A New Characterization of -Uniformly Rotund Banach Space

Abstract

A new characterization of -uniformly rotund Banach space with is given. Moreover, a corresponding result in the locally -uniformly rotund Banach space with is given.

1. Introduction

In the geometric theory of Banach spaces the concept of uniform convexity plays a very significant role and is frequently used in functional analysis. The concept of a uniformly rotund (or uniformly convex) Banach space was first introduced by Clarkson [1] in 1936 and this class of Banach space is very interesting and has numerous applications (cf. [2–8]). In 1979, Sullivan [9] introduced the -uniformly rotund spaces as a generalization of uniformly rotund Banach spaces. Indeed, the 1-uniformly rotund Banach spaces coincide with usually uniformly rotund Banach spaces.

The purpose of this paper is to give a character inequality of -uniformly rotund Banach space (or locally -uniformly rotund Banach space) with . Throughout the sequel, the symbol denotes a real Banach space and denotes its dual space. and denote, respectively, the unit ball and the unit sphere in . For , the -dimensional volume enclosed by is given by

A Banach space is said to be -uniformly rotund [9] if, for any , there is a , such that, for , if then .

A Banach space is said to be locally -uniformly rotund [9] if, for , , there is a , such that, for , if then .

2. A New Characterization of -Uniformly Rotund Banach Spaces

Theorem 1. Let , let be a Banach space, and let be an arbitrary bounded subset of . Then, is -uniformly rotund space if and only if, for any , there exists , such that the inequality holds for all and with .

In order to prove Theorem 1, we give three lemmas.

Lemma 2 (Yu [10]). is -uniformly rotund space if and only if for any sequences if , , , then

For the sake of completeness of this paper, here we present the proof of Lemma 2.

The sufficiency of Lemma 2 is clear.

The Proof of Necessity. Without loss of generality, we may assume that .

Suppose that , satisfying the conditions given in Lemma 2. Then, for each , by the assumption that , there exists , such that the inequality holds for all .

On the other hand, since , , so there exists , such that the inequality holds for all .

Therefore, by letting , , , we can deduce that By the assumption that is -uniformly rotund space, we may take for any . Therefore, by the above proof, there exists an corresponding to such that the inequality holds for all .

Furthermore, by using inequality (8), we easily obtain the desired result that

Lemma 3. Let , then one has where , and the sign of equality holds if and only if .

Proof. () When , we construct a function ; then Obviously, It is easy to see that the function attains its maximum value at point and . Hence ; that is, And the sign of equality holds if and only if .
() Suppose the conclusion of Lemma 3 is true when ; that is, the inequality holds and the sign of equality holds if and only if .
() When , we construct a multivariate function then Now, let us fix variables . Then the function attains its maximum value at point . Hence This shows that the inequality holds and the sign of equality holds if and only if .
Combining (), (), and (), we have and the sign of equality holds if and only if .

Lemma 4. Let , then one has where .

Proof. () When , the conclusion of Lemma 4 is obvious. When , we construct a function then It is easy to see that the function attains its maximum value at point . Hence
() Suppose the conclusion of Lemma 4 is true when ; that is, we have
() When , we construct a multivariate function then Now, let us fix variables . Then the function attains its maximum value at point . Hence,
By using inequality (24) we have
Combining (), (), and (), we have

Proof of Theorem 1.   
Proof of Sufficiency. Suppose that, for , there is a , such that the inequality holds for all with .
Let and ; then .
By the definition of -uniformly rotund space, we know that is -uniformly rotund space.
Proof of Necessity. Suppose inequality (2) is not true. Then there exist , , such that, for , there exist , satisfying . When , we have Take , . Then By Lemma 3 we know that It follows that
Now we prove that
If (35) does not hold, then there exists subsequence of satisfying Hence, by Lemma 4 we have
Considering a function and noticing that we know that is strictly increasing (decreasing) function when ) and attains its maximum value at point .
Hence this contradicts (34), so .
Similarly, we can deduce that . It follows that Let ; then This means that But This contradicts that is -uniformly rotund space from Lemma 2.

Theorem 5. Let , let be a Banach space, and let be an arbitrary bounded subset of . Then, is locally -uniformly rotund space if and only if, for any and , there exists , such that the inequality holds for all with .

The proof of Theorem 5 is greatly similar to the proof of Theorem 1.

In particular, considering the special cases of Theorems 1 and 5 when , we give a new characterization of uniformly rotund (resp., locally uniformly rotund) Banach space; that is, we have the following two corollaries.

Corollary 6. Let , let be a Banach space, and let be an arbitrary bounded subset of . Then, is uniformly rotund space if and only if, for any , there exists , such that the inequality holds for all and with .

Corollary 7. Let , let be a Banach space, and let be an arbitrary bounded subset of . Then, is locally uniformly rotund space if and only if, for any and , there exists , such that the inequality holds for all with .

Conflict of Interests

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

Acknowledgments

The authors thank the referee for very thorough reading the paper and suggesting a useful comment. The project is supported by the Natural Science Foundation of Inner Mongolia (Grant no. 2012MS1022) and Foundation of Inner Mongolia Normal University RCPY-2-2012-K-034.