# Local Uniform Kadec-Klee Property (LUKK) and Modulus of (LUKK)

Authors: Yunan Cui; Xiaoxia Wang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

A new geometry property and two new moduli are introduced in Banach space. First, the concept of local uniform Kadec-Klee property (LUKK) is introduced and the implication relationships between LUKK and local near uniform convexity LNUC, uniformly Kadec-Klee (UKK), (H) are investigated in Banach space. Furthermore, the modulus PXLε of (LUKK) and the modulus ΔXLε of LNUC are introduced and the relationship of size between PXLε and ΔXLε is also investigated in Banach space. Finally, several formulas for PXLε are calculated in classical Banach space lp.


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## Body

## 1. Introduction

LetX,· be a Banach space, X∗ be the dual space of X. By UX and SX, we denote the closed unit ball and the unit sphere of Banach space X, respectively. By coA and co¯A, we denote the convex hull and closed convex hull of the set A, respectively. xn⟶wx as n⟶∞ denotes xn is weakly converges to x as n⟶∞.It is well known that the condition equivalent to near uniform convexity (NUC) was independently formulated in [1] (see also [2]). Recall that a notion of noncompactness with Hausdorff measure and Kuratowski measure of a set (see [3]). Let A be a bounded subset of X. Fix ε∈0,1 for all convex closed sets A⊂UX with μA≥ε, we put
(1)ΔXε=inf1−dθ,A:A⊂UX,A=coA,μA≥ε,Δ^Xε=inf1−dθ,A:A⊂UX,A=coA,αA≥ε,where
(2)μA=infε>0:Acanbecoveredwithafinitenumberofsetsofradiismallerthanε,αA=infε>0:Acanbecoveredbyfinitelymanysetswithdiameter<ε.The functionsΔXε and Δ^Xε are called the moduli of noncompact convexity with Hausdorff measure and Kuratowski measure of X, respectively. It is clear that X is NUC if and only if ΔXε>0 for every ε∈0,1. Other properties of ΔXε were investigated in [4–7].Recall also that a function(3)ε1X=supε:ΔXε=0.J. Banas’ proved that ifε1X<1, then X is reflexive and has normal structure (see [3]).The modulus ofUKK was introduced in [8] by J. R. Partington that is PX:0,1⟶0,1(4)PXε=inf1−∥x∥:∃x∈UXandxn⊂UXs.t.xn⟶wxandsepxn≥ε,where sepxn=inf∥xn−xm∥:m≠n. He proved that X has UKK property if and only if PXε>0 whenever ε∈0,1. He also proved that the function PX is nondecreasing on [0,1].There are many recent papers concerning the Kadec-Klee property, such as Kadec¨CKlee property and fixed points and Dual Kadec-Klee property and fixed points studied by Jean Saint Raymond (see [9, 10]). In a recent paper [11], Maciej Ciesielski, Paweł Kolwicz, and Ryszard Płuciennik were interested in local approach to Kadec¨CKlee property in symmetric function spaces. Moreover, normal structure and moduli of UKK, NUC, and UKK∗ in Banach spaces have been deeply investigated by Satit Saejung and Ji Gao. The new kind of Banach spaces: semi−UKK, semi−NUC, modulus of semi−UKK, and modulus of semi−NUC are introduced in terms of this u-separation measure in their paper.(see [12]).Since Banach space is more extensive than Hilbert space, it is quite difficult to describe its geometry structure. An effective method is to introduce new geometric properties for Banach space and to define an appropriate function, usually called a modulus or a geometric constant. Because the range of values of these geometric constants directly determines the existence of some geometric properties; therefore, many scholars are interested to calculate the modules and constants of some specific spaces. The starting point of the present paper is the observation that theUKK property can be localized. We call this new property named the local uniform Kadec-Klee property (LUKK), and we observe that it lies strictly between H and UKK properties. By using the same localized method, we localize the modulus PXε of UKK introduced by J. R. Partington and the modulus of noncompact convexity with Hausdorff measure ΔXε obtain two new moduli PXLε and ΔXLε, and we observe that ΔXLε≥PXLε.


## 2. Preliminaries

Before starting with our results, we need to recall some notions and a lemma in [13–15].We say that a Banach spaceX has H property if for any x∈SX, xn⊂X, limn⟶∞∥xn∥=∥x∥ and xn⟶wx, then limn→∞∥xn−x∥=0.We say that a Banach spaceX has Kadec−Klee (UKK) property if for every ε>0 there exists 0<δ<1 such that xn⊂UX, xn⟶wx and sepxn≥ε then x∈Bδ0 where Bδ0=x:∥x∥≤δ.We say that a Banach spaceX has near uniform convexity property (NUC) if for every ε>0 there exists 0<δ=δε<1 such that xn⊂UX with sepxn≥ε then there is a N≥1 and scalars λ1,⋯,λN≥0 with ∑λn=1 such that ∥∑λnxn∥≤1−δ.We say that a Banach spaceX has local near uniform convexity property (LNUC) if for every ε>0 and x∈UX there exists δ=δx,ε>0 such that for any sequence xn⊂UX with sepxn≥ε then
(5)coxn,x∩Bδ0≠∅.i.e.,(6)∥λ0y+1−λ0x∥<δwherey∈coxn,for some λ0∈0,1 and y∈coxn (see [16]).Lemma 1.
LetX be a Banach Space, xn⊂X and xn⟶wx0. Then,
(7)x0=∩n=1∞co¯xkk≥n.


## 3. Materials and Methods

In this paper, we take Kutazrova and Bor-Luh Lin’s approach to localize theUKK property and obtain the LUKK property. By using the same localized method, we localize the modulus PXε of UKK which introduced by Partington and the modulus of noncompact convexity with Hausdorff measure ΔXε and obtain two new moduli PXLε and ΔXLε; then, we study the relationship of size between PXLε and ΔXLε in Banach space by using the Corollary of Hahn-Banach Theorem and the weak lower semi continuity of norm.


## 4. Results and Discussion

We begin this section by formulating some definitions.Definition 2.
A Banach spaceX is said to have local uniform Kadec−Klee property (LUKK), for every ε>0 and x∈SX there exists δ>0 such that if ∥xn∥≤1, xn⟶wx0, sepxn≥ε and ∥xn−x∥≥ε then
(8)cox0,x∩Bδ0≠∅whereBδ0=x:∥x∥≤δ,
i.e.,(9)∥λ0x0+1−λ0x∥<δ,for some λ0∈0,1.Definition 3.
LetX be a Banach Space. For every ε>0 and x∈SX(10)PXLε=inf1−∥cox,x0∥:∣xn∥≤1,xn⟶wx0,sepxn≥ε,∥xn−x∥≥ε,is said to be the modulus of LUKK property or local Partington’s coefficient.Definition 4.
For everyε>0 and x∈SX, we put
(11)ΔXLε=inf1−dθ,cox,coxn:∥xn∥≤1,αcox,coxn≥ε,∥xn−x∥≥ε,where
(12)αA=infε>0:Acanbecoveredbyfinitelymanysetswithdiameter<ε
ΔXLε is said to be the modulus of LNUC with Kuratowski measure.Corollary 5.
If a Banach spaceX has LUKK property, then X has H property.Proof.
We prove the contrapositive. SupposeX does not have H property, then there exists xn⊂SX and x0∈SX such that although xn⟶wx0 as n⟶∞, we still have xn½x0 what means there exists ε0>0 and nk>n for any n∈N such that ∥xnk−x∥≥ε0 this implies that sepxnk≥ε0 holds.X has LUKK property, for ε0>0 and x0∈SX mentioned above, there exists δ∈0,1 such that
(13)cox0,x0∩Bδ0≠∅,i.e.,(14)x0∩Bδ0≠∅,this shows ∥x0∥≤δ<1, a contradiction. Thus, the assumption does not hold.The following conclusion follows from the definitions ofLUKK and UKK.Corollary 6.
If Banach spaceX has UKK property, then X has LUKK property.It follows from previous Corollaries, we conclude the following Corollary.Corollary 7.
For every Banach spaceX, the implication UKK⟹LUKK⟹H holds.We are now ready to prove the main theorems of this paper.Theorem 8.
If a Banach spaceX is LNUC, then X has LUKK property.Proof.
Suppose thatX does not have LUKK property. Then, there exists ε0>0, x0∈SX and yn⊂UX with sepyn≥2ε0, yn⟶wy0 with
(15)cox0,y0∩B1−1/n0=∅.SinceX has LNUC property, for ε0>0 and x0∈SX mentioned above, there exists δ=δx0,ε0>0 such that cox0,coyn∩B1−δ0≠∅, which means for some λ0∈0,1 and y′∈coyn, we have
(16)λ0x0+1−λ0y′≤1−δ,from (15) it follows that for any λ∈0,1 we have
(17)λx0+1−λy0>1−1n⟶1,n⟶∞.Case (i) ify0=y′, then (16) contradicts with (17).Case (ii) ify0≠y′, since yn⟶wy0, then by Lemma 1, we have y0=∩n=1∞co¯ykk≥n, since y′∈coyn then for any x∈SX and λ∈0,1. Let
(18)Yxδ=y∈BX:1−δ<λx+1−λy<1,ψλ=λx+1−λy,L=λ∈0,1:ψλ≤1−δ,λ′=infλ∈0,1:ψλ≤1−δ,It is obvious thatψ0≤ψ1=1, L≠∅ , and ψλ′=1−δ. From (16), we get
(19)y′∈Yx0δ.For another facts, forλ∈0,1, let z0=λ′y′+1−λ′x0; then, we have
(20)∥λz0+1−λx0∥=∥λλ′y′+1−λλ′x0∥=ψλλ′>1−δ,i.e.,(21)∥λλ′y′+1−λλ′x0∥>1−δ.Thus,y′∈Yx0δ which contradicts with (19); therefore, the assumption is not true.Theorem 9.
For every Banach spaceX, we have PXLε≥ΔXLε.Proof.
Fixε∈0,1 and take an arbitrary sequence xn⊂UX with sepxn≥ε, xn⟶wx0 as n⟶∞, αxn≥ε. For every x∈SX and some λ∈0,1, we let
(22)yn=λx+1−λxn,y0=λx+1−λx0.By the corollary of Hahn-Banach theorem, there existsf0∈SX∗ such that f0yn=∥yn∥. Picking η>0 be small enough and considering the following set
(23)Df0,η=y∈UX:f0y≥∥y0∥−η.It is obvious that the setDf0,η is closed, convex, and
(24)dθ,Df0,η≥∥y0∥−η.Sincexn⟶wx0, then
(25)f0yn⟶f0y0=∥y0∥.Then, there existsn0∈N such that
(26)f0yn≥∥y0∥−ηwhenn≥n0,this implies that the set Yn0=yn:n≥n0 is a subset of Df0,η. Then, we get
(27)dθ,coYn0≥∥y0∥−η.SinceαYn0≥ε, then
(28)αcoYn0≥εandΔXLε≤1−dθ,coYn0.from (27) it follows that
(29)1−dθ,coYn0≤1−∥y0∥+η.Consequently,(30)ΔXLε≤1−dθ,coYn0≤1−∥y0∥+η,1−∥y0∥≥ΔXLε−η.Thus,PXLε≥ΔXLε−η. Since η>0 is small enough, then we get PXLε≥ΔXLε and the proof is complete.Theorem 10.
For Banach spacelp1<p<∞, we have PlpLε≥1−1−ε/2p1/p.Proof.
For everyε>0, x∈Slp and λ∈0,1, let xn⊂Ulp such that xn⟶wx0, sepxn≥ε and ∥xn−x∥≥ε. Let
(31)yn=λx+1−λxn,y0=λx+1−λx0,then yn⟶wy0. By the weak lower semicontinuity of norm function, we get
(32)∥y0∥≤inflimn∥yn∥.
Then, there exists a subsequenceynk⊆yn and K∈N such that ∥y0∥≤∥ynk∥ for all k>K. Hence,
(33)∥y0∥p≤min∥yni∥p,∥ynj∥p:i,j>k=∣∥yni∥p+∥ynj∥p∣−∣∥yni∥p−∥ynj∥p∣2≤2−∥yni−ynj∥p2.
Since(34)∥yni−ynj∥=λ∥xni−xnj∥≥λε,then we get
(35)∥y0∥p≤1−∥yni−ynj∥p2≤1−λεp2,∥y0∥≤1−λεp21/p,1−∥y0∥≥1−1−λpεp21/p,thus
(36)PlpLε≥1−1−λpεp21/pwhere1<p<∞,it follows that
(37)PlpLε≥1−1−εp21/pwhere1<p<∞.Theorem 11.
IfX is a reflexive Banach space, then for any ε∈0,2, we have ΔXLε≥PXLε/2.Proof.
Fixε∈0,2. Take xn⊂UX with αxn≥ε then αcoxn≥ε; here, we let
(38)yn=λx+1−λxn,for every x∈SX and λ∈0,1. Thus, there exists zn⊂coyn such that sepzn≥ε/2. By the reflexivity of X, there exists subsequence znk⊂zn⊂coyn and z0∈UX such that znk⟶wz0. It is obvious that
(39)∥coz0,x∥≤1−PXLε2,1−PXLε2≥sup∥αz0+1−αx∥:α∈0,1.
And consequently,(40)∥z0∥≤1−PXLε2andz0∈coznk,this implies that
(41)dθ,cocox,xn=dθ,coyn≤dθ,cozn≤dθ,coznk≤∥z0∥≤1−PXLε2.
From (41), it follows that
(42)1−dθ,cocox,xn≥PXLε2.
Thus,ΔXLε≥PXLε/2 for any ε∈0,2, and the proof is complete.


## 5. Conclusions

In this paper, we introduce a new geometric propertyLUKK that lies between two classical geometric properties (UKK) and (H). Moreover, two new moduli PXLε and ΔXLε for (LUKK) and LNUC are introduced in Banach spaces; these new notions introduced in our paper play a very significant role in some recent trends of the geometric theory of Banach spaces. Furthermore, we give some further facts concerning the implication between LUKK and LNUC. Moreover, the relationship of size between the moduli PXLε and ΔXLε is discussed in Banach spaces, and PXLε is calculated in classical Banach spaces lp meanwhile. We believe that these introduced concepts will be useful and can be used to further solve the problems of accurately reflecting the shape and geometric structure of the unit sphere in Banach space.



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