# Best Simultaneous Approximation in 2-Normed Almost Linear Space

DOI : 10.17577/IJERTV3IS10787

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#### Best Simultaneous Approximation in 2-Normed Almost Linear Space

Best Simultaneous Approximation in 2-Normed Almost Linear Space

T. Markandeya Naidu & Dr. D. Bharathi

1. Department Of Mathematics, P.V.K.N.Govt.Degree College, Chittoor,Andhra Pradesh, India 2 .Department Of Mathematics, S.V.University, Tirupathi, Andhra Pradesh ,India.

ABSTRACT

In this paper we establish some of the results of best simultaneous approximation in linear 2- normed 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

1. x +y X ,

2. (x + y) +z = x+(y + z) ,

3. x +y = y + x ,

4. There exists an element 0 X such that x+0=x ,

5. 1 x = x ,

6. (x + y)= x+ y ,

7. 0 x =0 ,

8. ( x )=( )x ,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 ||| . ||| : X R is said to be normed almost linear space if it satisfies the following properties

1. ||| x |||=0 if and only if x=0, ii) ||| x |||=II ||| x ||| ,

iii) ||| x-z ||||||x-y|||+|||y-z||| 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 (23) T.Markandeya naidu and Dr D.Bharathi introduced a new concept called 2-normed almost linear space and established some of the results of best approximation in 2-nomed almost linear space.In (22) S.Elumalai & R.Vijayaragavan established some of the results of best simultaneous approximation in linear space in the context of linear2-normed space. In this paper we extend some of the results of best simultaneous approximation on linear2- normed space in to 2-normed almost linear space.

2. PRELIMINARIES

Definition 2.1. Let X be an almost linear space of dimension> 1 and

||| .|||: X x X R be a real valued function.

If ||| . ||| satisfy the following properties

i) ||| , |||=0 if and only if and are linearly dependent,

ii) ||| , ||| = ||| , |||,

iii) ||| a , ||| = IaI ||| , |||,

iv) ||| , – ||| ||| , – ||| + ||| ,

– ||| for every , , , X and a R.

then (X,|||.|||) 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) = ||| x , a-g ||| for every x X\ 2.1

and

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

(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)= ||| x,a-g||| 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)={ 0G:|||x,a-0|||=dist(a,G),

for every x X\ }

2.4

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 2.4. 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

i) ||| x, a-g ||| = ||| x, -g ||| for each

2.5

ii) ||| x, a- ||| ||| x, – ||| for every x X\. 2.6

We have (G). If 1 2 then (2)

(1).

Definition 2.5. 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\ .

Theorem 3.1 Let ( X, ||| . ||| ) be a

2- normed almost linear space. Let G X and A be a bounded subset of X. Then the function (h,g) defined by sup ||| h, a-g |||,

h X\ , g G, a A is a continuous function on X.

Proof: For any a A and g , G we have

||| h, a-g ||| ||| h, a- ||| +

||| h, g ||| , h X\ . Then

||| h, a-g |||

{ ||| h, a- ||| + ||| h, g ||| }

a best simultaneous approximation g G to any given compact subset A of X.

#### Proof: Since A is compact there exist a constant t such that

||| a , b ||| t for all a A and b X. Now we define the subset H of G as G G(0,2t) then

inf. ||| b, a-h ||| =

||| b, a-h ||| , b X\ t Since h is compact the continuous function

(h,b) attains its minimum over H for some g

G.

which is the best simultaneous approximation to A.

#### Theorem 3.3 Let ( X, ||| . ||| ) be a

2- normed almost linear space and let G be a

convex subset of X and A X . If g, G are two best simultaneous approximations to A by elements of G. Then

= g + (1- ) ,0 1 is also best simultaneous approximation to A.

Proof: For x X\ ,

Now if ||| h, g ||| then (h,g) (h, ) +

||| x, a-

|||

By interchanging g and we obtain

= ||| x, a- g + (1- ) |||

(h, ) (h,g) + that implies

| (h,g) – (h, ) |<

=

||| x, (a-g) + (1- ) (a-) |||

That is (h,g) is continuous on X.

||| x, a-g ||| +

#### Theorem 3.2 Let ( X, ||| . ||| ) be a 2- normed almost linear space. Let G be a finite dimensional subspace of X. Then there exist

||| x, a-g ||| +

(1-) ||| x, a- |||

(1-) ||| x, a- |||

= (, ) + (1-) (, )

= (, )

3.1

(, ) = ||| x, a- |||

||| x, a- + ||| = k 3.4

2

Since A is compact there exist an element 0

such that

2

||| x, a- + |||

2

= ||| x, 0 – + |||

=k 3.5

. 3.3 ||| x, 0- ||| k and ||| x, 0-

||| k .

Then by strict convexity we have

||| x, 0-+0- ||| < 2 k

||| x, g- ||| 3.2

That is ||| x, 0

+ ||| < k

2

(, ) = ||| x, a- ||| This proves the result.

#### Proof : The existence of a best simultaneous approximation follows from the Theorem3.2.

Suppose and ( ) are best simultaneous approximations to A then for x X\ ,

||| x,a-g |||

Hence the proof.

#### Theorem 3.5 Let G be a closed and convex subset of a uniformly convex

1. Banach space X.Then for any compact subset A of X there exist unique best approximation to A from the element of G.

#### Proof:

Let k= ||| x,a-g |||: x X\ and

be any sequence of elements in G Such that

||| x,a- ||| =k.

Also = ||| x,a- ||| ,m1 and x

X\ .

Then k which implies

=

||| x, a- |||

||| x, |||1 for a A 3.6

= ||| x, a- |||

=k 3.3

Now we consider 1 + =

2

+ +

2 +

Then by theorem (3.3),

Let = + .

,

+

+

2

is also

since G is convex, , G.

the best simultaneous approximation. That is

Hence ||| x,a-, |||k and |||

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||| x,a- ||| .

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||| x,a-2 ||| and hence

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