- Research Article
- Open Access

# Nonsquareness and Locally Uniform Nonsquareness in Orlicz-Bochner Function Spaces Endowed with Luxemburg Norm

- Shaoqiang Shang
^{1}Email author, - Yunan Cui
^{2}and - Yongqiang Fu
^{1}

**2011**:875649

https://doi.org/10.1155/2011/875649

© Shaoqiang Shang et al. 2011

**Received:**5 July 2010**Accepted:**12 February 2011**Published:**9 March 2011

## Abstract

Criteria for nonsquareness and locally uniform nonsquareness of Orlicz-Bochner function spaces equipped with Luxemburg norm are given. We also prove that, in Orlicz-Bochner function spaces generated by locally uniform nonsquare Banach space, nonsquareness and locally uniform nonsquareness are equivalent.

## Keywords

- Banach Space
- Measurable Function
- Function Space
- Convergence Theorem
- Measurable Space

## 1. Introduction

A lot of nonsquareness concepts in Banach spaces are known (see [1]). Nonsquareness are important notions in geometry of Banach space. One of reasons is that these properties are strongly related to the fixed point property (see [2]). The criteria for nonsquareness and locally uniform nonsquareness in the classical Orlicz function spaces have been given in [3, 4] already. However, because of the complicated structure of Orlicz-Bochner function spaces equipped with the Luxemburg norm, the criteria for nonsquareness and locally uniform nonsquareness of them have not been found yet. The aim of this paper is to give criteria for nonsquareness and locally uniform nonsquareness of Orlicz-Bochner function spaces equipped with Luxemburg norm.

Let be a real Banach space. and denote the unit sphere and unit ball, respectively. Let us recall some geometrical notions concerning nonsquareness. A Banach space is said to be nonsquare if for any we have . A Banach space is said to be uniformly nonsquare if there exists such that for any , . A Banach space is said to be locally uniformly nonsquare if for any , there exists such that , where .

Let be set of real numbers. A function is called an -function if is convex, even, , and , and .

First let us recall a known result that will be used in the further part of the paper.

Lemma 1.1 (see [3]).

## 2. Main Results

Theorem 2.1.

is nonsquare if and only if

(a) ;

(b) is nonsquare.

In order to prove the theorem, we give a lemma.

Lemma 2.2.

Proof.

Case 1.

Case 2.

This implies . This completes the proof.

- (a)

Hence, . But , and we deduce that . Moreover, we have and , a contradiction with nonsquareness of .

It is easy to see , a contradiction!

We will derive a contradiction for each of the following two cases.

Case 1.

Since , we have . Hence, . This implies , a contradiction!

Case 2.

Since , we have . Hence, . This implies , a contradiction!

Theorem 2.3.

is locally uniformly nonsquare if and only if

(a) ;

(b) is locally uniformly nonsquare.

In order to prove the theorem, we give a lemma.

Lemma 2.4.

If is locally uniformly nonsquare, then

Case 1.

Case 2.

holds.

We will derive a contradiction for each of the following two cases.

Case 1.

This implies , a contradiction!

Case 2.

This implies , a contradiction! Hence, .

which contradict (2.32). Hence, .

Combing (b1) with (b2), we get . This completes the proof.

Proof of Theorem 2.3.

*Necessity*. By Theorem 2.1, . If (b) is not true, then there exist , such that and as . Pick such that . Put

It is easy to see , as . By Lemma 1.1, we have and as , a contradiction with locally uniform nonsquareness of .

*Sufficiency*. Suppose that there exist
,
such that
as
. We will derive a contradiction for each of the following two cases.

Case 1.

By Lemma 1.1, we have , as . This is in contradiction with .

Case 2.

for large enough. By Lemma 1.1, we have , as , which contradicts , for large enough. This completes the proof.

Corollary 2.5.

The following statements are equivalent:

(a) is locally uniformly nonsquare if and only if is nonsquare;

(b) is locally uniformly nonsquare.

## Declarations

### Acknowledgments

The authors would like to thank the anonymous referee for some suggestions to improve the manuscript. This work was supported by China Natural Science Fund under Grant no. 11061022.

## Authors’ Affiliations

## References

- James RC:
**Uniform nonsquare Banach space.***Annals of Mathematics*1964,**80**(3):542–550. 10.2307/1970663MathSciNetView ArticleMATHGoogle Scholar - García-Falset J, Llorens-Fuster E, Mazcuñan-Navarro EM:
**Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings.***Journal of Functional Analysis*2006,**233**(2):494–514. 10.1016/j.jfa.2005.09.002MathSciNetView ArticleMATHGoogle Scholar - Chen ST:
**Geometry of Orlicz spaces.***Dissertationes Math*1996,**356:**1–204.Google Scholar - Wu CX, Wang TF, Chen ST, Wang YW:
*Geometry Theory of Orlicz Spaces*. H.I.T Print House, Harbin, China; 1986.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.