Strongly Unique Best Coapproximation In Linear 2-Normed Spaces

Strongly Unique Best Coapproximation In Linear 2-Normed Spaces

R.Vijayaragavan

School of Advanced Sciences V I T University

Vellore-632014, Tamilnadu, India.

Abstract

This paper deals with some fundamental properties of the set of strongly unique best coapproximation in a linear 2-normed space.

AMS Subject classification: 41A50,41A52, 41A99.

Keywords: Linear 2-normed space, best coapproximation and strongly unique best coapproximation.

1. 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 2-normed 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 2-norm.

2. 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 real-valued function on X × X satisfying the following properties for every x, y, z in X .

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

   2. I x, y I=I y, x I,

   3. I x, y I= || I x, y I , where is a real number,

   4. I x, y + z II x, y I + I x, z I .

   Then I ., . I is called a 2-norm and the linear space X equipped with the 2-norm is called a linear 2-normed space. It is clear that 2-norm is non negative.

   The following important property of 2-norm 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 non-empty subset of a linear 2-normed 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 non-empty subset of a linear 2-normed 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 2-normed subspace. Unless otherwise stated all linear 2-normed spaces considered in this paper are real linear 2-normed spaces and all subsets and subspaces considered in this paper are existence subsets and existence subspaces with respect to strongly unique best coapproximation.

3. 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 2-normed space X and x X . Then the following statements hold.

1. TG(x) is closed if G is closed.

2. TG(x) is convex if G is convex.

3. 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.

1. 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.

2. 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 linear2-normed 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 2-normed space X, x X and k

X \ [G, x] . Then the following statements are equivalent for every y [k] .

1. g0 TG(x) .

2. g0 TG(x + y) .

3. g0 TG(x y) .

4. g0 + y TG(x + y) .

5. g0 + y TG(x y) .

6. g0 y TG(x + y) .

7. g0 y TG(x y) .

8. g0 + y TG(x) .

9. 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 2-normed 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, · · · .

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 2-normed 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, · · ·

1. g0 TG(x) .

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

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

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

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

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

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

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

9. 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 2-normed 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 2-normed 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, · · ·

(i) g0 TG+[k](x) .

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

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

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

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

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

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

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

4. 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 2-normed 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 2-normed 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|| 2||x g, k|| t||x g0, k||.

Proof. The proof is trivial.

Theorem 3.9. Let G be a subspace of a linear 2-normed space X and x X . Then the following statements hold.

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

2. 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.

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