ +  _ _ _ _   ∆ +   _ _ _ _ 

= + +  _ _ _ _ ≤ > = 

     _    _     _ _ _∆ __   _ _  ≤ + _ _ _ _ ≤    _ _ __   

∑



( )

1 1 2 2 1 2

0, 1, 2,

1

inf : , 1 for some 0, 1, 2, .

k r r H p m m

nk v k

p H s

k k k

k rk I

r

t y

y u k M z r

h ρ ρ ρ ρ − = ∈  

 _> ₌ 

       _ _ _ __ _  ∆  _ _ _ _ 

+ +  _ _ _ _ ≤ > = 

     _    _   

∑

 

(

)

( )

.

g_∆ x+y ≤g_∆ x +g_∆ y

Finally, we prove that the scalar multiplication is continuous. Let λ be a given non zero scalar in . Then the continuity of the product follows from the following expression.

( )

(

)

(

)

( )

1 1 1 1 1

inf : , 1 for some 0, 1, 2,

1

inf : , 1 for some 0

k r r k r r H p m m

nk v k

p H s

k k k

k r k I

H p m

m _p _H

nk v k s

k k k

k r k I

t x

g x x u k M z r

h

t x

x u k M z

h

λ

λ λ ρ ρ

ρ

λ λ ζ ζ

ζ − ∆ = ∈ − = ∈  _ _ _ _   ∆   _ _ _ _ 

= +  _ _ _ _ ≤ > = 

     _    _     _ _ _∆ __   _ _ 

= + _   _ ≤ >

     _ _ __   

∑



, r 1, 2,

   ₌        

where ζ ρ 0. λ

= > Since λ pr ≤max 1,

(

λ

)

suppr ,

( )

(

)

(

)

sup 1 max 1, 1

inf : , 1, for some 0, 1, 2, .

r k r r p H p m

nk v k

p H s

k k k I

r

g x

t x

u k M z r

h λ λ ρ ρ ρ ∆ − ∈ =  _ _ _ _   _∆   _ _   _ _ __ 

+  _ _ _ __ _ ≤ > = 

  _ _ __  

 

 

 

∑



This completes the proof of this theorem.

( )

M_k be a Musielak-Orlicz function, p=

( )

p_k be a bounded sequence of positive

real number and θ=

( )

k_r be a lacunary sequence. Then

( )

0

( )

,M p u s, , , , .,. m_v ,M p u s, , , , .,. _vm ,M p u s, , , , .,. m_v .

θ θ θ

σ σ σ

ω ∞ ω ω

  ∆ ⊂  ∆ ⊂  ∆

     

Proof 3. The inclusion θ,M p u s, , , , .,. 0

( )

m_v θ,M p u s, , , , .,.

( )

m_v

σ σ

ω ω

_ _ ∆ ⊂_ _ ∆ is obvious. Let

( )

, , , , , .,. m

k v

x ∈_ωθ M p u s __σ ∆ . Then there exists some positive number ρ1 such that

(

)

1 1

lim , 0

k

r

p m

nk v k s

k k r

k I r

t x le

u k M z

h ρ − →∞ _∈   _∆ ₋     _→       

∑

as r→ ∞, uniformly in n. Define ρ=2ρ1. Since Mk is non decreasing and convex for all k∈, we

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(

)

(

)

(

)

1 1 1 1 1 , , , ,

max 1, ,

k r k r k r k r p m

nk v k s k k k I r p m

nk v k s k k k I r p k k I r p m

nk v k k

k I r

t x

u k M z

h

t x le

D

u k M z

h

D le

M z

h

t x le

D

M z

h

le

D M z

ρ ρ ρ ρ ρ − ∈ − ∈ ∈ ∈   _∆              _∆ ₋     ≤           +  _ _        _∆ ₋     ≤           +  _ _     

∑

G            

where G=supk

( )

pk , max 1, 2

(

1

)

G

D= − by (1). Thus xk ,M p u s, , , , .,.

( )

vm

θ

σ ω

 

∈_ _ ∆ .

( )

Mk be a Musielak-Orlicz functions. If sup

( )

k

p

kMk z  < ∞ for all z>0, then

( )

,M p u s, , , , .,. m_v ,M p u s, , , , .,. m_v .

θ θ

σ σ

ω ω ∞

  ∆ ⊂  ∆

   

Proof 4. Let xk ,M p u s, , , , .,.

( )

mv

θ

σ ω

 

∈_ _ ∆ by using (1), we have

(

)

(

)

1 , , , . k r k r k r p m

nk v k s k k k I r p m

nk v k k k I r p k k I r t x

u k M z

h

t x le

D M z h le D M z h ρ ρ ρ − ∈ ∈ ∈   _∆              _∆ ₋     ≤           +  _ _  _ _  

∑

Since sup_k_M z

( )

_pk < ∞, we can take the sup_k_M z

( )

_pk =K . Hence we can get

( )

, , , , , .,. m

k v

x ∈_ωθ M p u s __σ ∆ .

This complete the proof.

Theorem 5. Let m≥1 be fixed integer. Then the following statements are equivalent: 1) _, _{, , , , .,.}

(

m 1

)

_, _{, , , , .,.}

( )

m

v v

M p u s M p u s

θ θ

σ σ

ω ∞ − ω ∞

  ∆ ⊂  ∆

    ,

2)

( )

1

( )

, , , , , .,. m , , , , , .,. m

v v

M p u s M p u s

θ θ

σ σ

ω − ω

  ∆ ⊂  ∆

    ,

3) _, _{, , , , .,.} 0

(

m1

)

_, _{, , , , .,.} 0

( )

m _.

v v

M p u s M p u s

θ θ

σ σ

ω − ω

  ∆ ⊂  ∆

   

Proof 5. Let x_k∈_ωθ,M p u s, , , , .,.__σ0

( )

∆_vm−1 . Then there exist ρ>0 such that

(

)

1

lim , 0.

k

r

p m

nk v k s k k r k I r t x

u k M z

h ρ − →∞ _∈   _∆     _→       

∑

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(

)

(

)

(

)

(

)

1 1 1 1 1 1 1 , 2 1 , 2 1 1 , , 2 2 k r k r k r r p m

nk v k s k k k I r p m m

nk v k k

s k k k I r p m m

nk v k nk v k

s s

k k k k

k I k I

r r

t x

u k M z

h

t x x

u k M z

h

t x t x

u k M z u k M z

h h ρ ρ ρ ρ − ∈ − − + − ∈ − − + − − ∈ ∈   _∆              _∆ _{− ∆}     = _ _ __          _∆    _∆        ≤ +              

∑

(

1

)

(

1

)

1 , , . k k k r r p p p m m

nk v k nk v k

s s

k k k k

k I k I

r r

t x t x

D D

u k M z u k M z

h ρ h ρ

− − + − − ∈ ∈     _∆    _∆        ≤ +              

∑

Taking limr→∞, we have

(

)

1 , 0, k r p m

nk v k s

k k k I

r

t x

u k M z

h ρ − ∈   _∆     ₌       

∑

i.e. 0

( )

1

, , , , , .,. m .

k v

x ∈_ωθ M p u s __σ ∆ − The rest of these cases can be proved in similar way.

( )

Mk and T =

( )

Tk be two Musielak-Orlicz functions. Then we have

1) θ,M p u s, , , , .,.

( )

_vm θ, , , , , .,.T p u s

( )

_vm θ,M T p u s, , , , .,.

( )

m_v .

σ σ σ

ω ∞ ω ∞ ω ∞

  ∆   ∆ ⊂ +  ∆

     

2) ,M p u s, , , , .,.

( )

vm , , , , , .,.T p u s

( )

mv ,M T p u s, , , , .,.

( )

mv .

θ θ θ

σ σ σ

ω ω ω

  ∆   ∆ ⊂ +  ∆

     

3) θ,M p u s, , , , .,. 0

( )

_vm θ, , , , , .,.T p u s 0

( )

_vm θ,M T p u s, , , , .,. 0

( )

m_v .

σ σ σ

ω ω ω

  ∆   ∆ ⊂ +  ∆

     

Proof 6. Let , , , , , .,.

( )

m , , , , , .,.

( )

m .

k v v

x θ M p u s θ T p u s

σ σ

ω ∞ ω ∞

   

∈_ _ ∆ _ _ ∆ Then

(

)

, 1 sup , k r p m

nk v k s

k k r n rk I

t x

u k M z

h ρ

−

∈

  _∆ 

   _{< ∞}

      

∑

and

(

)

, 1 sup , k r p m

nk v k s

k k r n r k I

t x

u k T z

h ρ

−

∈

  _∆ 

   _{< ∞}

  

 

 

∑

uniformly in n. We have

(

)

(

)

,

(

)

,

(

)

,

k k k

p p p

m m m

nk v k nk v k nk v k

k k k k

t x t x t x

M T z D M z D T z

ρ ρ ρ

  _∆    _∆    _∆ 

 ₊   _≤    ₊   

        

     

     

by (1). Applying r

k I∈

∑

and multiplying by k,1

r

u

h and

s

k− both side of this inequality, we get

(

)

(

)

(

)

(

)

1 , , , k r k k r r p m

nk v k s

k k k k I

r

p p

m m

nk v k nk v k

s s

k k k k

k I k I

r r

t x

u k M T z

h

t x t x

D D

u k M z u k T z

(9)

uniformly in n. This completes the proof 2) and 3) can be proved similar to 1).

Theorem 7. 1) The sequence spaces θ,M p u s, , , , .,.

σ

ω ∞

 

  and θ,M p u s, , , , .,. 0

σ ω

 

  are solid and hence

they are monotone.

2) The space θ,M p u s, , , , .,. σ ω

 

  is not monotone and neither solid nor perfect.

Proof 7. We give the proof for θ,M p u s, , , , .,. 0

σ ω

 

  . Let x_k θ,M p u s, , , , .,. 0

σ ω

 

∈   and

( )

αk be a

sequence of scalars such that α_k ≤1 for all k∈. Then we have

(

)

( )

1 1

, , 0

k k

r r

p p

nk k k nk k

s s

k k k k

k I k I

r r

t x t x

u k M z u k M z

h h

α

ρ ρ

− −

∈ ∈

     

≤ →

 _ _  _ _

 _ _  _ _

   

∑

(

r→ ∞

)

, uniformly in n. Hence

(

_kx_k

)

θ,M p u s, , , , .,. 0

σ

α _{∈ }ω _ for all sequence of scalars

( )

α_k with

1

k

α ≤ for all k∈, whenever x_k θ,M p u s, , , , .,. 0

σ ω

 

∈   . The spaces are monotone follows from the remark (1).

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