Best Approximation in 2-Normed almost Linear Space

Best Approximation in 2-Normed almost Linear Space

T. Markandeya Naidu1

Assistant Professor of Mathematics Department of Mathematics, P.V.K.N.Govt.Degree College, Chittoor, Affiliated to S.V.University,Tirupathi, Andhra Pradesh,India.

Dr. D. Bharathi 2

Associate professor of Mathematics Department of Mathematics, S.V.University,Tirupathi,

Andhra Pradesh,India.

ABSTRACT

In this paper we introduce a new concept called 2-normed almost linear space and establish some of the results of best approximation on normed almost linear space in the context of 2- normed almost linear space.

1 .INTRODUCTION

In (1) Gliceria Godini introduced the concept almost linear space which is defined as A non empty set X together with two mappings s: XxX X and m:RxX X Where s(x ,y)=x +y and m(, x) = x is said to be an almost linear space if it satisfies the following properties.

For every x, y, z X and for every , R i) x +y X,

ii) (x + y) +z = x+(y + z) ,iii) x +y = y + x, iv) There exists

an element 0 X such that x+0=x , v) 1 x = x ,

1. (x + y)= x+ y , vii) 0 x =0 , viii) ( x )=( )x and ix) ( + )x= x + x for 0,0.

   In (1 & 4) Gliceria Godini also introduced the concept normed almost linear space which is defined as an almost linear space X together with III

   . III : X R is said to be normed almost linear space if it satisfies the following properties

   1. III x III=0 if and only if x=0,

   2. III x III=II III x III ,

   3. III x-z IIIIIIx-yIII+IIIy-zIII for every x,y X and R.

      The concept of linear 2-normed space has been initially investigated by S. G hler(17)

      and has been extensively by Y.J.Cho,C.Diminnie, R.Freese and many other, which is defined as a linear space X over R with dim>1 together with II .II is called Linear 2-normed space if II . II satisfy the following properties

      1. II x, y II>0 and II x , y II=0 if and only if x and y are linearly dependent,

      2. II x , y II=II y , x II ,

      3. II x ,y II= II II x , y II and

      4. II x , y+z II=II x , z II + II y ,z II for every

         x,y,z X and R.

         In (2 & 3) G.Godini established some results of best approximation on normed linear space in the context of normed almost linear space. In this paper we extend some of the results of best approximation on normed almost linear space in to 2-normed almost linear space.

         1. PRELIMINARIES

            Definition 2.1. Let X be an almost linear space of dimension> 1 and III .III: X x X R be a real valued function. If III . III satisfy the following properties

            1. III , III=0 if and only if and are

               linearly dependent,

            2. III , III = III , III ,

            3. III a , III = IaI III , III ,

            4. III , – III III , – III + III , – III

            for every , , , X and a R.

            then (X,III.III) is called 2-normed almost linear space.

            Definition 2.2. Let X be a 2-normed almost linear space over the real field R and G a non empty subset of . For a bounded sub set A of X let us define

            (A) = III x , a-g III for every x X\ and 2.1

            (A) = 0 G: III x , a-0 III = (A)

            for every x X\ . 2.2

            (A) is called the chebyshev radius of A with respect to G and an element 0 (A) is called a best simultaneous approximation or chebyshev centre of A with respect to G.

            Definition 2.3. When A is a singleton say A= {a},

            a X\ then (A) is the distance of a to G, denoted by dist(a,G) and defined by

            dist(a,G)= IIIx,a-gIII for every x X\ 2.3

            and (A) is the set of all best approximations of

            a out of G denoted by (a) and defined by

            (a)={ 0 G : IIIx,a-0III=dist(a,G),

            for every x X\ } 2.4

            Definition 2.4. Let X be a 2-normed almost linear space. The set G is said to be proximinal if (a) is nonempty for each a X\ .

            It is well known that for any bounded subset A of X we have (A)= (0())= ( )

            (A)= (0 ())= ( )

            Where 0() stands for the convex hull of A and

            stands for the closure of A.

            Definition 2.5. Let X be a 2-normed almost linear

            spaces and G . We difine (G) X in the following way

            (G) if for each g such that the following conditions are hold

            1. III x, a-g III = III x, -g III

               for each 2.5

            2. III x, a- III III x, – III

            for every x X\ . 2.6

         2. MAIN RESULTS

Theorem 3.1 Let X be a 2-normed almost linear spaces and G a bounded weakly compact subset of

. Then G is proximinal in X.

Proof : Let G be a bounded compact subset of .

By definition of d(a,G), there exist a sequence

{ }, = 1 in G such that

lim III x, a- III = dist (a, G)

Since G is bounded for some > 0 there exist N such that

III x, a- III dist(a,G)+ for nN.

M for every n.

Where M=max( 1, 2), 1 = (a,G)+ and

2=max III x, a- III for nN. Now III x, IIIIIIx,a- III+IIIx,aIII

M+IIIx,aIII

This implies that { } and therefore converges weakly to g in G.

Hence we have IIIx,a-g III lim IIIx,a- III = dist(a,G)

But III x, a-g III dist(a,G)

Therefore we have III x, a-g III= dist (a, G) and so g

is a best approximation to a from G. Thus G is Proximinal in X.

Definition3.2 Let X be 2-normed almost linear space and G a non-empty subset of .

Let be the sub set of X defined in the following

way.

a if for each g and > 0 i=1,2, the relations III x , a-g III < 1 + 2 and (, 2 ) G

is non-empty implies (, 1) (, 2 )G is no-empty.

We observe that by definition of , G is a subset of .

Theorem 3.3 Let X be 2- normed almost linear space and G is a non-empty subset of .Then for each a

we have (a) is non-empty.

Proof: Let 1 G and a .

We have (G). 1 1

If 1 2 then (2) (1).

Let 1 = 2 and 2 = d(a,G)+ 2

1 1

Then we have III x, a-1III < 2 + ( d(a,G) + 2 ) and B ( a, 2) ) G .

Since a we get that B(1, 1 ) B(a, 2 ) ) G . Let us choose 2 ( B(1 , 1) B(a, 2) ) G

2

2

Then III x, 1-2III< 1 and III x, a-2III<dist(a,G)+ 1

If G is reflexive then by theorem (3.1) G is proximinal in X.

Definition: 3.5 Let X be a 2-normed almost linear

2

22 22

22 22

Let 1 = 1 and 2 = , + 1

spaces and G is subset of . We shall assign to each a (G) a non-empty subset (a) is subset of in

Again we have III x, a-2III< 1 + , + 1 and

the following way

22

B ( a, 2) G .

22

for gG , let (a) = { : } satisfying (i) and (ii)

Since a we have (B(2, 1 ) B(a, 2 ) ) G . Choose 3 ( B(2, 1 ) B(a, 2) ) G. We get

of definition (2.4).

Since a (G), the set (a) is non-empty.

Lemma 3.6 Let X (G), and g . then for each

1 1

IIIx, 2-3III< 22 and IIIx, a-3III < dist(a,G)+ 22 .

By continuing the above process at n stages we get

(a) we have

III x, a-gIII = III x, g III

<>2

2

III x, -+1 III< 1 and

= (a)III x , b-g III 3.2

2

2

III x, a-+1 III<dist(a, G)+ 1

By eq. (3.1) it follows that lim

3.1

III x, a-+1 III=

Consequently, the set (a) is the non empty bounded subset of , which is removable with

respect to G. if a , then (a) = {a}.

dist(a,G) and { } is a Cauchy sequence.

Since G is complete { } contains a sub-sequence

Proof:- Let a (G), gG and (a) By (i) of definition (2.4) we have

say { } which converges to in G.

0

Now lim III x, a- III = dist(a,G) implies III x, a-0III = dist(a,G).

Hence 0 (a) implies that (a) .

Theorem: 3.4 Let X be a 2-normed almost linear

spaces and G . If with respect to G, then for each a (G), the set (a) contains atmost one element. If in addition G is reflexive then for each a (G), the set (a) is singleton.

Proof: Let a (G) and suppose 1, 2 G such that III x, a- III = dist(a,G), i=1,2

Then III x, a-(1+ 2) III = dist(a,G).

2

2

2

since a (G), for the element (1+ 2) ,

III x, a-gIII = III x, -gIII .

Let b (a). By (ii) of definition (2.4) we have III x, a-gIII III x, b-gIII.

From this it follows that

III x, -gIII = III x, a-gIII III x, b-gIII.

Hence equation (3.2) follows since (a). Let now a is subset of (G) and 0 (a).

Now by (ii) of definition (2.4) for = a

we have

0= III x, a-aIII III x, 0 -aIII

This implies a=0 . Hence (a)={a}.

Theorem:3.7 Let X be a 2- normed almost linear

spaces, 1 G and let a (G) we have

0 such that

III x, a-(1 + 2) III = III x, 0 -(1+ 2) III and

dist(a,1)=1 ( (a)) 3.3

2 2 and 1 = 1 ( (a)) 3.4

0

0

III x, a-1III III x, 0 – III, i=1,2. Then dist(a,G) = III x, -(1+ 2) III

2

( III x, 0 -1 III+ III x, 0 -2 III)/2

dist a, G .

And so III x, 0 -1 III= III x, 0 -2 III

0

0

= III x, -(1+ 2) III

2

Proof:- Let g1 , since a (G) and 1 G by lemma(3.6)We have

III x, a-1III = (a)III x, b-1III

Now taking the infimum in both sides over all 1 G We get 1 IIIx,a-gIII = 1 (a)III b-gIII

By definition (A) we have

Since (0 -1)-( 0 -2)= 2 1 G and is strictly convex with respect to G it follows that 1 = 2 .

(A) =G III x, a-g III for every

x X\ 3.5

By definition dist(a,G)=G IIIx,a-gIII 3.6

Now by equations ( 3.5 & 3.6)

we get dist(a, 1) = 1 ( (a)).

Choose such that ( ) >0 then

1 III x, III 1.

Then it follows that 1 = 1

( (a)).

1( ) 1 (g ) and

Theorem 3.8 Let G be a one dimensional chebyshev sub-space of . Then (a) is a singleton for each a (G) .

Proof: Clearly G is proximinal in X, since G is one dimensional sub-space of .

Let now a (G) and suppose there exist

1, 2 (a), 1 2.

2

2

For (1+ 2) , 0 such that

Sep { 1( ) } 1.

Hence by uniformly kadec-klee (UKK) we obtain that

1 III x, g III 1-

a contradiction. Hence (a) is a compact.

Definition 3.10 Let X be a 2- normed almost linear

spaces and G , the pair (a, G) is said to have the property (P) if for every r>0 and any >0 here is a ()>0 and a function f: G X GG such that

for every ||< () we have f( , ) ( , )

III x, a-(1 + 2) III = III x, 0 -(1+ 2) III

1 2

and (1 ,r+()) (2 , + )

1

2 2

(f(1 , 2 ), + ))

III x, a- III III x , 0 -vIII for each

2

2

Since (1+ 2 ) (a),

It follows that

0

0

dist(a, G) = III x, -(1+ 2) III

2

III x, 0 -1 III+ III x, 0 -2 III

Theorem 3.11 Let X be a 2- normed almost linear

spaces and G a complete subset of . If pair(a, G) has the property (P), then G is proximinal in X.

Proof:-

For r= (A), , =1 find the corresponding 1)

( III x, a-1 III+ III x, a-2 III ) /2 2 2 1

= d(a,G).

2

2

And so III x, 0 -(1+ 2) III = III x, 0 – III , i=1,2. Since dim G = 1, we must have 1, 2 (0 ),

Then there is a point 1 G with A B(1,r+ 2)) Assume now that for an n N, the points

1 , 2 , , , , G

1 1 1

And the number , , , , , , , ) with the

a contradiction

2 4 2

Hence

=

implies (a) is a singleton.

property ( 1 ) 1 , A B(1,r+ 1 )) i=1,2,3, , , n.

1 2

2

2

1

2

Theorem 3.9 Let X be a 2-normed almost linear space such that is Banach space and the

III x, +1 III 2 , i=1,2, , , , n-1 have already been constructed.

Now for r and 1 find the corresponding 1 )

norm of is uniformly kadec-klee (UKK) and let

2 +1

2 +1

It is easy to see that it is possible to choose

G be a W-compact ,convex set. Then for each

1 < min 1 ), 1 )

a (G) the set (a) is compact and convex.

2 +1

2

2 +1

Proof: Clearly G is proximinal in X.

Let now a (G). If (a) is not compact then ther

exist a sequence { } (a) with

There is a point b G with A B(b,r+ 1 ))

2 +1

2 +1

Using the fact that (a,G) has the property (P) we obtain A B( ,r+ 1 )) B(b,r+ 1 ))

2

2 +1

Sep { } for some > 0.

Since (a) is W-compact, may assume that

g (a).

Since a (G), for g G there exist such that III x, a-g III = III x, -g III and III x, a- III and III x, a- III III x, – III ,n=1,2,3,.

Here r = III x, g-0 III III x, III

B( +1 , r+ 1 )) where +1 =f( ,b).

2 +1

2 +1

1

1

Then we get III x, +1 III 2

By continue the above process we get a cauchy sequence { } in G.

Now let the above sequence has the limit 0.

This implies 0 (A). Hence G is proximinal in X .

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