Journal of Function Spaces

1. Notations and Definitions

Let be a Banach space and let be its dual space. Let us denote by and the closed unit ball and the unit sphere of , and by the closed ball centered at of radius . Let . Let us denote by the set of all norm-attaining functionals in and .

For a subset , the metric projection is defined by , where . If for each , then is said to be proximinal. If is at most a singleton for each , then is said to be semi-Chebyshev. If is a simultaneously proximinal and semi-Chebyshev set, then is said to be a Chebyshev set.

Definition 1.1 (see [1]). One says that is strongly convex (resp., very convex/nearly strongly convex/nearly very convex) if for any and convergence as for some implies that as (resp., as is relatively compact/ is weakly relatively compact).

Definition 1.2 (see [2]). One says that is midpoint locally uniformly rotund (MLUR) (resp., weakly midpoint locally uniformly rotund (WMLUR)) if any , , , , as implies that as (resp., as ).

Wu and Li defined strong convexity in [3], and Wang and Zhang in [4] defined very convexity, nearly strong convexity, and nearly very convexity which are two generalizations of locally uniformly rotund (LUR) and weakly locally uniformly rotund (WLUR) spaces. By [3, 5], we have the following relations:

Sullivan defined very rotund space in [6]. A Banach space is said to be very rotund if no is simultaneously a norming element for some and , where . Z. H. Zhang and C. J. Zhang proved that very rotund space coincides with very convex space in [5]. In [3–5, 7], many results of these four classes of convexities were proved. In particular, Zhang and Shi proved that they have important applications in approximation theory in [1]. In [8], Bandyapadhyay et al. also proposed two generalizations of locally uniformly rotund space, which are called almost locally uniformly rotund space and weakly almost locally uniformly rotund space. A Banach space is said to be ALUR (resp., WALUR) if for any , , and , the condition implies (resp., ). Many properties of these two classes of convexities were studied in [8–10] too. Recently, we proved that almost locally uniformly rotund space is equivalent to strongly convex space and that weakly almost locally uniformly rotund space is equivalent to very convex space [7]. Thus, we unified the results of the studies about the strongly convex space (resp., very convex space) and the almost locally uniform rotundity (resp., weakly almost locally uniform rotundity). This shows that these convexities have important effects on and applications in geometry of Banach space and approximation theory.

A sequence is said to be minimizing for if as .

Definition 1.3 (see [10]). Let be a closed subset (resp., a weakly closed subset) of and .(1)One says that is approximatively (resp., weakly approximatively) compact for if every minimizing sequence for has a convergent (resp., weakly convergent) subsequence.(2)One says that is strongly Chebyshev (resp., weakly strongly Chebyshev) for if every minimizing sequence for is convergent (resp., weakly convergent).

If is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) for every , we say that is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) in .

Definition 1.4 (see [11]). Let , , and .(1) is said to be strongly unique at if there exists a constant such that for any .(2) is said to be strongly unique of order at if there exists an such that for any .(3) is said to be bounded strongly unique of order at if given any , there exists an such that for any .(4) is said to be locally strongly unique of order at if there exist and such that for any .

In order to study the uniqueness of best approximation in nonlinear approximation theory, Wulbert [12] defined the strong uniqueness of best approximation. Smarzewski [13] and Schmidt [14] defined the strongly unique of order and the bounded strongly unique of order , respectively. By [11], we know that the strongly unique of order and the bounded strongly unique of order all are generalizations of the strongly unique. The strongly unique of order and the bounded strongly unique of order imply the locally strongly unique of order , but converse implied relation is not generally true. When , the strongly unique of order and the bounded strongly unique of order all are strongly unique. The locally strongly unique of order 1 is not equivalent to the strongly unique.

Definition 1.5 (see [10]). For and , let one define the following maps: is called the duality map and is called the preduality map. Naturally, is defined only on .

Definition 1.6 (see [10]). The preduality map is said to be upper semicontinuous on if for any and any open set with , there exists such that whenever , where stands for norm or weak topology.

2. Convexity and Continuity of the Preduality Map

Using the Bronsted-Rockafeller Theorem (see [15, Theorem  3.18, page 51]), we can prove the following lemma.

Lemma 2.1. Suppose that , , , and , then there are and such that

Theorem 2.2. Let be a Banach space. is strongly convex if and only if the preduality map is singlevalued and continuous.

Proof. Necessity. Since strong convexity implies strict convexity, is singlevalued. Suppose that , with , then there exist and such that . We have that Since and , we have that Since is strongly convex, we deduce that as , that is, as . This means that is continuous.
Sufficiency. Let with as for some . Since is singlevalued, by Lemma 2.1, there exist and such that and Since is continuous, we have that as , that is, as . So as , which means that is strongly convex.

Using Lemma 2.1, in a similar way to prove Theorem 2.2, we can prove the following result.

Theorem 2.3. Let be a Banach space. is very convex if and only if the preduality map is singlevalued and weakly continuous.

Lemma 2.4. Let be a convex set of a strongly convex Banach space . The following are equivalent:(1) is proximinal;(2) is weakly approximatively compact;(3) is approximatively compact;(4) is strongly Chebyshev.

Proof. We only need to prove .
Let , such that In order to finish the proof, we will show that there exists some such that as .
Step 1. If , then and where . By the separation theorem [2] and definition of norm, there exists an such that for any . Hence, we have that This shows that . From , we get . Combining it with the condition it follows that as . Hence, as . Since is strongly convex, as , that is, as .Step 2. If , we set . It is clear that is proximinal and is a minimizing sequence for 0. By Step 1, there exists , such that as . This shows that as and .

Similarly to the proof of Lemma 2.4, we can prove the following result.

Lemma 2.5. Let be a convex set of a nearly strongly convex Banach space . The following are equivalent:(1) is proximinal;(2) is weakly approximatively compact;(3) is approximatively compact.

Lemma 2.6. Let , , then hyperplane is a proximinal convex subset in .

Proof. Let We will prove that is a proximinal convex subset in . For any , if , since is a closed subspace, we know that . Hence, . If , since , there exist and such that . For any , , and , we have that and . Therefore, we have that This means that . Hence, we have It follows that , which means that is proximinal set.
Furthermore, we will prove that hyperplane is a proximinal convex subset in . For , . For any , Since is a proximinal subset, there exists a such that Therefore, , which means that is proximinal set in .

Theorem 2.7. Let be a Banach space. is a nearly strongly convex if and only if the preduality map is upper semicontinuous on with norm compact images, where stands for norm topology.

Proof. Necessity. Arbitrarily take . Then by Lemma 2.6, we know that is a proximinal convex subset in . Since is nearly strongly convex, by Lemma 2.5, we know that is approximatively compact. Suppose that is not upper semicontinuous at , then for some open set in with , there exists such that as and for all . Let . Fix . Let , then is a minimizing sequence for . Since is approximatively compact, has a convergent subsequence. So has convergent subsequence converging to . Thus, , but is closed, which is a contradiction. By the assumption, we easily know that the image of is compact.
Sufficiency. Let with as for some . By Lemma 2.1, there exist and such that and Since is compact, is proximinal. Let such that . In virtue of being upper semicontinuous on and , for any , there is such that for all , which means that as . Combining this with the compactness of has convergent subsequence, and hence has convergent subsequence. By as , has convergent subsequence, which means that is nearly strongly convex.

Theorem 2.8. Let be a Banach space. is nearly very convex if and only if the preduality map is upper semicontinuous on with weakly compact images.

Proof. In a similar way to the proof of Theorem 2.7, the necessity can be proved.
Sufficiency. Let with as for some . By Lemma 2.1, there are and such that and Suppose that does not have weakly convergent subsequence. By (2.17), we know that does not have weakly convergent subsequence. Without loss of generality, we can assume that for some and . Since is weakly compact, there exists , such that . Combining upper semicontinuity () of with for some . So, there are subsequence and such that . Hence, a contradiction with (2.18).

3. Convexity and Proximinality

Theorem 3.1. Let be a weakly approximatively compact subset of nearly strongly convex Banach space , , and . If is the unique element of best approximation of , then the following are equivalent:(1) is strongly unique of order at ;(2) is bounded strongly unique of order at ;(3) is locally strongly unique of order at .

Proof. . Let be such that for any . For any and each , since by Lagrange’s mean value theorem, Set Then Because is independent of is bounded strongly unique of order at .
. From we can take such that Again, by Lagrange’s mean value theorem, we have Therefore, for any and . Set , then for any . That shows is strongly unique of order at .
Now, we only need to prove . Let such that for all and .
If the condition is not true, there exist , with such that then, because is bounded, that is, . This shows that is a minimizing sequence for . Since is weakly approximatively compact, there is a weakly convergent subsequence of . Without loss of generality, we can assume that Based on weak lower semi-continuity of norm, we get that Hence, , consequently, . Take , then Since is nearly strongly convex, there exists such that , that is, . It follows that . This is a contradiction with the fact that .

Corollary 3.2. Let be a weakly approximatively compact subset of strongly convex Banach space , , and , then the following are equivalent:(1) is strongly unique of order at ;(2) is bounded strongly unique of order at ;(3) is locally strongly unique of order at .

If is a convex set, we have the following result.

Theorem 3.3. Let be a convex subset of a nearly strongly convex Banach space , , and . If is the unique element of best approximation of , then the following are equivalent:(1) is strongly unique of order at ;(2) is bounded strongly unique of order at ;(3) is locally strongly unique of order at .

Proof. By the proof of Theorem 3.1, we have . Now, we only need to prove .
If the condition is not true, there exist with such that In the same way of the proof of in Theorem 3.1, we can also prove that is a minimizing sequence for . By Lemma 2.5, is approximatively compact. Hence, there exists a convergent subsequence of . Without loss of generality, we can assume that Consequently, , but , which is a contradiction.

Corollary 3.4. Let be a convex set of a strongly convex Banach space . and . The following are equivalent:(1) is strongly unique of order at ;(2) is bounded strongly unique of order at ;(3) is locally strongly unique of order at .

Acknowledgment

The authors very much appreciated the reviewer’s suggestions for the revision of this paper.