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 Authors : R. Vijayaragavan
 Paper ID : IJERTV2IS60047
 Volume & Issue : Volume 02, Issue 06 (June 2013)
 Published (First Online): 01062013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
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Strongly Unique Best Coapproximation In Linear 2Normed Spaces
R.Vijayaragavan
School of Advanced Sciences V I T University
Vellore632014, Tamilnadu, India.
Abstract
This paper deals with some fundamental properties of the set of strongly unique best coapproximation in a linear 2normed space.
AMS Subject classification: 41A50,41A52, 41A99.
Keywords: Linear 2normed space, best coapproximation and strongly unique best coapproximation.

INTRODUCTION
The problem of best coapproximation was first introduced by Franchetti and Furi [2] to study some characteristic properties of real Hilbert spaces and was followed up by Papini and Singer [12]. Subsequently, Geetha S.Rao and coworkers have developed this theory to a considerable extent [4,5,6,7,8,9]. This theory is largely concerned with the questions of existence, uniqueness and characterization of best coapproximation. Newman and Shapiro [11] studied the problems of strongly unique best approximation in the space of continuous functions under supremum norm. Geetha S.Rao, et al. [3,10] established many significant results in strongly unique best coapproximation in normed linear spaces. The notion of strongly unique best coapproximation in the context of linear 2normed spaces is introduced in this paper. Section 2 provides some important definitions and results that are required. Sections 3 delineates some fundamental properties of the set of strongly unique best coapproximation with respect to 2norm.

PRELIMINARIES
Definition 2.1. [ 1 ] Let X be a linear space over real numbers with dimension greater than one and let I ., . I be a realvalued function on X Ã— X satisfying the following properties for every x, y, z in X .

I x, y I= 0 if and only if x and y are linearly dependent,

I x, y I=I y, x I,

I x, y I=  I x, y I , where is a real number,

I x, y + z II x, y I + I x, z I .
Then I ., . I is called a 2norm and the linear space X equipped with the 2norm is called a linear 2normed space. It is clear that 2norm is non negative.
The following important property of 2norm was established by Cho [1].
Theorem 2.2. [ 1 ] For any points a, b X and any R ,
I a, b I=I a, b + a I .
Definition 2.3. Let G be a nonempty subset of a linear 2normed space X . An element
g0 G is called a best coapproximation to x X from G if for every g G ,
I g g0, k II x g, k I, for every k X \ [G, x],
where [G, x] represents a linear space spanned by elements of G and x .
The definition of strongly unique best coapproximation in the context of linear 2 normed space is introduced here for the first time as follows.
Definition 2.4. Let G be a nonempty subset of a linear 2normed space X . An element g0 G is called a strongly unique best coapproximation to x X from G, if there exists a constant t > 0 such that for every g G ,
I g g0, k II x g, k I t I x g0, k I, for every k X \ [G, x].
The set of all elements of strongly unique best coapproximations to x X from G is denoted by TG(x) .
The subset G is called an existence set if TG(x) contains at least one element for every x X . G is called a uniqueness set if TG(x) contains at most one element for every x X . G is called an existence and uniqueness set if TG(x) contains exactly one element for every x X .
For the sake of brevity, the terminology subspace is used instead of a linear 2normed subspace. Unless otherwise stated all linear 2normed spaces considered in this paper are real linear 2normed spaces and all subsets and subspaces considered in this paper are existence subsets and existence subspaces with respect to strongly unique best coapproximation.


SOME FUNDAMENTAL PROPERTIES OF TG(x)
Some basic properties of strongly unique best coapproximation are obtained in the following Theorems.
Theorem 3.1. Let G be a subset of a linear 2normed space X and x X . Then the following statements hold.

TG(x) is closed if G is closed.

TG(x) is convex if G is convex.

TG(x) is bounded.
Proof. (i). Let G be closed.
Let {gm} be a sequence in TG(x) such that gm g . To prove that TG(x) is closed, it is enough to prove that
g TG(x) .
Since G is closed, {gm} G and gm g , we have we have
g G . Since {gm} TG(x) ,
I g gm, k II x g, k I t I x gm, k I, for every k X\[G, x] and for some t > 0
I g gm + g g, k II x g, k I t I x gm, k I
I g g, k I I gm g, k II x g, k I t I x gm, k I, for every g G (3.1)
Since gm g , gm g 0 . So I gm g, k I 0 , as 0 and k are linearly dependent.
Therefore, it follows from (3.1) that
I g g, k II x g, k I t I x g, k I,
for every g G and for some t > 0 .
Thus
g TG(x) . Hence TG(x) is closed.

Let G be convex, g1, g2 TG(x) and (0, 1) . To prove that g1 + (1 )g2
TG(x) ,
let k X \ [G, x] . Then
I g (g1 + (1 )g2, k I
= I (g g1) + (1 )(g g2), k I
I g g1, k I +(1 ) I g g2, k I
I x g, k I t I x g1, k I
+(1 ) I x g, k I (1 )t I x g2, k I,
for every g G and for some t > 0.
= I x g, k I t(I x g1, k I + I (1 )x (1 )g2, k I)
I x g, k I t I x g1 + (1 )x (1 )g2, k I
= I x g, k I t I x (g1 + (1 )g2), k I .
Thus g1 + (1 )g2 TG(x) . Hence TG(x) is convex.

To prove that TG(x) is bounded, it is enough to prove for arbitrary g0, g0 TG(x) that I g0 g0, k I< c for some c > 0 , since I g0 g0, k I< c implies that
sup
g0 ,g0 TG(x)
I g0, g0, k I is finite and hence the diameter of TG(x) is finite.
Let g0, g0 TG(x) . Then there exists a constant t > 0 such that for every g G
and k X \ [G, x],
I g g0, k II x g, k I t I x g0, k I
and
I g g0, k II x g, k I t I x g0, k I . Now,
I x g0, k I I x g, k I + I g g0, k I
2 I x g, k I t I x g0, k I .
1+t
1+t
Thus I x g0, k I 2 I x g, k I, for every g G .
Hence I x g0, k I 2 d, where d = inf
I x g, k I .
1+t
2
Similarly, I x g , k I d.
gG
0
Therefore, it follows that
1 + t
4
4
I g0 g0, k I I g0 x, k I + I x g0, k I
1 + td
= C.
Whence TG(x) is bounded.
Let X be a linear2normed space, x X and [x] denote the set of all scalar multiplications of x
i.e., [x] = {x : R} .
Theorem 3.2. Let G be a subset of a linear 2normed space X, x X and k
X \ [G, x] . Then the following statements are equivalent for every y [k] .

g0 TG(x) .

g0 TG(x + y) .

g0 TG(x y) .

g0 + y TG(x + y) .

g0 + y TG(x y) .

g0 y TG(x + y) .

g0 y TG(x y) .

g0 + y TG(x) .

g0 y TG(x) .
Proof. The proof follows immediately by using Theorem 2.2.
Theorem 3.3. Let G be a subspace of a linear 2normed space X, x X and k X \ [G, x] . Then g0 TG(x) g0 TG(mx + (1 m)g0), for all R and m = 0, 1, 2, Â· Â· Â· .
Proof. Claim: g0 TG(x) g0 TG(x +(1 )g0) , for very R. Let g0 TG(x) . Then
I g g0, k II x g, k I t I x g0, k I , for all g G and for some t > 0 .
( \ ( \
( \ ( \
I g g0, k II x g, k I t I x g0, k I, for all g G.
0 0
0 0
( 1)g + g ( 1)g + g
0
0
I g , k II x , k I
t I x g0, k I, for all g G and
0, since ( 1)g0 + g G.
I g g0, k II x + (1 )g0 g, k I t I x + (1 )g0 g0, k I
g0 TG(x + (1 )g0, when /= 0.
If = 0 , then it is clear that g0 TG(x + (1 )g0).
The converse is obvious by taking = 1 . Hence the claim is true.
By repeated application of the claim the result follows.
Corollary 3.4. Let G be a subspace of a linear 2normed space X, x X and k X \ [G, x] . Then the following statements are equivalent for every y [k], R and m = 0, 1, 2, Â· Â· Â·

g0 TG(x) .

g0 TG(mx + (1 m)g0 + y).

g0 TG(mx + (1 m)g0 y).

g0 + y TG(mx + (1 m)g0 + y).

g0 + y TG(mx + (1 m)g0 y).

g0 y TG(mx + (1 m)g0 + y).

g0 y TG(mx + (1 m)g0 y).

g0 + y TG(mx + (1 m)g0).

g0 y TG(mx + (1 m)g0).
Proof. The proof follows from simple application of Theorem 2.2 and the Theorem 3.3.
Theorem 3.5. Let G be a subset of a linear 2normed space X, x X and
k X \ [G, x] . Then g0 TG(x) g0 TG+[k](x).
Proof. The proof follows from simple application of Theorem 3.2.
A corollary similar to that of Corollary 3.4 is established next as follows:
Corollary 3.6. Let G be a subspace of a linear 2normed space X, x X and k X \ [G, x] . Then the following statements are equivalent for every y [k], R and m = 0, 1, 2, Â· Â· Â·
(i) g0 TG+[k](x) .

g0 TG+[k](mx + (1 m)g0 + y).

g0 TG+[k](mx + (1 m)g0 y) .

g0 + y TG+[k](mx + (1 m)g0 + y) .
(v) g0 + y TG+[k](mx + (1 m)g0 y) .

g0 y TG+[k](mx + (1 m)g0 + y) .

g0 y TG+[k](mx + (1 m)g0 y) .

g0 + y TG+[k](mx + (1 m)g0) .

g0 y TG+[k](mx + (1 m)g0) .
Proof. The proof easily follows from Theorem 3.5 and Corollary 3.4.
Proposition 3.7. Let G be a subset of a linear 2normed space X, x X, k X \ [G, x] and 0 G . If g0 TG(x), then there exists a constant t > 0 such that I g0, k II x, k I t I x g0, k I .
Proof. The proof is obvious.
Proposition 3.8. Let G be a subset of a linear 2normed space X, x X and k X \ [G, x] . If g0 TG(x) , then there exists a constant t > 0 such that for all g G,
x g0, k 2x g, k tx g0, k.
Proof. The proof is trivial.
Theorem 3.9. Let G be a subspace of a linear 2normed space X and x X . Then the following statements hold.

TG(x + g) = TG(x) + g, for every g G .

TG(x) = TG(x), for every R .
Proof. (i). Let g be an arbitrary but fixed element of G .
Let g0 TG(x) . It is clear that g0 + g TG(x) + g .
To prove that TG(x) + g TG(x + g) , it is enough to prove that g0 + g TG(x + g) .
Now,
I g + g g0 g, k II x g, k I t I x g0, k I, for all g G
and for some t > 0.
I g + g (g0 + g), k II x + g (g + g), k I t I x + g (g0 + g), k I, for all g G.
g0 + g TG(x + g), since g g G.
Conversely, let g0 + g TG(x + g) .
To prove that TG(x + g) TG(x) + g , it is enough to prove that g0 TG(x) . Now,
I g g0, k I = I g + g (g0 + g), k I
I x + g (g + g), k I t I x + g (g0 + g), k I,
for all g G and for some t > 0.
g0 TG(x). Thus the result follows. (ii). The proof is similar to that of (i).
Remark 3.10. Theorem 3.9 can be restated as
TG(x + g) = TG(x) + g, for every g G.

Y.J.Cho, Theory of 2inner product spaces, Nova Science Publications, New York, 1994.

C.Franchetti and M.Furi, Some characteristic properties of real Hilbert spaces, Rev. Roumaine Math. Pures. Appl., 17 (1972), 10451048.

Geetha S.Rao and S.Elumalai, Semicontinuity properties of operators of strong best approximation and strong best coapproximation, Proc. Int. Conf. on Constructive Function Theory, Varna, Bulgaria (1981), 495498.

Geetha S.Rao and K.R.Chandrasekaran, Best coapproximation in normed linear spaces with property (), Math. Today, 2 (1984), 3340.

Geetha S.Rao, Best coapproximation in normed linear spaces in Approximation Theory, V.C.K.Chui, L.L.Schumaker and J.D.Ward (eds.), Academic Press, New York, (1986), 535538.

Geetha S.Rao and S.Muthukumar, Semicontinuity properties of the best coapproximation operator, Math. Today, 5 (1987), 3748.

Geetha S.Rao and K.R.Chandrasekaran, Characterization of elements of best coapproximation in normed linear spaces, Pure Appl. Math. Sci., 26 (1987), 139147.

Geetha S.Rao and M.Swaminathan, Best coapproximation and Schauder bases in Banach Spaces, Acta Scient. Math. Szeged, 54 (1990), 339354.

Geetha S.Rao and K.R.Chandrasekaran, HahnBanach extensions, best coapproximation and related results, in Approximation Theory and its Applications, Geetha S.Rao (ed.), New Age International Publishers, New Delhi, 1996, 5158.

Geetha S.Rao and R.Saravanan, Strongly unique best coapproximation,
kyungpook Math. J., 43 (2003), 000000.

D.J.Newman and H.S.Shapiro, Some theorems on Chebyshev approximation, Duke Math. J., 30 (1963), 673684.

P.L.Papini and I.Singer, Best coapproximation in normed linear spaces, Mh. Math.,
88 (1979), 2744.