Modular Spaces Topology

Modular Spaces Topology

Ahmed Hajji

Laboratory of Mathematics, Computing and Application, Department of Mathematics, Faculty of Sciences, Mohammed V-Agdal University, Rabat, Morocco

Email: [email protected]

Received April 29, 2013; revised May 29, 2013; accepted June 7, 2013

Copyright © 2013 Ahmed Hajji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In this paper, we present and discuss the topology of modular spaces using the filter base and we then characterize closed subsets as well as its regularity.

Keywords: Topology of Modular Spaces; ∆2-Condition; Filter Base

1. Introduction

In the theory of the modular spaces X_, the notion of

∆2-condition depends on the convergence of the

se-quences in modular space X_. More precisely, it reads:

for any sequence

 

x_{n n} in X_, if m

 

2 n 0 nli x  , we have lim

 

2x_n 0

n  . This condition has been used to study the topology of modular spaces, see J. Musielak [1], and to establish some fixed point theorems in modu-lar spaces, see [2-7]. Some fixed point theorems without ∆2-condition can be found in [8,9].

In this paper, we present a new equivalent form for the ∆2-condition in the modular spaces X which is used to show that the corresponding topology is separate and to establish some associated topological properties, includ-ing the characterization of the -closed subsets as well as its regularity. The present work is an improved Eng-lish version of a pervious preprint in French [10].

2. Preliminaries

We begin by recalling some definitions.

Definition 2.1 Let X be an arbitrary vector space over

K  or .

1) A functional :X 



0,



0

x

is called modular if

 

x 0

  implies .

a) 

 

 x 

 

x for any xX when K, and

b)

 

eit

 

x x

  for any real t when K.

c)  



xy





 

x 

 

y for  , 0 and

1

   .

2) If we replace c) by the following



x y



 

x

 

     y for  , 0 and

1

   , then the modular  is called convex. 3) For given modular  in X, the

 



X   x 0 as 0



X_ x  is called a modular

space.

4) a) If  is a modular in X, then

inf 0, x

x u u

u

 

     _  _{ } _

 

 

is a F-norm.

b) Let  be a convex modular, then

inf 0, 1x

x u

u

 

     _  _{ } _

 

 

is called the Luxemburg norm.

3. Topology

τ

in Modular Spaces

In this section, we introduce the property 0 for a

mod-ular  , which will be used to show that the corre-sponding topology, noted by  , on modular space X is separate, and to characterize their closed subsets.

We begin with the following

Proposition 3.1 Consider the family

 



B 0,r r 0



  , where



 

0,



 



B_ r  xX_  x r .

Then

1) The family  is a filter base.

2) Any element of  is balanced and absorbing. Furthermore, if  is convex, then any element of is convex.



Proof.

a)   because any B_

 

0,r  .

b) Let B



0,r1



and B



0,r2



zB_

 be in and set . Then, for any we have





r r1, 2



inf

r



0,r





2

 

 

12

z r r

z r r

 

  

 _{ } 

and therefore zB



0,r1



B



0,r . That is

 

0,



0,1





0, 2



.

B_ r B_ r B_ r

Hence  is a filter base for the existence of

 

0,

B_ r  such that

 

0,



0, 1





0, 2

B_ r B_ r B_ r



.

2) Let B

 

0,r .

 

0,

B r

a)  is balanced. Indeed, for given   ei with   and   1, and given xB

 

0,r ,

we have

 



ei



 

 

_<

x x x x

       r.

This means that xB_

 

0,r .

b) B_



0,r



is absorbing. Indeed, for given xX_

we have

 

. Whence, for all there 0 x 0

   

lim r> 0

exists  > 0, such that 0 < <  and  

 

x <r .

Hence, there exists > 0 such that xB_

 

0,r .

This shows that B_

 

0,r is absorbing.

Now, assume that  is in addition convex and let . For given

 

0,

B_ r  x y, B_



0,r



and  

 

0,1 ,

we have







x 1 y



  

x 1

  

y < ,

        r



, . then



1





0

x y B r

   

Thence B



0,r



is convex.

Definition 3.1 We say that  satisfies the property 0

 if for all > 0, there exist L> 0 and  > 0 such that 

   

y  x  for every x, y satisfying 

 

x L and 



xy



< .

Theorem 3.1 Assume that the modular  satisfies the property 0. Then X is a separate topological

vector space.

Proof. In Proposition 3.1, we have seen that the family

is a filter base, and furthermore any element of is balanced and absorbing. On the other hand, for any

, there exists 







0,

B r



0> 0 such that



0, 0





0, 0



 

0, .

B_  B_  B_ r

In fact, let  ; r> > 0 . Since  satisfies the property 0, there are L> 0 and  > 0, such that for

 

x <L

 and 



xy



< we have

 

y

 

x <

  . Thus, if we set





0 inf r , ,L ,

   

we see that for z  x y B



0,0



B



0,0



with

 

 

00.

x y

       _ 

We obtain y  z x B



0,0



. This implies



z x



0

    and 

 

x 0L. Thence

 

z

 

x 0 r .

            r This infers that zB_

 

0,r , and so



0, 0





0, 0



 

0,

B_  B_  B_ r .

Hence the family is a fundamental system of neighborhoods of zero, then the unique topology defined by in



 X is given by





 

, if ,

then such that ,

G G X x G

V x V G



    

    



 

so that X_ is a topological vector space.

To show that



X_,



is separate, let x, y in X_

such that x y and assume that for any Vx

neighbor-hood of x and Vy neighborhood of y we have VxVy .

So that one can consider

1 1

0, 0,

z x B y B

n n

 

      _  _{  }  _{ }

      

 

for certain *. Then

n









1

1_.

x z

n

y z

n





 _{ } 



 _{ } 

Since  satisfies the property 0, then there exist

for any  0, two reals L0 and  0, such that

 

 

< 2

y x 

  for every x, y satisfying 

 

x <L and 



yx



< . Now, set Y  y x and X  z x

and note that we have

 













1

1_.

X x z

n

Y X y z

n

 

 

 _{  } 



 _{   } 

It follows that for any n such that

1 _{inf , ,}

2

L n

 

  _

 



, we have

 









.

2 2 2

Y y x z x   

        

This infers that 



yx



 , for arbitrary 0. Thus, 



xy



0 and then x = y, a contradiction since

by hypothesis xy. Therefore there exist

x y

V V  .

τ Convergence and Characterization of τ-Closed Subsets of Xρ

We begin by recalling some needed definitions of the

 -convergence and the  -closed subsets of the the modular space X_ (see for examples [2-8]).

Definition 3.2 Let X_ be a modular space.

1) A sequence

 

x_{n n} in X_ is said to be -con-

vergent to x, denoted by xnx



, if as

.



xn x



0

   n 

2) A subset B of X_ is said to be -closed if for any sequence

 

x_{n n} B, such that x_nx



, we have

xB. We denote by B the closure of in the

sense of

B

.

3) A modular  is said to be satisfying the Fatou property, if 



xy



liminf



xnyn



as xn x





and yn y. 



In this section, we define the -convergence, the  - closed subsets of X_, and we show that the topology

defined by -closed in the definition before, noted by

1

 , and the topology  are the same topology.

The naturel convergence in the sense of the topology

 and  -closed subsets of X_ given by the follow-

ing definitions.

are

Definition 3.3 A sequence

 

x_{n n} in X_ is said to be convergent to x in the sense of the topology  (or simply  -convergent) if for any > 0 there exists

such that 0

N  xn x B

 

0, whenever 0.

Note that the property 0

>

n N

 is a necessary condition to show the uniqueness of the limit when exists. Thus, the

-convergence need the property 0 and it is easy to

see that -convergence and -convergence are equiva-lent.

Definition 3.4 Let  be a modular satisfying the property 0. A subset B of X is said to be - losed if

and only if the complimentary of B in c X_, noted by

B X

C _, is an element of  .

The following lemma shows that the property 0

makes sense in the theory of modular spaces.

Lemma 3.1 Let  be a modular and X_ be a modular space. Then  satisfies the -condition if and only if

2

  satisfies the property 0.

Proof. To prove “if”, let

 

x_{n n} be a sequence in X_ such that as . This implies

that for all

 

0

n

x

 

> 0

n 

 , there exists such that for any

we have 0

n

0 >

n n

 

inf , , .

2

n

x L 

  _  _

 

Now, take Xn xn and , for any . It

follows

2

n

Y  x_n nn₀

 

 





inf , , .

2

n n n n

X x Y X L 

     _  _

 

This yields

 

 

2

 

2

n n n

Y x  x

     when- ever nn0. Whence, the sequence





 

2xn



_n tends

to zero as n goes to , and therefore  satisfies the 2

 -condition.

For “only if”, let  be a modular satisfying the

2

 -condition, and suppose that there exists > 0 such that for any L> 0 and for any  > 0, there exist

,

x yX_ satisfying 

 

x < ,L <



xy



 and

 

y

 

x

   . In particular, for L 1 n



  there exist x yn, nX such that

 





 

 

1_{, and}1

,

n n n

n n

x y x

n n

y x

 

  

  

 

which implies 

 

x_n 0 and as . However, we have





0 n n

y x

  

n 

 















2



 

2

n n n n

n n n

y y x x

.

x y x

 

 

     

Now, since  satisfies the ∆2-condition, then

 

yn 0

  as n . It follows that

 

yn

 

xn 0 as n ,

    

which contradicts the fact that 

 

yn 

 

xn  > 0 for any n. Finally, for all > 0 , there are

and

> 0

L

> 0

 such that if 

 

x < and 



yx



< , we have 

 

y 

 

x <. This completes the proof of Lemma 3.1.

In the following theorem, we show that the -topol- ogy and the 1-topology are the same.

Theorem 3.2 Let  be a modular satisfying the ∆2- condition and FX_, then F is -closed if and only if F is -closed.

The following result is needed to show Theorem 3.2. Proposition 3.2 Let  be a modular satisfying the

∆2-condition and F a  -closed subset of X_. Then

 

0, , .

x   F  B_ x F  Proof. For xX, we have

 

 

 

 

, is an open set of the -topology

0, 0, ,

> 0, such that , . F F

X X

F X

x F x C C

B x B B x

B x F

 

C _

  





  

 

  

     

    

Finally,

 

> 0, , .

x  F  B_ x F  

Proof of Theorem 3.2. Let F be  -closed and n be a sequence in

 

xn F such that xn x



 . Then, for any > 0 , there exists 0 such that for every

, we have

n





0

>

n n x_nB x



, . This implies that

 

> 0,B x, F .

 

  

Whence, making use of Proposition 3.1, we get that xF.

Conversely, assume that F is not  -closed, then

F X

C

 is not an open set for the -topology. There exists then F

X

x C 

 satisfying

 

, F X

C B x



   and so

 

,

B x F  for any > 0. Therefore, for 1 k



there exists xk B x,1

k

  F   

 

n n

. Thence, the obtained

sequence x _F satisfies x_n x



 . This implies

xF , which is in contradiction with the fact that

F X

xC _. In conclusion, F is -closed.

Remark 3.1 Observe that

2

0

satisfies the -condition

satisfies the property .



 

 

As consequence, we see that under the assumption that

 satisfies the 0 property, we have

1 topology topology.

 

Then definitions of  -convergence and  -closed subsets of X_ need the hypothesis that  satisfies the ∆2-condition.

The following result shows that the modular space X_ is a regular space.

Theorem 3.3 Let  be a modular satisfying the ∆2- condition, A be a  -closed subset of X_ and x₀A. Then there exists an open neighborhood V_x0 of x₀

such that .

0

x

In order to show the theorem above, we need the following result.

V A 

Proposition 3.3 Let  be a modular satisfying the

∆2-condition and AX_. Then



x A,



inf





x y



,y A



     0

if and only if xA, where A is the closure of A for the -topology.

Proof. We have



x A,



inf





x y



,y A



     0.

Then for any 1 n

  , there exists ynA such that



x yn



< 1

n

  this implies that there exists a sequence

 

y_{n n}A such that y_n x. Whence



 xA. Inversely, let xA, then by Theorem 3.2, there ex-

ists a sequence

 

y_{n n}A such that y_n , there-

 x fore, for any 0 there exists n0 such that



x A,





x yn



; n 0.

     n Hence



x A,



0

 .

Proof of the Theorem 3.3. By Proposition 3.3, x0A

if and only if 



x A₀,



r>0. Next, since  satisfies the 2-condition then by Lemma 3.1, for ₃> 0

r

,

 

there exist L0, and  0 such that if 

 

x L and 



yx



 we have 

 

y 

 

x . More- over, there exists * such that

0

m  r inf



L,



m  whenever ₀. Now, let and we consider the open neighborhood of

mm m₁max 3,



0



0 m x

0 0

1

0, .

x

r

V x B

m

     _{ }  

Suppose next that Vx₀A  and let yVx₀A. Since A is closed we make use of Proposition 3.1 to

exhibit a sequence

 

y_{n n}A such that y_n y. So

  that one considers Xn y yn and Yn x0yn. Since

n

y A and x₀A , then . On the other hand, note that

 

n

Y r



 









1

inf , ,

n n

r

X y y L

m

     

whenever nn0 and







0







1

inf , .

n n

r

X Y x y L

m

      

Therefore

 





1

2

3 3

n n

r r r

r Y y y

m

  

      

whenever nn0, a contradiction. Thus Vx₀A .

Remark 3.2 If  satisfies Fatou property, then

 

0,

 



 



B r B_ 0,r  xX_  x r

is a closed ball of the topology  . We note by Bf

 

x r, 0

r

all closed ball centered at x with the radius (see

[7]).

3.3, and if the modular  satisfies Fatou property, then

By Proposition 3.1, there exists

 

1

0,

n n

r

y B

m

 

   _{ }  



such that yn . Moreover, the sequence 



0

x

V A .

Proof. Making appeal of Theorem 3.3, there exists

y



0 n



_{n 0}

0 0

1 0,

x

r

V x B

m

     _

  such that . Then, we

have

0 x

V A 

0 0

1 0,

x f

r

V x B

m

  

  _  

x

x y _V satisfying x₀ yn x₀



  y. Hence

. Indeed, let y Vx₀

  and

note that from Proposition 3.1, there exists a sequence

 

1 0,

n n f r

y B

m



   _

 



0

0 x .

x  y V 

Finally, we take the same arguments as in the proof of Theorem 3.3, we have

 such that

0 .

x

V A 

0 n ,

x y y

  

REFERENCES

which implies that . Indeed, it is easy to see

that and since

0

n

y y x

  



0 0 x

  



n n

Y y  y  satisfies the -condition we have also .

2

 X 2



y



y x₀







 

[1] J. Musielak, “Orlicz Spaces and Modular Spaces,” Lec-ture Notes in Mathematics, Vol. 1034, 1983.

0

n n

[2] A. Ait Taleb and E. Hanebaly, “A Fixed Point Theorem and Its Application to Integral Equations in Modular Function Spaces,” Proceedings of the American Mathe- matical Society, Vol. 128, 2000, pp. 419-426.

doi:10.1090/S0002-9939-99-05546-X Thence, for 0, there are L0 and  0 such

that _{[3] A. Razani and R. Moradi, “Common Fixed Point Theo-} rems of Integral Type in Modular Spaces,” Bulletin of the Iranian Mathematical Society, Vol. 35, No. 2, 2009, pp. 11-24.

 

inf , , ,

2

n

X L 

  _  _

 

and [4] A. Razani, E. Nabizadeh, M. B. Mohammadi and S. H. Pour, “Fixed Point of Nonlinear and Asymptotic Contrac- tions in the Modular Space,” Abstract and Applied Analy- sis, Vol. 2007, 2007, Article ID: 40575.





 

inf , , ,

2

n n n

Y X Y L 

    _  _

 

whenever nn0, then [5] A. P. Farajzadeh, M. B. Mohammadi and M. A. Noor, “Fixed Point Theorems in Modular Spaces,” Mathemati- cal Communications, Vol. 16, 2011, pp. 13-20.

 





0





inf , , ,

2 2 2 2

n n

Y y y x

L

 

   

 

    

 _{ }     

[6] M. A. Khamsi, “Nonlinear Semigroups in Modular Func-tion Spaces,” Thèse d'état, Département de Mathé-matiques, Rabat, 1994.

whenever nn0. Therefore

0

1 1

0, 0, .

n f

r r

y y x B B

m m

 



       _{ }  _{ }    

[7] M. A. Khamsi, W. Kozlowski and S. M.-Reich, “Fixed Point Theory in Modular Function Spaces,” Nonlinear Analysis, Theory, Methods and Applications, Vol. 14, No. 11, 1990, pp. 935-953.

[8] F. Lael and K. Nourouzi, “On the Fixed Points of Corre- spondences in Modular Spaces,” ISRN Geometry, Vol. 2011, 2011, Article ID: 530254.

doi:10.5402/2011/530254 It follows





0 0 0

1

0, ,

f

r

y x y x x B

m

       _{ }

  [9] M. A. Khamsi, “Quasicontraction Mappings in Modular Spaces without ∆2-Condition,” Fixed Point Theory and Applications, Vol. 2008, 2008, Article ID: 916187. and hence

[10] A. Hajji, “Forme Equivalente à la Condition ∆2 et

Cer-tains Résultats de Séparations dans les Espaces Modu-laires,” 2005. http://arXiv.org/abs/math.FA/0509482

0 0

1

0, .

x f

r

V x B

m

  

  _{ }  

Inversely, let

0 0

1

0, .

f

r

x y x B

m