数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (2): 461-474.doi: 10.1007/s10473-021-0210-5

• 论文 • 上一篇    下一篇

THE BALL-COVERING PROPERTY ON DUAL SPACES AND BANACH SEQUENCE SPACES

商绍强   

  1. College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China
  • 收稿日期:2020-01-06 修回日期:2020-05-20 出版日期:2021-04-25 发布日期:2021-04-29
  • 作者简介:Shaoqiang SHANG,E-mail:sqshang@163.com
  • 基金资助:
    This research is supported by the "China Natural Science Fund" under grant 11871181 and the "China Natural Science Fund" under grant 12026423.

THE BALL-COVERING PROPERTY ON DUAL SPACES AND BANACH SEQUENCE SPACES

Shaoqiang SHANG   

  1. College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China
  • Received:2020-01-06 Revised:2020-05-20 Online:2021-04-25 Published:2021-04-29
  • About author:Shaoqiang SHANG,E-mail:sqshang@163.com
  • Supported by:
    This research is supported by the "China Natural Science Fund" under grant 11871181 and the "China Natural Science Fund" under grant 12026423.

摘要: In this paper, we prove that $(X,p)$ is separable if and only if there exists a $w^{*}$-lower semicontinuous norm sequence $\{ {p_n}\} _{n = 1}^\infty $ of $(X^{*},p)$ such that (1) there exists a dense subset $G_{n}$ of $X^{*}$ such that $p_{n}$ is G$\mathrm{\hat{a}}$teaux differentiable on $G_{n}$ and $dp_{n}(G_{n})\subset X$ for all $n\in N$; (2) $p_n \leq p$ and $p_n \to p$ uniformly on each bounded subset of $X^{*}$; (3) for any $\alpha\in(0,1)$, there exists a ball-covering $\{ B({x_{i,n}^{*}},{r_{i,n}})\} _{i = 1}^\infty $ of $(X^{*},p_{n})$ such that it is $\alpha$-off the origin and ${x_{i,n}^{*}}\in G_{n}$. Moreover, we also prove that if $ X_{i}$ is a G$\mathrm{\hat{a}}$teaux differentiability space, then there exist a real number $\alpha > 0$ and a ball-covering $\mathfrak{B_{i}}$ of $X_{i}$ such that $\mathfrak{B_{i}}$ is $\alpha $-off the origin if and only if there exist a real number $\alpha > 0$ and a ball-covering $\mathfrak{B}$ of ${l^\infty }({X_i})$ such that $\mathfrak{B}$ is $\alpha$-off the origin.

关键词: Ball-covering property, separable space, G$\mathrm{\hat{a}}$teaux differentiable point, weak$^{*}$ exposed point

Abstract: In this paper, we prove that $(X,p)$ is separable if and only if there exists a $w^{*}$-lower semicontinuous norm sequence $\{ {p_n}\} _{n = 1}^\infty $ of $(X^{*},p)$ such that (1) there exists a dense subset $G_{n}$ of $X^{*}$ such that $p_{n}$ is G$\mathrm{\hat{a}}$teaux differentiable on $G_{n}$ and $dp_{n}(G_{n})\subset X$ for all $n\in N$; (2) $p_n \leq p$ and $p_n \to p$ uniformly on each bounded subset of $X^{*}$; (3) for any $\alpha\in(0,1)$, there exists a ball-covering $\{ B({x_{i,n}^{*}},{r_{i,n}})\} _{i = 1}^\infty $ of $(X^{*},p_{n})$ such that it is $\alpha$-off the origin and ${x_{i,n}^{*}}\in G_{n}$. Moreover, we also prove that if $ X_{i}$ is a G$\mathrm{\hat{a}}$teaux differentiability space, then there exist a real number $\alpha > 0$ and a ball-covering $\mathfrak{B_{i}}$ of $X_{i}$ such that $\mathfrak{B_{i}}$ is $\alpha $-off the origin if and only if there exist a real number $\alpha > 0$ and a ball-covering $\mathfrak{B}$ of ${l^\infty }({X_i})$ such that $\mathfrak{B}$ is $\alpha$-off the origin.

Key words: Ball-covering property, separable space, G$\mathrm{\hat{a}}$teaux differentiable point, weak$^{*}$ exposed point

中图分类号: 

  • 46B20