On Generalized Moduli of Quasi-Banach Space

Abstract

We shall discuss three generalized moduli such as generalized modulus of convexity, modulus of smoothness, and modulus of Zou-Cui of quasi-Banach spaces and give some important properties of these moduli. Furthermore, we establish relationships of these generalized moduli with each other.

1. Introduction

The study on Banach space geometry provides many fundamental notions and interesting aspects and sometimes has surprising results. The basic geometric properties such as convexity, smoothness, and nonsquareness have made great contributions to various fields of Banach space theory. Strict convexity of Banach spaces was first introduced in 1936 by Clarkson [1] (and independently by Akhiezer and Krein) as the property that the unit sphere contains no nontrivial line segments; that is, whenever . Clarkson [1] made use of these values to define the “uniform” version of convexity to look at how “convex” the unit ball is in a space. And the modulus of convexity provides a quantification of the geometric structure of the space from the viewpoint of convexity. A situation similar to this also occurs in smoothness and other properties. A Banach space is said to be smooth if each unit vector has a unique norm one support functional. In fact, this is equivalent to the statement that the norm is Gateaux differentiable. This allows us to quantify the geometric structure of the space from the viewpoint of smoothness, namely, the modulus of smoothness of a Banach space . An advantage of these quantifications is that the complete duality between uniform convexity and uniform smoothness can be easily deduced by the well-known Lindenstrauss formulas; that is, a Banach space is uniformly convex if and only if its dual space is uniformly smooth. The same statement still holds if is replaced with Thus quantifying geometric structures might lead to better results. Note that the same duality does not hold between strict convexity and smoothness in general, though one of those two properties of implies the other of . There are some other ideas to quantify geometric structures of Banach spaces.

In [2], the authors claim that modulus of convexity and generalized convexity mold have dual relationship, and generalized convexity mold has many excellent properties.

In [3], the authors study a generalized modulus of convexity where certain related geometrical properties of this modulus are analyzed in Banach spaces.

In [4, 5], the modulus of Yang-Wang was introduced in Banach spaces.

In [6], the modulus of Zuo-Cui was introduced in Banach spaces. The author proved many results with this special type of modulus.

The most recent research work at this topic can be consulted from [7, 8].

2. Preliminaries

There are lots of quantitative descriptions of geometrical properties of quasi-Banach spaces. The most common way for creating these descriptions is to define a real function (a modulus) and a suitable coefficient or constant closely related to this function, depending on the space structure under consideration. Some of the moduli and their related coefficients (or characteristics) for quasi-Banach spaces have also been investigated so far. These moduli are the attempts in order to get a better understanding of the two facts about the space:(i)The shape of the unit ball of the concerned space.(ii)The conditions and relations for convergence of sequences.

The most recent research work with these moduli is investigated by [7, 8].

Definition 1. For a quasi-Banach space , the modulus of convexity is a function defined asA characteristic or related coefficient of this modulus is

Definition 2. Let and . For a quasi-Banach space , the generalized modulus of convexity is a function defined asA characteristic or related coefficient of this modulus is

Definition 3. For a quasi-Banach space , the modulus of smoothness is a function defined asA characteristic or related coefficient of this modulus is

Definition 4. Let and . For a quasi-Banach space , the generalized modulus of smoothness is a function defined as where , , , and
A characteristic or related coefficient of this modulus is

Definition 5. Let and . For a quasi-Banach space , the modulus of Zuo-Cui is a function defined asA characteristic or related coefficient of this modulus is

3. Relations Concerning Generalized Modulus of Convexity

Lemma 6 (see [9]). Every convex function with convex domain in is continuous.

Proposition 7 (see [10]). Let be a uniformly convex space. Then for every , , and for arbitrary vectors, with , , and , where , there exists such that

Proposition 8 (see [10]). A nontrivial quasi-Banach space is uniformly nonsquare if and only if .

Proposition 9. Let be the modulus of convexity of a quasi-Banach space ; then

Proof. Let , , and . Then we haveand we have obtained the first inequality; now to prove the second one, for any , there exist with , , and such that ; then we haveThis completes the proof.

Corollary 10. Let be the characteristic of generalized modulus of convexity of a quasi-Banach space . Then

Theorem 11. A quasi-Banach space is uniformly nonsquare if and only if

Proof. By Proposition 8, is uniformly nonsquare if and only if , where , and by Proposition 9, if and only if .
Combining both of these results we get the proof.

Lemma 12. Let be a quasi-Banach space and . Then the following statements hold:(1) is convex and continuous function.(2) is a nondecreasing function.(3) is a nondecreasing function.

Proof. LetThenNowby using Proposition 9. Hence we get Since is convex, so it is continuous by Lemma 6.
Let and with . Let us consider Then with ; we havethus which implies that Let ; we have Since , so we have This completes the proof.

Theorem 13. Let be a uniformly convex space and . Then for every , and for arbitrary vectors, with , , and , there exists such that

Proof. Without loss of generality, assume that soNow by Proposition 7, there exist such that, for , Therefore, we getThis completes the proof.

4. Relations Concerning Generalized Modulus of Smoothness

Theorem 14. Let be a quasi-Banach space. Then for every , , and ,

Proof. Throughout the proof of the first part, we take TakeHence, we get This completes the first part of the proof.
To prove the second part, let us start as

Lemma 15. Let be a quasi-Banach space and . Then the following statements hold:(1) is convex and continuous function.(2) is a nondecreasing function.(3) is a nondecreasing function.

Proof. Since is convex, so it is continuous by Lemma 6.
Let , , and
Now let Now From (38) and (40), we get Similarly, Therefore, we havewhich implies thatthus we haveThis shows that is a nondecreasing function.
This part is an immediate consequence of Theorem 14.

Theorem 16. For a nontrivial quasi-Banach space with and , one has

Proof. Let and Then, we haveThis proves the first inequality; now to prove the second inequality, we proceed asAlsoTherefore,This completes the second inequality. Now, combining both of the above inequalities and taking the max of the latter, we get This completes the proof.

Theorem 17. A quasi-Banach space is uniformly smooth if and only if

Proof. One hasHence we getDividing by “” and taking supremum on both sides, we getthuswhich implies thatNowThis can be expressed asDividing by “” and taking supremum on both sidesthusTherefore, from (57) and (61) we get This completes the proof.

4.1. Relations Concerning Modulus of Zuo-Cui

Theorem 18. Let be a nontrivial quasi-Banach space and . Then

Proof. By using convexity of the function on , one can easily obtain so,which implies that For , we have .
For , we have , where

Proposition 19. For a Banach space ,

Proof. Consider a convex function defined by Let and . ThenThis shows that . Therefore, we have Since the opposite inequality holds obviously, to get the first inequality let Since is a convex and an even function, therefore . Now for we havewhich obtains the second inequality. This completes the proof.

Lemma 20. For any quasi-Banach space and any , then the following statements hold:(1) is a nondecreasing function.(2) is convex and continuous function.(3) is a nondecreasing function.

Proof. Let be a convex and even function. Let and . Then we havewhich implies that Hence . Since is convex, so it is continuous by Lemma 6.
Let , , and and . Then we haveThis shows that is convex.
Let and with . Then This shows that is nondecreasing. This completes the proof.

Proposition 21. Let be a quasi-Banach space, , and . Then the following conditions are equivalent:(1)(2).

Proof. Suppose on the contrary that ; it is enough to take . Since by using the definition of , for any , there exist such thatApplying convexity of the function , we get therefore from (79) Since is any arbitrary so which leads a contradiction.
Suppose on the contrary that ; it is enough to take . Again using the definition of , for all , there exist such that also using thus since is arbitrary, so which implies that So using the definition of we get , which leads to a contradiction.

Corollary 22. Let be a quasi-Banach space, , and . Then the following conditions are equivalent:(1) is uniformly nonsquare.(2)(3)

Theorem 23. A quasi-Banach apace is uniformly smooth if

Proof. Suppose that From Theorem 18, we know that which implies that dividing both sides by and applying the So by definition is uniformly smooth.

Theorem 24. Let be a quasi-Banach space. Then for

Proof. If , then holds true.
Let and . Then we have and we haveTherefore, we have

Theorem 25. Let be a quasi-Banach space. Then

Proof. Sincetherefore, we have

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Dong-A University research fund.