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)  - . When people focus on the proximinal subspace, they find the (strong) proximinlality of unit ball BY of a subspace Y is stronger than the (strong) proximinality in Y. In , Saidi showed for any nonreflexive space Y there is a Banach space X such that Y is isometrically isomorphic to a subspace Z of X such that Z is proximinal in X but not ball proximinal in X. ( , Example 3.3) showed that the space Z is strongly proximinal in X. Based on these results, the notion of ball proximinality, which focuses the problem of best approximation from linear subspaces on non-linear convex sets was introduced in 2007 in the paper  by Bandyopadhyay et al. Later, this new concept has been extensively studied  - . To characterize ball proximinal and strongly ball proximinal hyperplanes, Indumathi and Prakash  introduced the so called E-proximinality. Then Lin et al.  generalize those results from E-proximinal hyperplanes to E-proximinal subspaces. Lalithambigai  study the ball proximinality of equable spaces and prove an equable subspace is strongly ball proximinal.
In general, there are a few results about stability of the ball proximinality. Firstly, Bandyopadhyay et al.  showed if ( ) and ( ) are two sequences of Banach spaces such that is a subspace of that is ball proximinal in for each n, then the -direct sum is ball proximinal in . Paul  showed stability of ball proximinality and strongly ball proximinality in spaces of Bochner integrable functions. Then, fruitful results about ball proximinality and strong ball proximinality were obtained in . For example, it has been proved if E is a Banach space with a uniformly monotone 1-unconditional basis (e.g. for ) or E is , then is strongly ball proximinal in , where is a subspace of that is strongly ball proximinal in for each n.
For , 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 and to find the proper conditions for a Banach space X such that the unit ball of is strongly proximinal. In this paper, we can see for , is strongly proximinal because for these with , they have the common for strong ball proximinality, then we can get the strong ball proximinality of . Paul  developed the notion of “uniform proximinality” of a closed convex set in a Banach space and gave some examples to have this property. Also, we can give another example. That is motivated by the proof in , we show that equable subspace Y of a Banach space X is uniform ball proximinality.
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 and . For and , we set .
Let C be a nonempty closed convex subset of X. For any and , denote the sets:
where is the distance of x to C, that is .
Definition 1  :
1) A subset C is said to be proximinal if for every x in X, the set .
2) A subset C is said to be strongly proximinal if for any and any , there exists such that for any with , then there is with and .
3) A subset C is said to be uniformly proximinal if for any and , there exists such that for any , and any with , then there is with and .
From the Definition 1, we can see uniformly proximinal strongly proximinal proximinal. For any Banach space X, it is easy to see is
proximinal. Since for any , and
But, from the example by Godefroy in ( , Pg. 87) it is clear that the closed unit ball of a Banach space not necessarily have strongly proximinal property.
Definition 2  : Let X be a Banach space,
1) X is said to be strongly ball proximinal if the unit ball BX is strongly proximinal.
2) X is said to be uniformly ball proximinal if the unit ball BX is uniformly proximinal.
Definition 3 : Let X be a Banach space and Y be subspace of X. We say Y is an equable subspace of X if for every there is a and a map such that for every , and
Remark 1: In Theorem 2.6 , it has been proved if Y is an equable subspace of X. Then Y is strongly ball proximinal in X.
Let , we give the next lemma which is the remark 2.3 in  by using translation invariance of the Banach space and (2) in Definition 3.
Lemma 1 : Let Y be an equable subspace of a Banach space X, for any , there is such that for any real scalar , y and z in Y, there is
with , then
Additionally, if both y and z are in , then .
Next, to avoid confusion, we use or for some real numbers, or for the vectors in Banach space.
Let , is the Banach space of all sequences of real so that . For , is the Banach space of sequ- ences such that .
Let ( ) be a sequence of Banach spaces. For , -direct sum denote the collection of elements ( ) such that and the sequence . Thus the norm of ( ) is
If for any n, , we can simply denote by .
3. Main Results
In this section, we will give our main results. For Theorem 1, we can see is strongly ball proximinal. This result is using the “uniformly” strongly ball proximinal of the which is showed by Lemma 2. For Theorem 2, we prove when Y is an equable subspace in Banach space X, BY is uniformly proximinal.
Lemma 2: For every , if with , then exist , such that for every , when , we have .
Proof: In this proof, we simplified the norm by the symbol .
Since , so
If with , then by (1)
using (4) and (5),
By (5) and (6), we get
So for any , when and using (7), we have
then we compute the norm by
thus according to (8), we have
The last inequality is because . Then we have
which means .
From the Lemma 2, let , then and , this means when satisfied , 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 we have
Proof: If , then . Thus we can assume for any , .
Then , so by (1)
For another side, for any , since , thus
by the arbitrary of , we have
Now, we can give the proof of Theorem 1.
Theorem 1: Let , then is strongly ball proximinal.
Proof: For every , if with , without loss of generality, we can assume , thus
Then for all , such that
where the is same as the Lemma 2. From (9) and (10), we can see for any ,
so we will divide into three cases to choose so that and
Case 1. , it is simple to choose .
Case 2. and .
Since , so , then for this , since
Let , then by the Lemma 2
and we also have
Case 3. and .
Let , then ,
and we have
Thus for any case, we can find the proper such that meet the requirements of (11), which means 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 and there exists , where is from the equability of Y which depends on . Then for any ,
. For any with , we will show there is such that
Note for the above fixed x and y, there is
Since Y is equable subspace of X, then Y is strongly ball proximinal by the above Remark 1, thus . So we can choose . Thus
Therefore, by (13) and (14) we have
Let , then using (3) in the Lemma 1, there is such that
Note, both y and are in , thus again by Lemma 1. Using the equability of Y and Lemma 1, it is easy to see
thus we have
According to (15) and (16), we have found the proper to satisfy (12). Thus we complete the proof.
In this paper, we can see for these with , they have the common for strong ball proximinality, then we can get the strong ball proximinality of . 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.
 Bandyopadhyay, P., Li, Y., Lin, B.-L. and Naraguna, D. (2008) Proximinality in Banach Spaces. Journal of Mathematical Analysis and Applications, 341, 309-317.
 Luo, X.-F., Tao, J. and Wei, M. (2019) Characterizations of Generalized Proximinal Subspaces in Real Banach Spaces. Results in Mathematics, 74, Paper No. 88.
 Cheng, L.X., Luo, Z.H., Zhang, W. and Zheng, B.T. (2016) On Proximinality of Convex Sets in Superspaces. Acta Mathematica Sinica (Engl. Ser.), 32, 633-642.
 Saidi, F.B. (2005) On the Proximinality of the Unit Ball of Proximinal Subspaces in Banach Spaces; a Counterexample. Proceedings of the American Mathematical Society, 133, 2697-2703.