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The Extreme Points of the Unit Ball of the James space $J$ and its dual spaces

Spiros A. Argyros, Manuel González

TL;DR

The paper advances the descriptive theory of extreme points for James space and its duals by giving a new proof of Bellenot's extremal-point characterization for $Ext(B_J)$ and an explicit transfinite description of the norm on $J^{**}$, enabling a parallel characterization for $Ext(B_{J^{**}})$. It also provides a complete description of $Ext(B_{J^*})$, including its non-closedness and the relationship to $Ext(B_J)$, revealing deep connections between the extreme-point structures of $J$, $J^*$, and $J^{**}$. The authors develop the innovative machinery of $x$-norming partitions and finest partitions to organize extreme-point analysis, and prove that every nonzero element admits a finest partition, with extremality characterized by equality of the $J$-norm and the $", 2$-norm. Furthermore, the work shows that $Ext(B_J)$ and $Ext(B_{J^*})$ are tightly coupled via norming sets and that certain vectors can possess multiple, even uncountably many, $x$-norming partitions, highlighting rich combinatorial structure within James spaces. These results sharpen the understanding of quasi-reflexive Banach spaces and their duals, with potential implications for the geometry of unit balls in related spaces.

Abstract

We provide a new proof of S. Bellenot's characterization of the extreme points of the unit ball $B_J$ of James quasi-reflexive space $J$. We also provide an explicit description of the norm of $J^{**}$ which yields an analogous characterization for the extreme points of $B_{J^{**}}$. In the last part of the paper we describe the set of all extreme points of $B_{J^*}$ and its norm closure. It is remarkable that the descriptions of the extreme points of $B_J$ and $B_{J^*}$ are closely connected.

The Extreme Points of the Unit Ball of the James space $J$ and its dual spaces

TL;DR

The paper advances the descriptive theory of extreme points for James space and its duals by giving a new proof of Bellenot's extremal-point characterization for and an explicit transfinite description of the norm on , enabling a parallel characterization for . It also provides a complete description of , including its non-closedness and the relationship to , revealing deep connections between the extreme-point structures of , , and . The authors develop the innovative machinery of -norming partitions and finest partitions to organize extreme-point analysis, and prove that every nonzero element admits a finest partition, with extremality characterized by equality of the -norm and the -norm. Furthermore, the work shows that and are tightly coupled via norming sets and that certain vectors can possess multiple, even uncountably many, -norming partitions, highlighting rich combinatorial structure within James spaces. These results sharpen the understanding of quasi-reflexive Banach spaces and their duals, with potential implications for the geometry of unit balls in related spaces.

Abstract

We provide a new proof of S. Bellenot's characterization of the extreme points of the unit ball of James quasi-reflexive space . We also provide an explicit description of the norm of which yields an analogous characterization for the extreme points of . In the last part of the paper we describe the set of all extreme points of and its norm closure. It is remarkable that the descriptions of the extreme points of and are closely connected.
Paper Structure (16 sections, 46 theorems, 93 equations)

This paper contains 16 sections, 46 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some properties of $J$ and $J^*$
  4. Notation
  5. A characterization of the extreme points of $B_J$
  6. The existence of $x$-norming partitions
  7. Properties of $x$-norming families
  8. On the extreme points of $B_J$
  9. Further properties of $Ext(B_J)$
  10. The finest $x$-norming partition
  11. The norm of $J^{**}$ and the extreme points of $B_{J^{**}}$
  12. A norming set for $J^{**}$
  13. The norm of $J^{**}$
  14. The extreme points of $B_{J^{**}}$
  15. The extreme points of $B_{J^*}$
  16. ...and 1 more sections

Key Result

Proposition 3.2

Let $x\in J$ with $\textrm{supp}(x) = \mathbb{N}$ and $(A_n)$ be a sequence of non-empty, finite subsets of $\mathbb{N}$ pointwise convergent to $A\subset \mathbb{N}$. Then $\lim_{n\to\infty} \|x\|_{A_n} = \|x\|_A$ for every $x \in J$.

Theorems & Definitions (97)

  • Definition 1.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • ...and 87 more