^{1}

^{*}

^{2}

In this paper, we prove when these x ∈
*l*
_{2} with
, they have the common
*δ* for strongly ball proximinal. By using this property, we can prove the strong ball proximinality of
*l*
_{∞}(
*l*
_{2}). Also, we show that equable subspace
*Y* of a Banach space
*X* is actually uniform ball proximinality.

The best approximation is one of the most important concepts in approximation theory, and it plays an important role in many scientific fields. For example, it has full application in Banach space geometry theory, smooth analysis, function approximation, optimization theory and other disciplines. Many researchers have conducted a lot of in-depth study on the proximinal set (especially the proximinal subspace) [_{Y} of a subspace Y is stronger than the (strong) proximinality in Y. In [

In general, there are a few results about stability of the ball proximinality. Firstly, Bandyopadhyay et al. [

For E = l ∞ , it seems difficult to get a general answer to the stability of strong ball proximinality. So it is possible to consider some special cases as l ∞ ( X ) and to find the proper conditions for a Banach space X such that the unit ball of l ∞ ( X ) is strongly proximinal. In this paper, we can see for X = l 2 , l ∞ ( X ) is strongly proximinal because for these x ∈ l 2 with d ( x , B l 2 ) > 1 12 , they have the common δ for strong ball proximinality, then we can get the strong ball proximinality of l ∞ ( l 2 ) . Paul [

We will now present the notations and definitions that would be used throughout the paper. Let X denotes a real Banach space. Also, we assume that all subspaces are closed. The closed unit ball of X is denoted by B X and B X = { x ∈ X : ‖ x ‖ ≤ 1 } . For x ∈ X and r > 0 , we set B ( x , r ) = { y ∈ X : ‖ y − x ‖ ≤ r } .

Let C be a nonempty closed convex subset of X. For any x ∈ X and δ > 0 , P C ( x ) denote the sets:

P C ( x ) = { y ∈ C : ‖ x − y ‖ = d ( x , C ) } ,

where d ( x , C ) is the distance of x to C, that is d ( x , C ) = inf { ‖ x − z ‖ : z ∈ C } .

Definition 1 [

1) A subset C is said to be proximinal if for every x in X, the set P C ( x ) ≠ ∅ .

2) A subset C is said to be strongly proximinal if for any ϵ > 0 and any x ∈ X , there exists δ > 0 such that for any y ∈ C with ‖ x − y ‖ < d ( x , C ) + δ , then there is y ′ ∈ C with ‖ y − y ′ ‖ < ϵ and ‖ x − y ′ ‖ ≤ d ( x , C ) .

3) A subset C is said to be uniformly proximinal if for any ϵ > 0 and R > 0 , there exists δ > 0 such that for any x ∈ X , d ( x , C ) ≤ R and any y ∈ C with ‖ x − y ‖ < R + δ , then there is y ′ ∈ C with ‖ y − y ′ ‖ < ϵ and ‖ x − y ′ ‖ ≤ R .

From the Definition 1, we can see uniformly proximinal ⇒ strongly proximinal ⇒ proximinal. For any Banach space X, it is easy to see B X is

proximinal. Since for any x ∈ X \ B X , x ‖ x ‖ ∈ P B X ( x ) and

d ( x , B X ) = ‖ x − x ‖ x ‖ ‖ = ‖ x ‖ − 1. (1)

But, from the example by Godefroy in ( [

Definition 2 [

1) X is said to be strongly ball proximinal if the unit ball B_{X} is strongly proximinal.

2) X is said to be uniformly ball proximinal if the unit ball B_{X} is uniformly proximinal.

Definition 3 [

B ( 0,1 ) ∩ B ( y ,1 + δ ) ⊂ B ( ψ ϵ ( y ) y ,1 ) . (2)

Remark 1: In Theorem 2.6 [

Let Ψ ϵ ≜ ψ ϵ ( y ) y , we give the next lemma which is the remark 2.3 in [

Lemma 1 [

w ∈ Y with w = r { Ψ ϵ ( z − y r ) + y r } , then

B ( y , r ) ∩ B ( z , r ( 1 + δ ) ) ⊂ B ( w , r ) . (3)

Additionally, if both y and z are in B Y , then w ∈ B Y .

Next, to avoid confusion, we use a n or b n for some real numbers, x n or y n for the vectors in Banach space.

Let 1 ≤ p < ∞ , l p is the Banach space of all sequences x = ( a n ) n = 1 ∞ of real so that ‖ x ‖ p = ( ∑ n = 1 ∞ | a n | p ) 1 / p < ∞ . For p = ∞ , l ∞ is the Banach space of sequ- ences such that ‖ x ‖ ∞ = sup { | a n | : n ∈ N } < ∞ .

Let ( X n ) be a sequence of Banach spaces. For 1 ≤ p ≤ ∞ , l p -direct sum ( ∑ ⊕ X n ) l p denote the collection of elements ( x n ) such that x n ∈ X n and the sequence ( ‖ x n ‖ X n ) n = 1 ∞ ∈ l p . Thus the norm of ( x n ) is

‖ ( x n ) ‖ ( ∑ ⊕ X n ) l p = ‖ ( ‖ x n ‖ X n ) ‖ p .

If for any n, X n = X , we can simply denote ( ∑ ⊕ X ) l p by l p ( X ) .

In this section, we will give our main results. For Theorem 1, we can see B l ∞ ( l 2 ) is strongly ball proximinal. This result is using the “uniformly” strongly ball proximinal of the B l 2 which is showed by Lemma 2. For Theorem 2, we prove when Y is an equable subspace in Banach space X, B_{Y} is uniformly proximinal.

Lemma 2: For every 0 < ε < 1 / 2 , if x = ( a n ) n = 1 ∞ ∈ l 2 with d ( x , B l 2 ) > 1 / 12 , then exist 0 < δ < ϵ 2 3 , such that for every y = ( b n ) n = 1 ∞ ∈ B l 2 , when ‖ x − y ‖ 2 < d ( x , B l 2 ) + δ , we have ‖ x ‖ x ‖ − y ‖ 2 < ϵ .

Proof: In this proof, we simplified the l 2 norm ‖ ⋅ ‖ 2 by the symbol ‖ ⋅ ‖ .

Since ϵ 2 3 > δ , so

‖ x ‖ ϵ 2 > 3 ‖ x ‖ δ > 3 ( ‖ x ‖ − 1 ) δ = 2 ( ‖ x ‖ − 1 ) δ + ( ‖ x ‖ − 1 ) δ . (4)

If x = ( a n ) n = 1 ∞ ∈ l 2 with d ( x , B l 2 ) > 1 / 12 , then by (1)

‖ x ‖ − 1 = d ( x , B l 2 ) > 1 12 > δ , (5)

using (4) and (5),

‖ x ‖ ε 2 > 2 ( ‖ x ‖ − 1 ) δ + δ 2 .

Thus

( ‖ x ‖ − 1 ) 2 + ‖ x ‖ ϵ 2 > ( ‖ x ‖ − 1 ) 2 + 2 ( ‖ x ‖ − 1 ) δ + δ 2 = [ ( ‖ x ‖ − 1 ) + δ ] 2 . (6)

By (5) and (6), we get

[ d ( x , B l 2 ) + δ ] 2 = [ ( ‖ x ‖ − 1 ) + δ ] 2 < ( ‖ x ‖ − 1 ) 2 + ‖ x ‖ ϵ 2 . (7)

So for any y = ( b n ) n = 1 ∞ ∈ B l 2 , when ‖ x − y ‖ < d ( x , B l 2 ) + δ and using (7), we have

‖ x − y ‖ 2 < ( ‖ x ‖ − 1 ) 2 + ‖ x ‖ ϵ 2 , (8)

then we compute the l 2 norm by

‖ x ‖ 2 = ∑ n = 1 ∞ | a n | 2 , ‖ x − y ‖ 2 = ∑ n = 1 ∞ | a n − b n | 2 ,

thus according to (8), we have

∑ n = 1 ∞ a n 2 − 2 ∑ n = 1 ∞ a n b n + ∑ n = 1 ∞ b n 2 < ( ‖ x ‖ − 1 ) 2 + ‖ x ‖ ϵ 2 = ‖ x ‖ ϵ 2 − ( ‖ x ‖ − 1 ) + ‖ x ‖ 2 − ‖ x ‖ ≤ ‖ x ‖ ϵ 2 − ( ‖ x ‖ − 1 ) ∑ n = 1 ∞ b n 2 + ∑ n = 1 ∞ a n 2 − ∑ n = 1 ∞ a n 2 ‖ x ‖ .

The last inequality is because y ∈ B l 2 . Then we have

∑ n = 1 ∞ a n 2 ‖ x ‖ − 2 ∑ n = 1 ∞ a n b n + ‖ x ‖ ∑ n = 1 ∞ b n 2 < ‖ x ‖ ϵ 2 ,

which means ‖ x ‖ x ‖ − y ‖ 2 < ϵ 2 .

From the Lemma 2, let y ′ = x ‖ x ‖ , then { y ′ } = P B l 2 ( x ) and ‖ y − y ′ ‖ < ϵ , this means when x ∈ l 2 satisfied d ( x , B l 2 ) > 1 / 12 , there is a “uniformly” strongly

ball proximinal for these x. The next lemma is simple which is also needed in Theorem 1, but we give the proof for the completeness.

Lemma 3: Let X be a Banach space, for ( x n ) ∈ l ∞ ( X ) we have

d ( ( x n ) , B l ∞ ( X ) ) = sup { d ( x n , B X ) : n ∈ N } .

Proof: If x n ∈ B X , then d ( x n , B X ) = 0 . Thus we can assume for any n ∈ N , d ( x n , B X ) ≠ 0 .

Then x n ∈ X \ B X , so by (1)

d ( x n , B X ) = ‖ x n − x n ‖ x n ‖ ‖ = ‖ x n ‖ − 1.

Thus

d ( ( x n ) , B l ∞ ( X ) ) = inf { ‖ ( x n ) − ( y n ) ‖ l ∞ ( X ) : ( y n ) ∈ B l ∞ ( X ) } ≤ ‖ ( x n ) − ( x n ‖ x n ‖ ) ‖ l ∞ ( X ) = ‖ ( ‖ x n − x n ‖ x n ‖ ‖ ) ‖ ∞ = ‖ ( d ( x n , B X ) ) ‖ ∞ = sup { d ( x n , B X ) : n ∈ N } .

For another side, for any ( y n ) ∈ B l ∞ ( X ) , since d ( x n , B X ) ≤ ‖ x n − y n ‖ , thus

‖ ( d ( x n , B X ) ) ‖ ∞ ≤ ‖ ( ‖ x n − y n ‖ ) ‖ ∞ ,

by the arbitrary of ( y n ) ∈ B l ∞ ( X ) , we have

sup { d ( x n , B X ) : n ∈ N } ≤ d ( ( x n ) , B l ∞ ( X ) ) .

Now, we can give the proof of Theorem 1.

Theorem 1: Let X = l 2 , then l ∞ ( X ) is strongly ball proximinal.

Proof: For every 0 < ε < 1 / 2 , if ( x n ) ∈ l ∞ ( X ) with ‖ ( x n ) ‖ l ∞ ( X ) = r , without loss of generality, we can assume r = 2 , thus

d ( ( x n ) , B l ∞ ( X ) ) = sup { d ( x n , B X ) : n ∈ N } = 1. (9)

Then for all ( y n ) ∈ B l ∞ ( X ) , such that

‖ ( x n ) − ( y n ) ‖ l ∞ ( X ) = sup { ‖ x n − y n ‖ 2 : n ∈ N } < 1 + δ 2 8 , (10)

where the δ is same as the Lemma 2. From (9) and (10), we can see for any n ∈ N ,

d ( x n , B X ) ≤ 1 , ‖ x n − y n ‖ 2 < 1 + δ 2 8 ,

so we will divide into three cases to choose y ′ n ∈ B X so that ( y ′ n ) ∈ B l ∞ ( X ) and

‖ ( y n ) − ( y ′ n ) ‖ l ∞ ( X ) ≤ ε , ‖ ( x n ) − ( y ′ n ) ‖ l ∞ ( X ) ≤ 1. (11)

Case 1. ‖ x n − y n ‖ 2 ≤ 1 , it is simple to choose y ′ n = y n .

Case 2. ‖ x n − y n ‖ 2 > 1 and d ( x n , B X ) ≥ 1 − δ 2 .

Since δ < ϵ 2 3 < 1 12 , so d ( x n , B X ) ≥ 23 24 > 1 12 , then for this x n ∈ X , since

‖ y n − x n ‖ 2 < 1 + δ 2 8 ≤ d ( x n , B X ) + δ 2 + δ 2 8 < d ( x n , B X ) + δ .

Let y ′ n = x n ‖ x n ‖ 2 , then by the Lemma 2

‖ y n − y ′ n ‖ 2 < ϵ

and we also have

‖ y ′ n − x n ‖ 2 = ‖ x n ‖ 2 − 1 = d ( x n , B X ) ≤ 1.

Case 3. ‖ x n − y n ‖ 2 > 1 and d ( x n , B X ) < 1 − δ 2 .

Let y ′ n = ( 1 − δ 2 ) y n + δ 2 x n ‖ x n ‖ 2 , then y ′ n ∈ B X ,

‖ x n − y ′ n ‖ 2 ≤ ( 1 − δ 2 ) ‖ x n − y n ‖ 2 + δ 2 ‖ x n − x n ‖ x n ‖ 2 ‖ 2 ≤ ( 1 − δ 2 ) ( 1 + δ 2 8 ) + δ 2 d ( x n , B X ) = ( 1 − δ 2 ) ( 1 + δ 2 8 ) + δ 2 ( 1 − δ 2 ) ≤ 1

and we have

‖ y n − y ′ n ‖ 2 = ‖ δ 2 y n − δ 2 x n ‖ x n ‖ 2 ‖ 2 ≤ δ 2 + δ 2 = δ < ϵ .

Thus for any case, we can find the proper ( y ′ n ) ∈ B l ∞ ( X ) such that ( y ′ n ) meet the requirements of (11), which means l ∞ ( X ) is strongly ball proximinal.

Now we will show the uniformly ball proximinal of the equable subspace Y in Banach space X.

Theorem 2: Let Y be an equable subspace of X. Then Y is uniformly ball proximinal in X.

Proof: For any ϵ > 0 and R > 0 there exists η = R δ ( ϵ R ) ≜ R δ , where δ ( ϵ R ) is from the equability of Y which depends on ϵ R . Then for any x ∈ X ,

d ( x , B Y ) ≤ R . For any y ∈ B Y with ‖ x − y ‖ < R + η = R ( 1 + δ ) , we will show there is y ′ ∈ B Y such that

‖ x − y ′ ‖ ≤ R , ‖ y − y ′ ‖ ≤ ϵ . (12)

Note for the above fixed x and y, there is

x ∈ B ( y , R ( 1 + δ ) ) . (13)

Since Y is equable subspace of X, then Y is strongly ball proximinal by the above Remark 1, thus P B Y ( x ) ≠ ∅ . So we can choose y 1 ∈ P B Y ( x ) . Thus

‖ x − y 1 ‖ ≤ d ( x , B Y ) ≤ R . (14)

Therefore, by (13) and (14) we have

x ∈ B ( y 1 , R ) ∩ B ( y , R ( 1 + δ ) ) .

Let ϵ ′ = ϵ R , then using (3) in the Lemma 1, there is y ′ = R { Ψ ϵ ′ ( y − y 1 R ) + y 1 R } such that

x ∈ B ( y ′ , R ) ⇒ ‖ x − y ′ ‖ ≤ R (15)

Note, both y and y 1 are in B Y , thus y ′ ∈ B Y again by Lemma 1. Using the equability of Y and Lemma 1, it is easy to see

‖ y − y ′ R ‖ = ‖ y − y 1 R − Ψ ϵ ′ ( y − y 1 R ) ‖ ≤ ϵ ′ ,

thus we have

‖ y − y ′ ‖ ≤ R ϵ ′ ≤ ϵ . (16)

According to (15) and (16), we have found the proper y ′ to satisfy (12). Thus we complete the proof.

In this paper, we can see for these x ∈ l 2 with d ( x , B l 2 ) > 1 12 , they have the common δ for strong ball proximinality, then we can get the strong ball proximinality of l ∞ ( l 2 ) . Also, we give an example of uniform ball proximinality. That is the equable subspace Y of a Banach space X.

This work is supported by Huaqiao University High-level Talents Research Initiative Project (11BS220). The authors would also like to thank the Editor-in-Chief, the Associate Editor, and the anonymous reviewers for their careful reading of the manuscript and constructive comments.

The authors declare no conflicts of interest regarding the publication of this paper.

Wang, B. and Liu, X.Y. (2020) A Note on Ball Proximinality. Applied Mathematics, 11, 1196-1203. https://doi.org/10.4236/am.2020.1111081