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

Abstract

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

1. Introduction

Let be a Banach space, be the dual space of . By and , we denote the closed unit ball and the unit sphere of Banach space , respectively. By and , we denote the convex hull and closed convex hull of the set , respectively. as denotes is weakly converges to as .

It is well known that the condition equivalent to near uniform convexity () 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 be a bounded subset of . Fix for all convex closed sets with , we put where

The functions and are called the moduli of noncompact convexity with Hausdorff measure and Kuratowski measure of , respectively. It is clear that is if and only if for every . Other properties of were investigated in [4–7].

Recall also that a function

J. Banas’ proved that if , then is reflexive and has normal structure (see [3]).

The modulus of was introduced in [8] by J. R. Partington that is where . He proved that has property if and only if whenever . He also proved that the function 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 , , and in Banach spaces have been deeply investigated by Satit Saejung and Ji Gao. The new kind of Banach spaces: , , modulus of , and modulus of 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 the property can be localized. We call this new property named the local uniform Kadec-Klee property (), and we observe that it lies strictly between and properties. By using the same localized method, we localize the modulus of introduced by J. R. Partington and the modulus of noncompact convexity with Hausdorff measure obtain two new moduli and , and we observe that .

2. Preliminaries

Before starting with our results, we need to recall some notions and a lemma in [13–15].

We say that a Banach space has property if for any , , and , then .

We say that a Banach space has () property if for every there exists such that , and then where .

We say that a Banach space has near uniform convexity property () if for every there exists such that with then there is a and scalars with such that .

We say that a Banach space has local near uniform convexity property () if for every and there exists such that for any sequence with then

i.e., for some and (see [16]).

Lemma 1. Let be a Banach Space, and . Then,

3. Materials and Methods

In this paper, we take Kutazrova and Bor-Luh Lin’s approach to localize the property and obtain the property. By using the same localized method, we localize the modulus of which introduced by Partington and the modulus of noncompact convexity with Hausdorff measure and obtain two new moduli and ; then, we study the relationship of size between and 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 space is said to have local uniform property (), for every and there exists such that if , , and then i.e., for some .

Definition 3. Let be a Banach Space. For every and is said to be the modulus of property or local Partington’s coefficient.

Definition 4. For every and , we put where is said to be the modulus of with Kuratowski measure.

Corollary 5. If a Banach space has property, then X has property.

Proof. We prove the contrapositive. Suppose does not have property, then there exists and such that although as , we still have what means there exists and for any such that this implies that holds.

has property, for and mentioned above, there exists such that

i.e., this shows , a contradiction. Thus, the assumption does not hold.

The following conclusion follows from the definitions of and .

Corollary 6. If Banach space has property, then X has property.

It follows from previous Corollaries, we conclude the following Corollary.

Corollary 7. For every Banach space , the implication holds.

We are now ready to prove the main theorems of this paper.

Theorem 8. If a Banach space is , then has property.

Proof. Suppose that does not have property. Then, there exists , and with , with

Since has property, for and mentioned above, there exists such that , which means for some and , we have from (15) it follows that for any we have

Case (i) if , then (16) contradicts with (17).

Case (ii) if , since , then by Lemma 1, we have , since then for any and . Let

It is obvious that , , and . From (16), we get

For another facts, for , let ; then, we have

i.e.,

Thus, which contradicts with (19); therefore, the assumption is not true.

Theorem 9. For every Banach space , we have .

Proof. Fix and take an arbitrary sequence with , as , . For every and some , we let

By the corollary of Hahn-Banach theorem, there exists such that . Picking be small enough and considering the following set

It is obvious that the set is closed, convex, and

Since , then

Then, there exists such that this implies that the set is a subset of . Then, we get

Since , then from (27) it follows that

Consequently,

Thus, . Since is small enough, then we get and the proof is complete.

Theorem 10. For Banach space , we have .

Proof. For every , and , let such that , and . Let then . By the weak lower semicontinuity of norm function, we get Then, there exists a subsequence and such that for all . Hence, Since then we get thus it follows that

Theorem 11. If is a reflexive Banach space, then for any , we have .

Proof. Fix . Take with then ; here, we let for every and . Thus, there exists such that . By the reflexivity of , there exists subsequence and such that . It is obvious that And consequently, this implies that From (41), it follows that Thus, for any , and the proof is complete.

5. Conclusions

In this paper, we introduce a new geometric property that lies between two classical geometric properties () and (). Moreover, two new moduli and for () and 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 and . Moreover, the relationship of size between the moduli and is discussed in Banach spaces, and is calculated in classical Banach spaces 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

The authors are grateful to the referee for comments which improved the paper. This paper is supported by “The National Science Foundation of China” (11871181); “The Science Research Project of Inner Mongolia Autonomous Region” (NJZY18253); “The Science Research Project of Ordos Institution of Applied Technology” (KYYB2017014).