On the weak uniform rotundity of Banach spaces

Wen D Chang,Ping Chang

doi:10.1155/s0161171203206359

Wen D Chang, Ping Chang

Open Access

https://doi.org/10.1155/s0161171203206359

Copy DOI

Abstract

We prove that ifXi,i=1,2,…,are Banach spaces that are weak* uniformly rotund, then theirlpproduct space(p>1)is weak* uniformly rotund, and for any weak or weak* uniformly rotund Banach space, its quotient space is also weak or weak* uniformly rotund, respectively.

Highlights

Let A ⊂ X be a closed subset and X/A denote the quotient space
We prove that if Xi, i = 1, 2, . . . , are Banach spaces that are weak∗ uniformly rotund, their lp product space (p > 1) is weak∗ uniformly rotund, and for any weak or weak∗ uniformly rotund Banach space, its quotient space is weak or weak∗ uniformly rotund, respectively
A Banach space X is URA, where A is a nonempty subset of X∗, if and only if for any pair of sequences {xn} and {yn} in S(X), if xn + yn → 2, f → 0 for all f in A

Summary

Introduction

Let A ⊂ X be a closed subset and X/A denote the quotient space. We use S(X) for the unit sphere in X and Plp (Xi) for the lp product space. A Banach space X is URA , where A is a nonempty subset of X∗, if and only if for any pair of sequences {xn} and {yn} in S(X), if xn + yn → 2, f (xn − yn) → 0 for all f in A . A Banach space X is WUR (weakly uniformly rotund) if and only if X is URX∗ .

Results

Conclusion