# Approximative Compactness and Radon-Nikodym Property inw∗ Nearly Dentable Banach Spaces and Applications

Authors: Shaoqiang Shang; Yunan Cui
Journal: Journal of Function Spaces
 (2015)

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## Abstract

Authors definew∗ nearly dentable Banach space. Authors study Radon-Nikodym property, approximative compactness and continuity metric projector operator in w∗ nearly dentable space. Moreover, authors obtain some examples of w∗ nearly dentable space in Orlicz function spaces. Finally, by the method of geometry of Banach spaces, authors give important applications of w∗ nearly dentability in generalized inverse theory of Banach space.


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## Body

## 1. Introduction

Let(X,∥·∥) be a real Banach space. S(X) and B(X) denote the unit sphere and the unit ball, respectively. By X∗ we denote the dual space of X. Let N,R, and R+ denote the set of natural numbers, reals, and nonnegative reals, respectively. A(x)={x∗:x∗∈S(X∗),x∗(x)=1=x} and Br(x)={y∈X:y-x<r}. By xn→wx  denote that {xn}n=1∞ is weakly convergent to x. xn∗→w∗x∗ denote that {xn∗}n=1∞ is weakly∗ convergent to x∗. The weak topology of X is denoted by σ(X∗,X) or (X,w). The weak∗ topology of X∗ is denoted by σ(X,X∗) or (X∗,w∗). C(Cw¯,Cw∗¯) denote closed hull of C (weak closed hull, closed hull weak∗ closed hull). dist(x,C) denote distance of x and C.The study of geometric and topological properties of unit balls of Banach spaces plays a central rule in the geometry of Banach spaces. Almost all properties of Banach spaces, such as convexity, smoothness, nonsquareness, and the Radon-Nikodym property, can be viewed as properties of the unit ball. We should mention here that there are many topics studying behavior of ball collections. For example, the Mazur intersection property, the packing sphere problem of unit balls, the measure of noncompactness with respect to topological degree, and the ball topology have also brought great attention of many mathematicians. In 2006, Cheng defined ball-covering property of Banach space (see [1]). In [2], Chen and Cheng proved analytical characterizations of Mazur intersection property.Starting with a different viewpoint, a notion of nearly dentable space is introduced by Shang et al. in [3].Definition 1.
Banach spaceX is called nearly dentable if, for any f∈S(X∗) and any open set UAf⊃Af, we have Af≠∅ and Af∩co¯(B(X)∖UAf)=∅, where Af={x:f(x)=x=f=1}.By James theorem, it is easy to see that nearly dentable space is reflexive.Definition 2.
A nonempty subsetC of X is said to be approximatively compact if for any {yn}n=1∞⊂C and any x∈X satisfying x-yn→inf⁡y∈Cx-y(n→∞), then {yn}n=1∞ has a subsequence converging to an element in C. X is called approximatively compact if every nonempty closed convex subset of X is approximatively compact.Let us present the history of the approximative compactness and related notions. This notion has been introduced by Jefimow and Stechkin in [4] as a property of Banach spaces, which guarantees the existence of the best approximation element in a nonempty closed convex set C for any x∈X. In [3] it was established (without proof) that the approximative compactness of C in a rotund Banach space X gives also the continuity of the projector operator PC. Fully rotund Banach spaces were defined by Fan and Glicksberg in [5]. In [6] it has been proved that fully rotund Banach spaces are approximately compact. In [7] it has been proved that approximative compactness of a Banach space X means that X is reflexive and it has the Kadec-Klee property (called also the property H). In 2007, Chen et al. (see [8]) proved that a nonempty closed convex C of a midpoint locally uniformly rotund space is approximately compact if and only if C is a Chebyshev set and the metric projection operator PC is continuous. In 1972, Oshman (see [9]) proved that the metric projection operator PC is upper semicontinuous if C is approximatively compact subset of a Banach space X.LetD be a nonempty open convex subset of X and f a real-valued continuous convex function on D. f is said to be Gateaux differentiable at the point x in D if the limit ∗f′xy=limt→0⁡fx+ty-fxtexists for all y∈X. When this is the case, the limit is a continuous linear function of y, denoted by f′(x). If the difference quotient in ∗ converges to f′(x)(y) uniformly for y in the unit ball, then f is said to be Fréchet differentiable at x. X is called a weak Asplund space (Asplund space) or said to have the weak Asplund property if, for every f and D as above, f is “generically” Gateaux (Frechet) differentiable; that is, there exists a dense Gδ subset G of D such that f is Gateaux (Frechet) differentiable at each point of G.The paper is organized as follows. In Section1, authors define w∗ nearly dentable Banach space and some necessary definitions and notations are collected. In Section 2, authors prove that if X∗ is a w∗ nearly dentable space, then X is an Asplund space if and only if AE(x)={x∗∈S(E∗):x∗(x)=1=x} is a separable subset of E∗, where E is a separable closed subspace of X. In Section 3, authors prove that the following statements are equivalent: (1) every weak∗ hyperplane of X∗ is approximatively compact; (2) if C is a weak∗ open convex set, then Cw∗¯ is approximatively compact; (3)  X∗ is w∗ nearly dentable space and A(x) is compact for any x∈S(X). Moreover, authors also prove that if X∗ is w∗ nearly dentable space, then (1) for any weak∗ open convex set C metric projector operator PC¯w∗ is upper semicontinuous; (2) For any weak∗ hyperplane D∗ metric projector operator PD∗ is upper semicontinuous. In Section 4, authors prove that there exists a Banach space X such that X∗ is not w∗ nearly dentable Banach space. Moreover, authors also prove that there exists a Banach space X such that X∗ is w∗ nearly dentable Banach space and not nearly dentable Banach space. The main results in this paper have important applications to the ill-posed equation theory and the generalized inverse theory. In Section 5, authors give an important application of w∗ nearly dentability in generalized inverse theory of Banach space.Definition 3.
X
∗ is called w∗ nearly dentable space, if, for any x∈S(X) and open set UA(x)⊃A(x), we have A(x)∩co¯w∗(B(X∗)∖UA(x))=∅.Let us recall some definitions and some lemmas which will be used in the further part of the paper.Definition 4 (see [8]).
A subsetC is said to be proximinal if PC(x)≠∅ for all x∈X. C is said to be semi-Chebyshev if PC(x) is at most a singleton for all x∈X. C is said to be Chebyshev if it is proximinal and semi-Chebyshev.Definition 5.
A Banach spaceX is said to have the Radon-Nikodym property whenever if (T,Σ,μ) is a nonatomic measure space and v is a vector measure on Σ with values in X which is absolutely continuous with respect to μ and has bounded variation; then there exists f∈L1(X) such that, for any A∈Σ, (1)vA=∫Aftdt.It is well known that X is reflexive if and only if each closed convex subset of X is proximinal. It is well known that X is Asplund space if and only if X∗ has the Radon-Nikodym property.Definition 6 (see [10]).
Set-valued mappingF:X→Y is called upper semicontinuous at x0, if, for each norm open set W with F(x0)⊂W, there exists a norm neighborhood U of x0 such that F(x)⊂W for all x in U. F is called lower continuous at x0, if, for any y∈F(x0) and any {xn}n=1∞ in X with xn→x0, there exists yn∈F(xn) such that yn→y as n→∞. F is called continuous at x0, if F is upper semicontinuous and is lower continuous at x0.Definition 7.
H
∗
⊂
X
∗ is called weak∗ hyperplane, if there exist x∈S(X) and λ∈R such that H∗={x∗∈X∗:x∗(x)=λ}.Lemma 8 (see [11]).
Suppose thatX is a Banach space. Then the following statements are equivalent:(1)
Xis Asplund space.(2)
IfE is a separable closed subspace of X, then E∗ is a separable space.(3)
X∗ has the Radon-Nikodym property.Lemma 9 (see [12]).
LetB⊂S(X∗) and B∩A(x) be not an emptyset for all x∈S(X). Assume that B⊂co¯F+αB(X∗) for some countable set F⊂X∗ and some 0<α<1. Then X∗ is the closed linear span of F.


## 2.w∗ Nearly Dentability and Radon-Nikodym Property in Banach Space

Theorem 10.
Suppose thatX∗ is a w∗ nearly dentable space. Then the following statements are equivalent:(1)
X is a Asplund space.(2)
X∗ has the Radon-Nikodym property.(3)
IfE is a separable closed subspace of X, then E∗ is a separable space.(4)
IfE is a separable closed subspace of X, then AE(x)={x∗∈S(E∗):x∗(x)=1=x} is a separable subset of E∗.Proof.
(
4
)
⇒
(
1
) Let E be a separable closed subspace of X and {xn}n=1∞ be a dense subset of S(E). Then, for any x∈S(E), we have (2)inf⁡⁡dist⁡AExn,AEx:n∈N=0.In fact, suppose that inf⁡{dist⁡(AE(xn),AE(x)):n∈N}>0. Then, for any ε>0, there exist x∗∈A(x) and xi∈{xn}n=1∞ such that inf⁡{dist⁡(A(xn),A(x)):n∈N}+ε>xi∗-x∗, where xi∗∈A(xi). This implies that(3)inf⁡⁡dist⁡Axn,Ax:n∈N+ε>xi∗-x∗≥xi∗-x∗E≥xi∗E-x∗E≥inf⁡⁡dist⁡AExn,AEx:n∈N.Since ε is arbitrary, we obtain that(4)inf⁡⁡dist⁡Axn,Ax:n∈N≥inf⁡⁡dist⁡AExn,AEx:n∈N>0.This implies that there exists r>0 such that inf⁡{dist⁡(A(xn),A(x)):n∈N}≥2r. Hence we define an open set (5)UAx=⋃x∗∈Axy∗∈X∗:y∗-x∗<r.Since X∗ is a w∗ nearly dentable space, we have A(x)∩co¯w∗(B(X∗)∖UA(x))=∅. Since co¯w∗(B(X∗)∖UA(x)) is a weak∗ compact set, by the separation theorem of locally convex space, there exists x∈S(X) such that (6)inf⁡⁡xx∗:x∗∈Ax>sup⁡⁡xx∗:x∗∈co¯w∗BX∗∖UAx.Since the sequence {xn}n=1∞ is a dense subset of S(E), we may assume without loss of generality that xn→x as n→∞. Pick xn∗∈A(xn). Then (7)xn∗x-1=xn∗x-xn∗xn≤xn∗·xn-x⟶0asn⟶∞.Moreover, we may assume without loss of generality that if m≠n, then xn∗≠xm∗. Since B(X∗) is weak∗ compact, there exists x0∗∈B(X∗) such that x0∗ is a weak∗ accumulation point of {xn∗}n=1∞. This implies that x0∗(x)=1. Then x0∗∈A(x). Therefore, by formula (6), we obtain that (8)xx0∗>sup⁡⁡xx∗:x∗∈co¯w∗BX∗∖UAx.Since A(xn)⊂co¯w∗(B(X∗)∖UA(x)) and x0∗∈A(x), by formula (6), we have x(x0∗)>sup⁡{x(x∗):x∗∈{xα∗}}, which contradicts xα∗→w∗x0∗. Hence formula (2) is true. Since AE(x) is a separable subset of E∗, we obtain that there exists a countable dense subset Dn of AE(xn) for each n∈N. Then D=∪n=1∞Dn is a countable set. Pick α∈(0,1). Therefore, by formula (2), we obtain that there exist α(x)∈AE(x) and n(x)∈N such that dist⁡(AE(xn(x)),α(x))<α for any x∈S(E). Hence, for every x∈S(E), we obtain that (9)B=αx:x∈SE⊂SE∗,B∩AEx≠∅for any x∈S(E). Since dist⁡(AE(xn(x)),α(x))<α and Dn(x) is a countable dense subset of AE(xn(x)), we obtain that(10)dist⁡αx,D≤dist⁡αx,Dnx=dist⁡αx,AExnx<αfor any x∈S(E). This implies that (11)αx∈D+αBE∗⊂co¯D+αBE∗for any x∈S(E). Noticing that B={α(x):x∈S(E)}, we have (12)B⊂D+αBE∗⊂co¯D+αBE∗.By Lemma 9, we obtain that E∗=span⁡¯(D), and so E∗ is separable.
This means that ifE is a separable closed subspace of X, then E∗ is a separable space. By Lemma 15, we obtain that X is Asplund space and X∗ has the Radon-Nikodym property.
(
1
)
⇒
(
4
). Since X∗ has the Radon-Nikodym property, we obtain that if E is a separable closed subspace of X, then E∗ is a separable space. Hence AE(x)={x∗∈S(E∗):x∗(x)=1=x} is a separable subset of E∗.
Therefore, by Lemma8, we obtain that Theorem 10 is true, which completes the proof.


## 3.w∗ Nearly Dentability and Approximative Compactness and Continuity of Metric Projector Operator in Banach Spaces

Theorem 11.
Suppose thatX is a Banach space. Then the following statements are equivalent:(1)
Everyweak∗ hyperplane of X∗ is approximatively compact.(2)
IfC⊂X∗ is a weak∗ open convex set, then Cw∗¯ is approximatively compact.(3)
X∗ is w∗ nearly dentable space and A(x) is compact for any x∈S(X).Proof.
(
3
)
⇒
(
2
). (a) Let xn∗(x0)→1 as n→∞, where xn∗∈S(X∗) and x0∈S(X). Since B(X∗) is weak∗ compact, there exists x0∗∈B(X∗) such that x0∗ is a weak∗ accumulation point of {xn∗}n=1∞. Hence there exists a net {xα∗}⊂{xn∗}n=1∞ such that xα∗→w∗x0∗ and x0∗∈A(x0). We next will prove that the sequence {xn∗}n=1∞ is relatively compact. The proof requires considering few cases separately.
Case
I
(
{
x
n
∗
}
n
=
1
∞
∩
A
(
x
0
)
=
∅
). Suppose that {xn∗}n=1∞ is not relatively compact. Then there exists a subsequence {xn∗}n=1∞ denoted again by {xn∗}n=1∞ such that {xn∗}n=1∞ does not contain convergent subsequence. This implies that every point of A(x0) is not accumulation point of {xn∗}n=1∞. It is easy to see that for any x∗∈A(x0), there exists εx∗>0 such that {xn∗}n=1∞∩{y∗:y∗-x∗<εx∗}=∅. Put (13)UAx0=⋃x∗∈Ax0y∗:y∗-x∗<εx∗.Then, it is easy to see that UA(x0)∩{xn∗}n=1∞=∅. Moreover, since B(X∗) is a weak∗ compact convex set, we obtain that co¯w∗(B(X∗)∖UA(x0)) is a weak∗ compact convex set. Since X∗ is a w∗ nearly dentable space, we obtain that A(x0)∩co¯w∗B(X∗)∖UA(x0)=∅. By separation theorem, there exist y∈X and r>0 such that (14)inf⁡⁡x∗y:x∗∈Ax0-r≥sup⁡⁡x∗y:x∗∈co¯w∗BX∗∖UAx0.From the previous proof, there exists {xα∗}⊂{xn∗}n=1∞ such that xα∗→w∗x0∗∈A(x0). By {xn∗}n=1∞∩UA(x0)=∅, we have (15)xα∗⊂xn∗n=1∞⊂BX∗∖UAx0⊂co¯w∗BX∗∖UAx0.Therefore, by formulas (14) and (15), we obtain that x0∗(y)-r≥sup⁡{x∗(y):x∗∈{xα∗}}. This implies that xα∗→w∗x0∗∈A(x0) is impossible, a contradiction. Hence {xn∗}n=1∞ is relatively compact.
Case
II
(
{
x
n
∗
}
n
=
1
∞
∩
A
(
x
0
)
≠
∅
). From the previous proof, {xn∗}n=1∞∖A(x0) is relatively compact. Since A(x0) is compact, we obtain that {xn∗}n=1∞∩A(x0) is relatively compact. Noticing that {xn∗}n=1∞={xn∗}n=1∞∩A(x0)∪{xn∗}n=1∞∖A(x0), we obtain that {xn∗}n=1∞ is relatively compact.
LetC be a weak∗ open convex set and H∗=Cw∗¯. Then, for any x∗∉H∗. we may assume without loss of generality that x∗=0. We next will prove that if 0-yn∗→dist⁡(x∗,H∗) as n→∞, then the sequence {yn∗}n=1∞ has a subsequence converging to an element in H∗, where {yn∗}n=1∞⊂H∗. Since H∗ is a weak∗ closed set, by proof of theorem 2.1 of [13], we obtain that H∗ is a proximinal set and dist⁡(0,H∗)=r>0. Pick y0∗∈PH∗(0). Then Br(0)¯∩C=∅. In fact, suppose that Br(0)¯∩C≠∅. Then there exist z∗∈Br(0)¯∩C and δ>0 such that B2δ(z∗)⊂C and z∗>4δ. Since (16)z∗-δz∗z∗-z∗=1-z∗-δz∗z∗=δ,we obtain that (17)z∗-δz∗z∗∈C⊂H∗,dist⁡0,H∗≤0-z∗-δz∗z∗<z∗≤r,a contradiction. Hence(18)r=dist⁡0,PH∗0=y0∗,Br0¯∩C=∅,Br0¯∩H∗=PH∗0.Noticing that (X∗,w∗)∗=X and separation theorem of locally convex space, there exists x∈S(X) such that (19)sup⁡⁡xy∗:y∗∈C≤inf⁡⁡xy∗:y∗∈Br0¯=Br0w∗¯=-x·y0∗.This implies that (20)sup⁡⁡xy∗:y∗∈H∗≤inf⁡⁡xy∗:y∗∈Br0=-x·y0∗.Since y0∗∈PH∗(0)⊂H∗, we obtain that (21)-x·y0∗≤xy0∗≤sup⁡⁡xy∗:y∗∈H∗≤inf⁡⁡xy∗:y∗∈Br0=-x·y0∗,-x·y0∗=xy0∗=sup⁡xy∗:y∗∈H∗.This means that the inequality x(y0∗)≥x(yn∗) holds. Therefore(22)0-y0∗=x0-y0∗≤x0-yn∗≤0-yn∗⟶dist⁡0,H∗=0-y0∗=y0∗.Hence yn∗→y0∗ and x(0-yn∗)→0-y0∗ as n→∞. Furthermore, we have(23)x-yn∗yn∗+yn∗y0∗=1y0∗-1yn∗·xyn∗⟶0asn⟶∞,which shows that x(-yn∗/yn∗)→1asn→∞. From the previous proof, we obtain that the sequence {-yn∗/yn∗}n=1∞ is relatively compact. Noticing that yn∗→y0∗ as n→∞, we obtain that the sequence {yn∗}n=1∞ is relatively compact. This implies that the sequence {yn∗}n=1∞ has a subsequence converging to an element in H∗. Hence we obtain that H∗ is approximatively compact.
(
2
)
⇒
(
1
). Let D∗={x∗∈X∗:x∗(x0)=1} be a weak∗ hyperplane. Then it is easy to see that D1∗=x∗∈X∗:x∗x0≥1 and D2∗={x∗∈X∗:x∗(x0)≤1} are weak∗ closed convex sets. Moreover, it is easy to see that if y∗(x0)>1, then PD∗(y∗)=PD2∗(y∗) and if y∗(x0)<1, then PD∗(y∗)=PD1∗(y∗). Since D1∗ and D2∗ are approximatively compact, we obtain that D∗ is approximatively compact.
(
1
)
⇒
(
3
). For any x∈S(X), we define a weak∗ hyperplane H∗={x∗:x∗(x)=x∗=1}. Let {yn∗}n=1∞⊂A(x). Then 0-yn∗=dist⁡(0,H∗)=1. Hence {yn∗}n=1∞ has a subsequence converging to y0∗∈H∗. Moreover, it is easy to see that A(x) is a closed set. Hence A(x) is compact. We claim that(24)r=rx,UAx=inf⁡⁡dist⁡x∗,Ax:x∗∈BX∗∖UAx>0for any open set UA(x)⊃A(x). In fact, suppose that r=0. Then there exists {xn∗}n=1∞⊂B(X∗)∖UA(x) such that dist(xn∗,A(x))→0 as n→∞. Since A(x) is compact, there exists {yn∗}n=1∞⊂A(x) such that xn∗-yn∗=dist⁡(xn∗,A(x))→0. Since A(x) is compact, without loss of generality, we may assume that {yn∗}n=1∞ is a Cauchy sequence. Let yn∗→y0∗ as n→∞. Then y0∗∈A(x). Hence there exists c>0 such that Bc(y0∗)⊂UA(x). Moreover, we have (25)0≤xn∗-y0∗≤xn∗-yn∗+yn∗-y0∗⟶0asn⟶∞.Therefore, xn∗∈Bc(y0∗)⊂UA(x), when n is large enough, which contradicts {xn∗}n=1∞⊂B(X∗)∖UA(x). Hence, for any open set UA(x)⊃A(x), we have r=r(x,UA(x))>0. We next will prove that there exists m>0 such that 1>y∗(x)+m whenever y∗∈B(X∗)∖UA(x). Otherwise, there exists {hn∗}n=1∞⊂B(X∗)∖UA(x) such that hn∗(x)→1. It is easy to see that hn∗→1. Let zn∗=hn∗/hn∗. Then it is easy to see that zn∗(x)→1. Hence we define a weak∗ hyperplane H∗={x∗∈X∗:x∗(x)=1}. Then it is easy to see that dist(0,H∗)=1. Hence, for any e∗∈H∗¯w∗, there exists a net {eα∗}α∈Δ⊂H∗ such that eα∗→w∗e∗. Moreover, it is easy to see that 1=eα∗(x)=e∗(x). This implies that e∗∈H∗. Hence we obtain that H∗ is a weak∗ closed set. By the proof of Theorem 2.1 of [13], we know that if H∗ is a weak∗ closed set, then H∗ is a proximinal set. Hence there exists yn∗∈H∗ such that zn∗-yn∗=dist⁡(zn∗,H∗). Pick x0∗∈H∗. Then H∗-x0∗={x∗∈X∗:x∗(x)=0}. Since(26)zn∗-yn∗=dist⁡zn∗,H∗=dist⁡zn∗-x0∗,H∗-x0∗=zn∗-x0∗x⟶0,we obtain that (27)yn∗≤zn∗+zn∗-yn∗⟶1=dist⁡0,H∗.Therefore, by yn∗≥1 and formula (27), we obtain that (28)limn→∞⁡0-yn∗=limn→∞⁡yn∗=1=dist⁡0,H∗.Since every weak∗ hyperplane of X∗ is approximatively compact, we obtain that {yn∗}n=1∞ has a subsequence converging to an element in H∗. By formula (26), we obtain that {zn∗}n=1∞ has a subsequence converging to a element in H∗. Hence we may assume without loss of generality that zn∗→z0∗. Then z0∗≤1. By zn∗(x)→z0∗(x) and zn∗(x)→1, we have z0∗(x)=1. This implies that z0∗∈A(x). Then dist(zn∗,A(x))→0 as n→∞. By zn∗=hn∗/hn∗ and hn∗→1, we obtain that dist(hn∗,A(x))→0 as n→∞. Moreover, we have proved that r=r(x,UA(x))>0 for any open set UA(x)⊃A(x), a contradiction. This implies that there exists m>0 such that 1>y∗(x)+m whenever y∗∈B(X∗)∖UA(x). Hence, for any x∗∈A(x), we have (29)1-m≥sup⁡⁡y∗x:y∗∈BX∗∖UAx=sup⁡⁡y∗x:y∗∈coBX∗∖UAx=sup⁡⁡y∗x:y∗∈co¯w∗BX∗∖UAx.This implies that x∗∉co¯w∗(B(X∗)∖UA(x)). Therefore, by the arbitrariness of x∗, we have A(x)∩co¯w∗(B(X∗)∖UA(x))=∅. Hence X∗ is w∗ nearly dentable space. This completes the proof.Theorem 12.
Suppose thatX∗ is w∗ nearly dentable space. Then, (1)
for anyweak∗ open convex set C, metric projector operator PCw∗¯ is upper semicontinuous;(2)
for anyweak∗ hyperplane D∗, metric projector operator PD∗ is upper semicontinuous.Proof.
(
1
) Let C be a weak∗ open convex set and H∗=Cw∗¯. Suppose that metric projector operator PH∗ is not upper semicontinuous. Then we without loss of generality may assume that metric projector operator PH∗ is not upper semicontinuous in orgin. Hence there exist a sequence {xn∗}n=1∞ and an open set UPH∗(0)⊃PH∗(0) such that PH∗(xn∗)⊄UPH∗(0)=∅, and xn∗→0 as n→∞. This implies that there exists yn∗∈PH∗(xn∗) such that yn∗∉UPH∗(0). We will derive a contradiction for each of the following two cases.
Case
I. Let 0∈H∗. Then, by xn∗→0, we have dist(xn∗,H∗)→0 as n→∞. Moreover, by yn∗∈PH∗(xn∗), we have xn∗-yn∗=dist⁡(xn∗,H∗)→0 as n→∞. Therefore, (30)0≤∥yn∗∥≤∥xn∗∥+∥xn∗-yn∗∥⟶0asn⟶∞.This implies that yn∗→0 as n→∞. Hence there exists n∈N such that yn∗∈UPH∗(0), a contradiction.
Case
II. Let 0∉H∗. Then d=dist⁡(0,H∗)>0, Bd(0)¯∩C=∅, and Bd(0)¯∩H∗=PH∗(0). Therefore, by (X∗,w∗)∗=X and the separation theorem of locally convex space, there exists x∈S(X) such that sup⁡{x(y∗):y∗∈Bd(0)¯}≤inf⁡{x(y∗):y∗∈C}. This implies that (31)sup⁡⁡xy∗:y∗∈Bd0¯≤inf⁡⁡xy∗:y∗∈H∗=d.Therefore, by PH∗(0)⊂H∗, we obtain that x∗(x)=d for any x∗∈PH∗(0). Since PH∗(0)⊂Bd(0)¯, we obtain that {x∗:x∗(x)=d,x∗∈Bd(0)¯}⊃PH∗(0). Since the distance function is continuous, we obtain that(32)xn∗-yn∗=dist⁡xn∗,H∗⟶dist⁡0,H∗=dasn⟶∞.Moreover, we have (33)0≤yn∗-xn∗-yn∗≤xn∗⟶0asn⟶∞.Therefore, by formulas (32) and (33), we have yn∗→d. By yn∗∉UPH∗(0), we get that every subsequence {yn∗}n=1∞ does not converge to point of PH∗(0). Moreover, by yn∗∉UPH∗(0), we have yn∗>d. Hence, for any y∗∈PH∗(0), there exists εy∗>0 such that {yn∗}n=1∞∩{x∗:x∗-y∗<εy∗}=∅. Since H∗ is closed convex set and ({x∗:x∗(x)=d,x∗∈Bd(0)¯}∖PH∗(0))∩H∗=∅, we obtain that for any y∗∈{x∗:x∗(x)=d,x∗∈Bd(0)¯}∖PH∗(0), there exists εy∗>0 such that dist(y∗,H∗)=2εy∗. This implies that {yn∗}n=1∞∩{x∗∈X∗:x∗-y∗<εy∗}=∅. Noticing that (34)x∗:x∗x=d,x∗∈Bd0¯∖PH∗0∪PH∗=x∗:x∗x=d,x∗∈Bd0¯,we obtain that, for any y∗∈{x∗:x∗(x)=d,x∗∈Bd(0)¯}, there exists εy∗>0 such that(35)yn∗n=1∞∩x∗∈X∗:x∗-y∗<εy∗=∅.Therefore, by yn∗→d and yn∗>d, we have (36)dyn∗·yn∗-yn∗=dyn∗-1·yn∗⟶0asn⟶∞.Let us define sequence {zn∗}n=1∞, where (37)zn∗=nn+1·dyn∗·yn∗n∈N.Then (38)zn∗=nn+1·dyn∗·yn∗=nn+1·dyn∗·yn∗=ndn+1<d.Noticing that x∈S(X), we obtain that zn∗(x)<d. This implies that zn∗∉{x∗:x∗(x)=d,x∗∈Bd(0)¯}. Since yn∗→d, we have (39)zn∗-yn∗=nn+1·dyn∗·yn∗-yn∗=nn+1·dyn∗-1·yn∗⟶0.Hence there exists ny∗∈N such that zn∗-yn∗<(1/4)εy∗ whenever n>ny∗. Therefore, by formula (35), we have yn∗-y∗≥εy∗. Hence if n>ny∗, then (40)y∗-zn∗≥y∗-yn∗-yn∗-zn∗≥εy∗-14εy∗=34εy∗.Since zn∗∉x∗:x∗x=d,x∗∈Bd0¯ and y∗∈{x∗:x∗(x)=d,x∗∈Bd(0)¯}, we have min⁡{y∗-z1∗,y∗-z2∗,…,y∗-zny∗∗}>0. Put(41)ηy∗=12min⁡⁡34εy∗,min⁡⁡y∗-z1∗,y∗-z2∗,…,y∗-zny∗∗.Then, by formulas (40) and (41), we have {zn∗}n=1∞∩x∗:x∗-y∗<εy∗=∅. Moreover, by the arbitrariness of y∗, we have(42)zn∗n=1∞∩⋃y∗∈x∗:x∗x=d,x∗∈Bd0¯x∗:x∗-y∗<εy∗=∅.Put (43)PH∗0¯¯=x∗:x∗x=d,x∗∈Bd0¯,VPH∗0¯¯=⋃y∗∈PH∗0¯¯x∗:x∗-y∗<εy∗.Then, by formulas (42) and (43), we have {zn∗}n=1∞∩VPH∗(0)¯¯=∅.
Since the distance function is continuous, we havexn∗-yn∗=dist⁡(xn∗,H∗)→dist⁡(0,H∗) as n→∞. Moreover, we have xn∗→0 as n→∞. This implies that 0-yn∗→dist⁡(0,H∗) as n→∞. Therefore, by zn∗-yn∗→0 as n→∞, we obtain that (44)0-zn∗≤0-yn∗+yn∗-zn∗⟶dist⁡0,H∗asn⟶∞.Since 0-zn∗≥dist⁡(0,H∗), we obtain that 0-zn∗→dist⁡(0,H∗) as n→∞. By (45)1dyn∗x=1,zn∗=nn+1·dyn∗·yn∗,we have zn∗(x)/d→1 as n→∞. Since B(X∗) is weak∗ compact, there exists z0∗/d∈B(X∗) such that z0∗/d is a weak∗ accumulation point of {zn∗/d}n=1∞. Hence there exists a net {zα∗}α∈Δ⊂{zn∗}n=1∞ such that zα∗→w∗z0∗ and z0∗/d∈A(x). Since VPH∗(0)¯¯⊃{x∗:x∗(x)=d,x∗∈Bd(0)¯}, we obtain that (46)1dVPH∗0¯¯⊃1dx∗∈Bd0¯:x∗x=d=x∗∈BX∗:x∗x=1=Ax.Let UA(x)=(1/d)VPH∗(0)¯¯. Then, by {zn∗}n=1∞∩VPH∗(0)¯¯=∅ and {zn∗}n=1∞⊂Bd(0)¯, we have zn∗∈Bd(0)¯∖VPH∗(0)¯¯. This implies that zn∗∈Bd(0)¯ and zn∗∉VPH∗(0)¯¯. Therefore, (47)1dzn∗∈1dBd0¯=BX∗,1dzn∗∉1dVPH∗0¯¯=UAx.This implies that zn∗/d∈B(X∗)∖UA(x). Since X∗ is a w∗ nearly dentable space, we obtain that A(x)∩co¯w∗B(X∗)∖UA(x)=∅. Since B(X∗) is w∗ compact, it is easy to see that co¯w∗(B(X∗)∖UA(x)) is w∗ compact. Moreover, since A(x) is w∗ closed convex set, by the separation theorem of locally convex space, there exist y∈X and r>0 such that (48)inf⁡⁡x∗y:x∗∈Ax-r≥sup⁡⁡x∗y:x∗∈co¯w∗BX∗∖UAx.Therefore, by zα∗→w∗z0∗, we obtain that zα∗/d→w∗z0∗/d. Noticing that zn∗/d∈B(X∗)∖UA(x), we obtain that (49)1dzα∗α∈Δ⊂1dzn∗n=1∞⊂BX∗∖UAx⊂co¯w∗BX∗∖UAx.Therefore, by formulas (48)-(49) and z0∗/d∈A(x), we have (z0∗/d)(y)-r≥sup⁡x∗(y):x∗∈zα/d∗α∈Δ, which contradicts zα∗/d→w∗z0∗/d. Hence the metric projector operator PH∗ is upper semicontinuous.
By the proof(3)⇒(1) of Theorem 11, it is easy to see that (2) is true. This completes the proof.


## 4. Some Examples ofw∗ Nearly Dentable Space in Orlicz Function Space

In the section, authors prove that there exists a Banach spaceX∗ such that X∗ is not a w∗ nearly dentable Banach space. Moreover, authors also prove that there exists a Banach space X∗ such that X∗ is a w∗ nearly dentable Banach space and not a nearly dentable Banach space. First, we give the definition of Orlicz function space.Definition 13.
M
:
R
→
R is called an N-function if it has the following properties: (1)
M is even, convex, and M(0)=0;(2)
M(u)>0 for all u≠0;(3)
limu→0M(u)/u=0 and limu→∞M(u)/u=∞.Letp(u) denote the right derivative of M(u) at u∈R+ and let q(v) be the generalized inverse function of p(u) defined on R+ by q(v)=sup⁡u≥0{u≥0:p(u)≤v}. Then we call N(v)=∫0|v|‍q(s)ds the complementary function of M. It is well known that there holds the Young inequality uv≤M(u)+N(v) and uv=M(u)+N(v)⇔u=|q(v)|signv or v=|p(u)|signu. Moreover, it is well known that M and N are complementary to each other. A continuous function M:R→R is called strictly convex if 2M(u+v)/2<M(u)+M(v) for all u≠v.Let(G,Σ,μ) be a finite nonatomic and complete measure space. Denote by p and q the right derivative of M and N, respectively. We define (50)ρMx=∫GMxtdt,LM=xt:ρMλx<∞for someλ>0,EM=xt:ρMλx<∞∀λ>0.It is well known that the Orlicz function space LM is a Banach space when it is equipped with the Luxemburg norm (51)x=inf⁡⁡λ>0:ρMxλ≤1or equipped with the Amemiya-Orlicz norm(52)x0=inf⁡k>0⁡1k1+ρMkx.LM, EM denote Orlicz function spaces equipped with the Luxemburg norm. LM0, EM0 denote Orlicz function spaces equipped with the Orlicz norm. It is well known that (EN)∗=LM0, (EN0)∗=LM.Definition 14 (see [13]).
We say that anN-function M∈Δ2(N∈∇2) if there exist K>2 and u0≥0 such that M(2u)≤KM(u) whenever u≥u0.Lemma 15 (see [13, 14]).
LetLM be Orlicz function space. Then the following statements are equivalent:(1)
Everyweakly∗ hyperplane of LM is approximatively compact;(2)
EN0 is smooth space and M∈Δ2.Lemma 16 (see [15]).
L
M
0(LM) has the Radon-Nikodym property if and only if M∈Δ2.Lemma 17 (see [15]).
L
M
0 is a strictly convex space if and only if M is strictly convex.Lemma 18 (see [15]).
L
M
(
L
M
0
) is reflexive if and only if M∈Δ2∩∇2.Theorem 19.
If everyweakly∗ hyperplane of LM0 is approximatively compact, then M∈Δ2.Proof.
Since everyweak∗ hyperplane of LM0 is approximatively compact, we obtain that LM0 has the Radon-Nikodym property (see [14]). Therefore, by Lemma 16, we have M∈Δ2. This completes the proof.Example 20.
IfM is strictly convex and M∉Δ2, then LM0 is not w∗ nearly dentable. In fact, suppose that LM0 is w∗ nearly dentable. Since M is strictly convex, we obtain that LM0 is a strictly convex space by Lemma 17. Since (EN)∗=LM0, we obtain that EN is a smooth space. This implies that if E is a separable closed subspace of EN, then AE(x)={x∗∈S(E∗):x∗(x)=1=x} is a singleton. Therefore, by Theorem 10, we obtain that LM0 has the Radon-Nikodym property. Then, by Lemma 16, we obtain that M∈Δ2, a contradiction.Example 21.
IfM is strictly convex, M∈Δ2 and N∉Δ2, then LM is w∗ nearly dentable not nearly dentable. In fact, by Theorem 19, we obtain that every weakly∗ hyperplane of LM0 is approximatively compact. Therefore, by Theorem 11, we obtain that LM is w∗ nearly dentable. Since N∉Δ2, we obtain that LM is not reflexive. This implies that LM is not nearly dentable.


## 5. Applications to Theory of Generalized Inverses of Banach Space

LetT be a linear bounded operator from X into Y. Let D(T),R(T) and N(T) denote the domain, range, and null space of T, respectively. If N(T)≠{0} or R(T)≠Y, the operator equation Tx=y is generally ill-posed. In applications, one usually looks for the best approximative solution (b.a.s.) to the equation (see [16]).A pointx0∈D(T) is called the best approximative solution to the operator equation Tx=y, if (53)Tx0-y=inf⁡⁡Tx-y:x∈DT,x0=min⁡⁡v:v∈DT,Tv-y=inf⁡x∈DT⁡Tx-y,where y∈Y (see [16]). Nashed and Votruba [16] introduced the concept of the (set-valued) metric generalized inverse T.Definition 22.
LetX,Y be Banach spaces and let T be a linear operator from X to Y. The set-valued mapping T∂:Y→X defined by (54)T∂y=x0∈DT:x0is a best aproximation solution toTx=yfor any y∈D(T∂) is called the (set-valued) metric generalized inverse of T, where(55)DT∂=y∈Y:Tx=yhas a best aproximation solution inX.Definition 23.
H
(
A
,
B
) is said to be Hausdorff distance of A and B if (56)HA,B=max⁡⁡sup⁡a∈A⁡inf⁡b∈Ba-b,sup⁡b∈B⁡inf⁡a∈Aa-b.During the last three decades, the linear generalized inverses of linear operators in Banach spaces and their applications have been investigated by many authors. In 2008, Chen et al. [8], under some assumption on the geometric structure of Banach space, gave the criteria for the existence of continuous metric generalized inverse. In 2005, Wei et al. [17] presented componentwise condition numbers for the problems of Moore-Penrose generalized matrix inverses and linear least squares. For the selection of the metric generalized inverse of linear operator in Banach space, in 2008, Hudzik et al. [18] obtained a bounded homogeneous selection.Theorem 24.
LetX∗ be a w∗ nearly dentable space, Y∗ a nearly dentable space, N(T) a one-dimensional subspace of X, and R(T∗) a closed semi-Chebychev set. Then D((T∗)∂)=X∗. Moreover, if for any y0∗∈(T∗)∂(x0∗) there exists f∈S(X∗∗) such that Af is compact and y0∗/y0∗∈Af, then(1)
(T∗)∂(x0∗) is compact and (T∗)∂ is upper semicontinuous at x0∗;(2)
(T∗)∂ is lower semicontinuous at x0∗ if and only if the function g(x∗)=H(T∗)∂(x∗),(T∗)∂(x0∗) is lower semicontinuous at x0∗.Proof.
SinceR(T∗) is closed, we obtain that R(T∗)=N(T)⊥={x∗∈X∗:x∗(x)=0,x∈N(T)} by the closed range theorem. Since dimN(T)=1, there exists x∈S(X)∩N(T) such that R(T∗)={x∗∈X∗:x∗(x)=0}. This implies that R(T∗) is a weakly∗ hyperplane of X∗. By proof of Theorem 2.1 of [13], we obtain that R(T∗) is a proximinal set. Since R(T∗) is a semi-Chebychev set, we obtain that R(T∗) is a Chebychev set. Let x∗∈X∗. Then there exists a unique z∗∈R(T∗) such that z∗∈PR(T∗)(x∗). Hence there exists y∗∈Y∗ such that (T∗)-1(z∗)=y∗+N(T∗). Since Y∗ is a nearly dentable space, we obtain that Y∗ is reflexive. This implies that N(T∗) is a proximinal set. Hence we obtain that D((T∗)∂)=X∗.
(
1
) We will prove that the set (T∗)∂(x0∗) is compact. Let xn∗→x0∗. Since X∗ is a w∗ nearly dentable space and R(T∗) is a weakly∗ hyperplane of X∗, by Theorem 12, we obtain that metric projector operator PR(T∗) is upper semicontinuous. Moreover, since R(T∗) is a Chebychev set, we obtain that metric projector operator PR(T∗) is continuous; that is, PR(T∗)(xn∗)→PR(T∗)(x0∗) as n→∞. Since T is linear bounded operator, we obtain that N(T∗) is a closed subspace. Put (57)T∗¯:Y∗NT∗⟶RT∗,T∗¯y∗=T∗y∗.Then, by the inverse operator theorem, we obtain that(58)yn∗=T∗¯-1PRT∗xn∗⟶T∗¯-1PRT∗x0∗=y0∗asn⟶∞.This implies that [yn∗-y0∗]→0 as n→∞. Since, for any y0∗∈(T∗)∂(x0∗), there exists f∈S(X∗∗) such that y0∗/y0∗∈Af, we have 0∉(T∗)∂(x0∗). Moreover, we may assume without loss of generality that yn∗-y0∗→0 as n→∞. Therefore, by the definition of set-valued metric generalized inverse, we obtain that (59)T∗∂xn∗=yn∗-PNT∗yn∗,T∗∂x0∗=y0∗-PNT∗y0∗.Suppose that PN(T∗)(y0∗) is not compact. Then there exists a sequence {zn∗}n=1∞⊂PN(T∗)(y0∗) such that {zn∗}n=1∞ has no Cauchy subsequence. Since Y∗ is a nearly dentable space, we obtain that Y∗ is reflexive. Hence we may assume without loss of generality that zn∗→wz0∗. Noticing that PN(T∗)(y0∗) is a closed convex set, we obtain that PN(T∗)(y0∗) is a weakly closed convex set. Then z0∗∈PN(T∗)(y0∗). Put (60)hn∗=y0∗-zn∗y0∗-zn∗,h0∗=y0∗-z0∗y0∗-z0∗.Noticing that zn∗∈PN(T∗)(y0∗), z0∗∈PN(T∗)(y0∗), and 0∉(T∗)∂(x0∗), we obtain that y0∗-zn∗=y0∗-z0∗≠0. By zn∗→wz0∗, we obtain that hn∗→wh0∗. Moreover, it is easy to see that y0∗-z0∗h0∗∈(T∗)∂(x0∗). Then there exists f∈S(X∗∗) such that Af is compact and h0∗∈Af. We will derive a contradiction for each of the following two cases.
Case
I. Let {hn∗}n=1∞∩Af be a finite set. Then we may assume without loss of generality that {hn∗}n=1∞∩Af=∅. Hence, for any y∗∈Af, there exists ηy∗>0 such that {hn∗}n=1∞∩B(y∗,ηy∗)=∅, where f(h0∗)=1=f. Hence there exists an open set UAf=∪y∗∈AfB(yn∗,ηy∗) such that UAf∩{hn∗}n=1∞=∅. Since Y∗ is a nearly dentable space, we obtain that Af∩co¯(B(X)∖UAf)=∅. By the separation theorem, there exists g∈Y∗∗ such that (61)inf⁡⁡gy∗:y∗∈Af-sup⁡⁡gy∗:y∗∈co¯BX∖UAf>0.Noticing that {hn∗}n=1∞⊂co¯(B(X)∖UAf) and h0∗∈Af, we obtain that g(h0∗)-sup⁡g(y∗):y∗∈{hn∗}n=1∞>0, which contradict hn∗→wh0∗ as n→∞. This implies that the sequence {hn∗}n=1∞ has a Cauchy subsequence.
Case
II. Let {hn∗}n=1∞∩Af be an infinite set. Then there exists a subsequence {hnl∗}l=1∞ of {hn∗}n=1∞ such that {hnl∗}l=1∞⊂Af. Since Af is a compact set, we obtain that {hnl∗}l=1∞⊂Af is relatively compact. This implies that {hn∗}n=1∞ has a Cauchy subsequence. Hence PN(T∗)(y0∗) is compact. By (T∗)∂(x0∗)=y0∗-PN(T∗)(y0∗), we obtain that (T∗)∂(x0∗) is compact.
We next will prove that the set-valued mapping(T∗)∂ is upper semicontinuous at x0∗. Let W be an open set and PN(T∗)(y0∗)⊂W. We claim that there exists δ>0 such that(62)⋃z∗∈PNT∗y0∗y∗∈Y∗:y∗-z∗<2δ⊂W.Otherwise, there exists a sequence {hn∗}n=1∞⊂PN(T∗)(y0∗) such that B(hn∗,1/n)⊄W. Let {hnk∗}k=1∞ be a Cauchy sequence. Noticing that hn∗→wh0∗, we have hnk∗→h0∗ as k→∞. Then (63)limk→∞⁡znk∗=y0∗-y0∗-z0∗h0∗∈PRT∗y0∗.Hence there exists r>0 such that By0∗-y0∗-z0∗h0∗,r⊂W. Since znk∗→y0∗-y0∗-z0∗h0∗, there exists k0∈N such that (64)y0∗-y0∗-z0∗h0∗-znk0∗<14r,1nk0<14r.Hence, if y∗-znk0∗≤1/nk0, then (65)y∗-y0∗-y0∗-z0∗h0∗≤y∗-znk0∗+znk0∗-y0∗-y0∗-z0∗h0∗≤1nk0+14r<r.This means that if y∗-znk0∗≤1/nk0, then y∗∈B(znk0∗,1/nk0)⊂B(y0∗-y0∗-z0∗h0∗,r)⊂W, a contradiction. Hence there exists δ>0 such that (66)⋃z∗∈PNT∗y0∗y∗∈Y∗:y∗-z∗<2δ⊂W.Let (67)W1=⋃z∗∈PNT∗y0∗y∗∈Y∗:y∗-z∗<δ⊃PNT∗y0∗.Since Y∗ is a w∗ nearly dentable space, we obtain that the metric projector operator PN(T∗) is upper semicontinuous. Since metric projector operator PN(T∗) is upper semicontinuous and yn∗-y0∗→0, there exists n1∈N such that PN(T∗)(yn∗)⊂W1 whenever n>n1. This means that y0∗-PN(T∗)(yn∗)⊂y0∗-W1 whenever n>n1. Moreover, by yn∗-y0∗→0, there exists n2∈N such that yn∗-y0∗<δ/4 whenever n>n1. Let N1=max⁡{n1,n2}. We claim that yn∗-PN(T∗)(yn∗)⊂y0∗-W whenever n>N1. In fact, if z∗∈yn∗-PN(T∗)(yn∗), then there exists hn∗∈PN(T∗)(yn∗) such that z∗=yn∗-hn∗. Let ln∗=yn∗-y0∗. Then z∗=y0∗-(h∗-ln∗) and ln∗<δ/4. Since hn∗∈PN(T∗)(yn∗)⊂W1, there exists vn∗∈PN(T∗)(y0∗) such that hn∗-vn∗<δ. Then(68)hn∗-ln∗-vn∗≤hn∗-vn∗+ln∗≤δ+14δ<2δ.This implies that hn∗-ln∗∈W. Then z∗=y0∗-(hn∗-ln∗)∈y0∗-W. This implies that yn∗-PN(T∗)(yn∗)⊂y0∗-W. Therefore, by formula (59), we have (T∗)∂(xn∗)⊂y0∗-W.
LetV be an open set and (T∗)∂(x0∗)⊂V. Then, by formula (59), we have y0∗-V⊃PN(T∗)(y0∗). Hence there exists natural number N2 such that (T∗)∂(xn∗)⊂y0∗-(y0∗-V) whenever n>N2. This implies that (T∗)∂(xn∗)⊂V whenever n>N2. Hence we obtain that the set-valued mapping T∂ is upper semicontinuous at x0∗.
(
2
) Let (T∗)∂ be lower semicontinuous at x0∗. It is easy to see that g(x0∗)=0. Suppose that g(x∗) is not lower semicontinuous at x0∗. Then there exist a sequence {xn∗}n=1∞ and ε0>0 such that xn∗→x0∗ and(69)lim⁡inf⁡n→∞⁡gxn∗=lim⁡inf⁡n→∞⁡HT∗∂xn∗,T∗∂x0∗>ε0.Since (T∗)∂ is lower semicontinuous at x0∗, we obtain that, for any y∗∈(T∗)∂(x0∗), there exists yn∗∈(T∗)∂(xn∗) such that yn∗→y∗ as n→∞. This implies that for any y∗∈(T∗)∂(x0∗), we have(70)limn→∞⁡inf⁡z∗∈T∗∂xn∗⁡z∗-y∗=0.We claim that (71)limn→∞⁡sup⁡y∗∈T∗∂x0∗⁡inf⁡z∗∈T∗∂xn∗z∗-y∗=0.In fact, suppose that formula (71) is not true. Since (T∗)∂(x0∗) is a compact set, there exist yn∗∈(T∗)∂(x0∗), zn∗∈(T∗)∂(xn∗), and r>0 such that (72)inf⁡z∗∈T∗∂xn∗⁡z∗-yn∗=zn∗-yn∗,lim⁡sup⁡n→∞⁡zn∗-yn∗>2r.Since (T∗)∂(x0∗) is compact, we may assume that yn∗→y0∗∈(T∗)∂(x0∗) as n→∞. Therefore, by formula (72), we have limsup⁡n→∞zn∗-y0∗>r. Moreover, by formula (71), we obtain that(73)0=lim⁡sup⁡n→∞⁡inf⁡z∗∈T∗∂xn∗⁡z∗-y0∗=lim⁡sup⁡n→∞⁡zn∗-y0∗>r,a contradiction. Hence formula (71) is true.
Since(T∗)∂ is upper semicontinuous at x0∗, we obtain that for any ε>0, there exists N1 such that(74)T∗∂xn∗⊂⋃y∗∈T∗∂x0∗z∗∈Y∗:z∗-y∗<εwhenever n>N1. This implies that for any yn∗∈(T∗)∂(xn∗), we obtain that dist(yn∗,(T∗)∂(x0∗))→0 as n→∞. Hence, for any yn∗∈(T∗)∂(xn∗), we obtain that (75)limn→∞⁡inf⁡z∗∈T∗∂x0∗⁡yn∗-z∗=limn→∞⁡dist⁡yn∗,T∗∂x0∗=0.We claim that (76)limn→∞⁡sup⁡y∗∈T∗∂xn∗⁡inf⁡z∗∈T∗∂x0∗⁡y∗-z∗=0.Otherwise, there exist yn∗∈(T∗)∂(xn∗), zn∗∈(T∗)∂(x0∗), and d>0 such that (77)inf⁡z∗∈T∗∂x0∗⁡yn∗-z∗=yn∗-zn∗,lim⁡sup⁡n→∞⁡yn∗-zn∗>2d.Moreover, we have (78)limn→∞⁡yn∗-zn∗=limn→∞⁡inf⁡z∗∈T∗∂x0∗⁡yn∗-z∗=limn→∞⁡dist⁡yn∗,T∗∂x0∗=0,a contradiction. This implies that formula (76) is true. Therefore, by formulas (71) and (76), we have(79)lim⁡sup⁡n→∞⁡HT∗∂xn∗,T∗∂x0∗=0,which contradict formula (69). Hence (80)lim⁡inf⁡x∗→x0∗⁡gx∗=lim⁡inf⁡x∗→x0∗⁡HT∗∂x∗,T∗∂x0∗=0.This implies that the function g(x∗)=H((T∗)∂x∗),(T∗)∂(x0∗) is lower semicontinuous at x0∗.
On the other hand, ifg(x∗) is lower semicontinuous at x0∗, then liminf⁡x∗→x0∗g(x∗)=liminf⁡x∗→x0∗H((T∗)∂(x∗),(T∗)∂(x0∗))=0. Hence, if xn∗→x0∗, then(81)limn→∞⁡sup⁡y∗∈T∗∂x0∗⁡inf⁡z∗∈T∗∂xn∗⁡z∗-y∗=0.Hence, for any y∗∈(T∗)∂(x0∗), there exists zn∗∈(T∗)∂(xn∗) such that(82)limn→∞⁡zn∗-y∗=limn→∞⁡inf⁡z∗∈T∗∂xn∗⁡z∗-y∗=0.This implies that (T∗)∂ is lower semicontinuous at x0∗. This completes the proof.Theorem 25.
LetX∗ be a w∗ nearly dentable space, Y∗ approximatively compact, N(T) a one-dimensional subspace, and R(T∗) a closed semi-Chebychev set. Then(1)
D((T∗)∂)=X∗ and (T∗)∂ is upper semicontinuous;(2)
(T∗)∂ is lower semicontinuous at x0∗ if and only if the function g(x∗)=H((T∗)∂(x∗),(T∗)∂(x0∗)) is lower semicontinuous at x0∗.Proof.
Letx0∗∈X∗. Since Y∗ is approximatively compact, we obtain that, for any y0∗∈(T∗)∂(x0∗), there exists f∈S(X∗∗) such that Af is compact and y0∗/y0∗∈Af. Moreover, since Y∗ is approximatively compact, we obtain that Y∗ is a nearly dentable space. By Theorem 24, Theorem 25 is true, which completes the proof.



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*Source: 921456-2015-05-06.xml*