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 Total Downloads : 329
 Authors : T. Markandeya Naidu, Dr. D. Bharathi
 Paper ID : IJERTV2IS121284
 Volume & Issue : Volume 02, Issue 12 (December 2013)
 Published (First Online): 30122013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Best Approximation in 2Normed 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 2normed 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 ,

(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

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

III x III=II III x III ,

III xz IIIIIIxyIII+IIIyzIII for every x,y X and R.
The concept of linear 2normed 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 2normed space if II . II satisfy the following properties

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

II x , y II=II y , x II ,

II x ,y II= II II x , y II and

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 2normed almost linear space.

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

III , III=0 if and only if and are
linearly dependent,

III , III = III , III ,

III a , III = IaI III , III ,

III , – III III , – III + III , – III
for every , , , X and a R.
then (X,III.III) is called 2normed almost linear space.
Definition 2.2. Let X be a 2normed 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 , ag III for every x X\ and 2.1
(A) = 0 G: III x , a0 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,agIII 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,a0III=dist(a,G),
for every x X\ } 2.4
Definition 2.4. Let X be a 2normed 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 2normed 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

III x, ag III = III x, g III
for each 2.5

III x, a III III x, – III
for every x X\ . 2.6


MAIN RESULTS



Theorem 3.1 Let X be a 2normed 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,ag III lim IIIx,a III = dist(a,G)
But III x, ag III dist(a,G)
Therefore we have III x, ag 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 2normed almost linear space and G a nonempty 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 , ag III < 1 + 2 and (, 2 ) G
is nonempty implies (, 1) (, 2 )G is noempty.
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 nonempty subset of .Then for each a
we have (a) is nonempty.
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, a1III < 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, 12III< 1 and III x, a2III<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 2normed 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 nonempty subset (a) is subset of in
Again we have III x, a2III< 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 nonempty.
Lemma 3.6 Let X (G), and g . then for each
1 1
IIIx, 23III< 22 and IIIx, a3III < dist(a,G)+ 22 .
By continuing the above process at n stages we get
(a) we have
III x, agIII = III x, g III
<>2
2
III x, +1 III< 1 and
= (a)III x , bg 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 subsequence
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, a0III = dist(a,G).
Hence 0 (a) implies that (a) .
Theorem: 3.4 Let X be a 2normed 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, agIII = III x, gIII .
Let b (a). By (ii) of definition (2.4) we have III x, agIII III x, bgIII.
From this it follows that
III x, gIII = III x, agIII III x, bgIII.
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, aaIII 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, a1III 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, a1III = (a)III x, b1III
Now taking the infimum in both sides over all 1 G We get 1 IIIx,agIII = 1 (a)III bgIII
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, ag III for every
x X\ 3.5
By definition dist(a,G)=G IIIx,agIII 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 subspace of . Then (a) is a singleton for each a (G) .
Proof: Clearly G is proximinal in X, since G is one dimensional subspace 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 kadecklee (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, a1 III+ III x, a2 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 2normed almost linear space such that is Banach space and the
III x, +1 III 2 , i=1,2, , , , n1 have already been constructed.
Now for r and 1 find the corresponding 1 )
norm of is uniformly kadecklee (UKK) and let
2 +1
2 +1
It is easy to see that it is possible to choose
G be a Wcompact ,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 Wcompact, may assume that
g (a).
Since a (G), for g G there exist such that III x, ag 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, g0 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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