On Generalized Moduli of Quasi-Banach Space

Young Chel Kwun,Absar Ul Haq,Waqas Nazeer,Hussain Minhaj Uddin Ahmad Qadri,Shin Min Kang

doi:10.1155/2018/7324783

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.

Highlights

The study on Banach space geometry provides many fundamental notions and interesting aspects and sometimes has surprising results
The modulus of convexity provides a quantification of the geometric structure of the space from the viewpoint of convexity
A Banach space X is said to be smooth if each unit vector has a unique norm one support functional

Summary

Introduction

The study on Banach space geometry provides many fundamental notions and interesting aspects and sometimes has surprising results. The modulus of convexity provides a quantification of the geometric structure of the space from the viewpoint of convexity. A Banach space X is said to be smooth if each unit vector has a unique norm one support functional. 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 X. In [3], the authors study a generalized modulus of convexity where certain related geometrical properties of this modulus are analyzed in Banach spaces. The most recent research work at this topic can be consulted from [7, 8]

Preliminaries

Relations Concerning Generalized Modulus of Convexity

Relations Concerning Generalized Modulus of Smoothness

Relations Concerning Modulus of Zuo-Cui