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

Spiros A. Argyros  National Technical University of Athens, Faculty of Applied
Sciences, Department of Mathematics, Zografou Campus, 157 80, Athens, Greece.
<EMAIL_ADDRESS>and  Manuel González  Department of Mathematics, Faculty
of Sciences, University of Cantabria, Avenida de los Castros s/n, 39071
Santander, Spain<EMAIL_ADDRESS>
###### 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.

2020 Mathematics Subject Classification: Primary 46B03, 46B20.
The second author was supported in part by MICINN (Spain), Grant
PID2019-103961GB-C22.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Some properties of $J$ and $J^{*}$
    2. 2.2 Notations
  3. 3 A characterization of the extreme points of $B_{J}$
    1. 3.1 The existence of norming partitions
    2. 3.2 Properties of $x$-norming partitions
    3. 3.3 On the extreme points of $B_{J}$
    4. 3.4 Further properties of $Ext(B_{J})$
    5. 3.5 The finest $x$-norming partition
  4. 4 The norm of $J^{**}$ and the extreme points of $B_{J^{**}}$
    1. 4.1 A norming set for $J^{**}$
    2. 4.2 The norm of $J^{**}$
    3. 4.3 The extreme points of $B_{J^{**}}$
  5. 5 The extreme points of $B_{J^{*}}$
  6. 6 Vectors with multiple $x$-norming partitions

##  1\. Introduction

The James space, constructed by R.C. James ([13]) more than seventy years ago,
showed the existence of a non-reflexive Banach space isomorphic to its second
dual, and a quasi-reflexive, non-reflexive Banach space. This space played a
central role in the development of Banach space theory and was extensively
studied by many authors. We mention James’ paper [14] and also the works of A.
Andrew ([2], [3]), S. Bellenot ([4], [5], [6]) and P.G. Casazza et al. ([8],
[9]). A comprehensive presentation of the properties of James’ space and its
dual space can be found in the monograph ([11]) by H. Fetter and B. Gamboa de
Buen.

In ([13]), James’ space was constructed using the squared variation norm
(Equation 2.1) which makes the unit vector basis $(e_{n})_{n\in\mathbb{N}}$ a
shrinking (Schauder) basis. We denote this space by $J_{s}$. In this paper we
consider a space $J$ isometric to $J_{s}$ for which $(e_{n})_{n\in\mathbb{N}}$
is a boundedly complete basis. It is defined as follows:

A sequence of real numbers $x=(x(n))_{n\in\mathbb{N}}$ is in $J$ when

(1.1) $\|x\|_{J}=\sup\biggl{\\{}\biggl{(}\sum_{i=1}^{n}\big{|}\sum_{k\in
I_{i}}x(k)\big{|}^{2}\biggr{)}^{1/2}:\;\\{I_{i}\\}_{i=1}^{n}\text{ disjoint
intervals of }\mathbb{N}\biggr{\\}}<\infty.$

This norm appeared later in the literature, and R.C. James ([14]) attributed
its definition to J. Lindenstrauss ([15]). Since
$\|x\|_{2}=\left(\sum_{k=1}^{\infty}|x(k)|^{2}\right)^{1/2}\leq\|x\|_{J}$, the
space $J$ is a dense linear subspace of $\ell_{2}$.

In this paper we characterize the extreme points of the unit balls of $J$,
$J^{*}$ and $J^{**}$.

In Section 2 we present some properties of $J$ and $J_{s}$, and we introduce
some notation.

Section 3 is devoted to prove the following characterization of the set of
extreme points of the unit ball of $J$, denoted $Ext(B_{J})$, first proved by
S. Bellenot ([5]):

Theorem 3.13. A vector $x\in J$ is in $Ext(B_{J})$ if and only if
$\|x\|_{J}=\|x\|_{2}=1$.

As a consequence of this characterization, $Ext(B_{J})$ is norm-closed
(Corollary 3.16). Theorem 3.13 is not explicitly stated in [5], where the set
of the extreme points of the unit ball of $J_{s}$ was characterized (see
Corollary 3.22). For doing this, Bellenot developed a maximum path algorithm
for graphs using a language vaguely related to that of Banach space theory.
Here we follow a different approach using the concept of _$x$ -norming
partition_ for $x\in J$, which is a (finite or infinite) family
$\\{I_{i}\\}_{i\in F}$ of disjoint intervals of $\mathbb{N}$ on which $x$
achieves its norm, in the sense that

$\|x\|_{J}=\bigl{(}\sum_{i\in F}\bigl{|}\sum_{n\in
I_{i}}x(n)\bigr{|}^{2}\bigr{)}^{1/2}.$

Note that we can assume that $\\{I_{i}\\}_{i\in F}$ is a partition of
$\mathbb{N}$. We show that every $x\in J$ admits at least one $x$-norming
partition (Corollary 3.4), although a vector $x$ can have more than one
$x$-norming partition (see Section 6).

When we discovered our first proof of Theorem 3.13, we were unaware of the
fact that Bellenot had already proved it. Our proof was based on some
properties of the $x$-norming partitions (Proposition 3.6). Bellenot’s concept
of ”maximal cost paths” helped us reveal some deeper regularity properties.
The key result (Corollary 3.9) shows that given $x\in J$ and two $x$-norming
partitions $\mathcal{I}$ and $\mathcal{L}$, if $I\in\mathcal{I}$ and
$L\in\mathcal{L}$, then either $I\subset L$ or $L\subset I$ or $I\cap
L=\emptyset$. As a consequence, there exists another $x$-norming partition
that refines both $\mathcal{I}$ and $\mathcal{L}$ (Corollary 3.10), and there
exists an $x$-norming partition $\mathcal{M}$ which refines any other
$x$-norming partition (Proposition 3.24). We call $\mathcal{M}$ the _finest
$x$-norming partition_. It corresponds to Bellenot’s maximal cost path and its
existence is the main result of [5]. Our proof of Theorem 3.13 is based on
these regularity properties.

Given a norm-one vector $x\in J$, every $x$-norming partition
$\\{I_{i}\\}_{i\in F}$ generates a extreme point of $B_{J}$ as follows. If
$b_{i}=\sum_{n\in I_{i}}x(n)$ for $i\in F$, then $\sum_{i\in
F}b_{i}e_{k_{i}}\in Ext(B_{J})$ for every strictly increasing sequence
$(k_{i})_{i\in F}$ in $\mathbb{N}$ (Corollary 3.18).

We also describe $Ext(B_{J})$ without explicitly using the norm of $J$. Given
$x\in\ell_{2}$ with $\|x\|_{2}=1$, $x\in Ext(B_{J})$ if and only if
$\bigl{|}\sum_{n\in I}x(n)\bigr{|}^{2}\leq\sum_{n\in I}|x(n)|^{2}$ for every
interval $I$ of $\mathbb{N}$ (Corollary 3.20).

In Section 4 we consider the set $\mathcal{D}\subset B_{J^{*}}$ (see Equation
2.2) and we define $\mathcal{D}_{1}$ as the w∗-closure of $\mathcal{D}$ in
$J^{*}$. We show that $\mathcal{D}_{1}$ norms $J^{**}$. This fact allows us to
give an explicit description of the norm of $J^{**}$, which is a natural
extension of the norm of $J$. Denoting $e_{\omega}=w^{*}$-$\lim_{n}e_{n}\in
J^{**}$, the space $J^{**}$ is generated by the transfinite
$(e_{p})_{p\in\omega+1}$ with the following norm:

$\|x^{**}\|_{**}=sup\biggl{\\{}\bigl{(}\sum_{i\in F}|\sum_{n\in
I_{i}}x^{**}(n)|^{2}\bigr{)}^{1/2}:\\{I_{i}\\}_{i\in F}\ \text{ disjoint
intervals of}\ \omega+1\biggr{\\}}.$

In a similar way, we can define the norm of the transfinite even dual spaces
of $J$. Moreover, the points of $Ext(B_{J^{**}})$ are described similarly to
those of $Ext(B_{J})$.

Section 5 is devoted to the study of $Ext(B_{J^{*}})$. Note that for an
interval $I$ of $\mathbb{N}$, $I^{*}(x)=\sum_{n\in I}x(n)$ defines $I^{*}\in
J^{*}$ with $\|I^{*}\|_{*}=1$. We consider the set

$\mathcal{D}_{1}=\bigl{\\{}\sum_{i\in F}\alpha_{i}I_{i}^{*}:\\{I_{i}\\}_{i\in
F}\text{ disjoint intervals of }\mathbb{N},\,\bigl{(}\sum_{i\in
F}\alpha_{i}^{2}\bigr{)}^{1/2}\leq 1\bigr{\\}},$

which is a $w^{*}$-closed subset of $B_{J^{*}}$ that norms $J$ ( [14]). Hence
$Ext(B_{J^{*}})\subset\mathcal{D}_{1}$ ([12]).

As we have mentioned, the characterization of $Ext(B_{J^{*}})$ is closely
related to that of $Ext(B_{J})$. The following result indicates this fact.

Proposition 5.3. Let $x^{*}=\sum_{i\in
F}\alpha_{i}I_{i}^{*}\in\mathcal{D}_{1}$. Then $\|x^{*}\|_{*}=1$ if and only
if $\sum_{i\in F}\alpha_{i}e_{i}$ is an extreme point of $B_{J}$

Therefore every $x^{*}=\sum_{i\in F}\alpha_{i}I_{i}^{*}\in Ext(B_{J^{*}})$
must satisfy the above condition. Furthermore we show that such an $x^{*}$
with $\alpha_{i}I_{i}^{*}\neq 0$ must also satisfy the property that
$\cup_{i\in F}I_{i}$ is an interval (Proposition 5.4). These two requirements
characterize the extreme points of $B_{J^{*}}$.

Theorem 5.10. A vector $x^{*}\in J^{*}$ is in $Ext(B_{J^{*}})$ if and only if
it satisfies

(i) $x^{*}=\sum_{i\in F}\alpha_{i}I_{i}^{*}$ with $\alpha_{i}I_{i}^{*}\neq 0$
for every $i\in F$, and

(ii) $x=\sum_{i\in F}\alpha_{i}e_{i}\in Ext(B_{J})$ and $\cup_{i\in F}I_{i}$
is an interval of $N$.

In contrast to $Ext(B_{J})$, $Ext(B_{J^{*}})$ is not norm closed. In
Proposition 5.13 we characterize the elements of the norm-closure of
$Ext(B_{J^{*}})$.

In Section 6 we show examples of vectors $x\in J$ such that the cardinal of
the set of $x$-norming partitions takes all possible values: any
$k\in\mathbb{N}$, $\aleph_{0}$ or $\mathfrak{c}$ (Proposition 6.1).

Acknowledgments. We thank I. Gasparis for his help in this research. In
particular, his suggestion to use Choquet’s theorem in Section 5 made it more
clear. We also thank W.B. Johnson and P. Motakis for their comments.

##  2\. Preliminaries

###  2.1. Some properties of $J$ and $J^{*}$

James space [13] can be described using two norms. The first norm is the
original one in [13]. It is called the _square variational norm,_ and it is
defined as follows. For $x\in c_{00}$ we set

(2.1)
$\|x\|_{s}=\frac{1}{\sqrt{2}}\max\biggl{\\{}\bigl{(}\sum_{k=1}^{m}\bigl{|}x(n_{k})-x(n_{k-1})\bigr{|}^{2}\bigr{)}^{1/2}:1\leq
n_{1}<\cdots<n_{m}\biggr{\\}},$

where we assume $n_{0}=0$ and $x(0)=0$.

We denote by $J_{s}$ the completion of $(c_{00},\|\cdot\|_{s})$, where
$c_{00}$ is the space of finitely non-zero real sequences. The unit vector
basis $(e_{n})_{n\in\mathbb{N}}$ is a shrinking basis for $J_{s}$.

The second norm $\|\cdot\|_{J}$ (Equation 1.1) appeared for the first time in
([14]). James attributed its definition to J. Lindenstrauss ([15]). The unit
vector basis $(e_{n})_{n\in\mathbb{N}}$ is a boundedly complete basis for $J$.

A vector $x=\sum_{n}a_{n}e_{n}\in J$ will often be denoted as a scalar
sequence $x\colon\mathbb{N}\to\mathbb{R}$, where $x(n)=a_{n}$ for all
$n\in\mathbb{N}$.

The spaces $J$ and $J_{s}$ are isometric. Indeed, it is easy to check that the
linear map $T:J\to J_{s}$ defined by $T(x)=(\sum_{k=n}^{\infty}x(k))_{n}$ is
an onto isometry. The inverse $T^{-1}$ maps $(a_{n})_{n}\in J_{s}$ to
$(a_{n}-a_{n+1})_{n}\in J$.

James [14] showed that $J^{*}$ is not isomorphic to any subspace of $J_{s}$,
and Andrew [2] showed that no non-reflexive subspace of $J_{s}$ is isomorphic
to a subspace of $J^{*}$. We also mention two important results concerning the
local structure of $J$ and $J^{*}$: James [14] proved that $\ell_{\infty}$ is
finitely representable in $J$, and G. Pisier [16] showed that $J^{*}$ has
cotype $2$, hence $\ell_{\infty}$ is not finitely representable in $J^{*}$.

###  2.2. Notations

For a Banach space $X$, $B_{X}$ and $S_{X}$ are the closed unit ball and the
unit sphere of $X$. The unit vector basis of $J$ is denoted $(e_{n})$.

The _support of $x\in J$_ is the set
$\mathrm{supp}(x)=\\{n\in\mathbb{N}:x(n)\neq 0\\}$. The range of $x$, denoted
by $\mathrm{ran}(x)$, is the minimal interval (finite or infinite) containing
$\mathrm{supp}(x)$. Given $A\subset\mathbb{N}$ and
$x\in\mathbb{R}^{\mathbb{N}}$, we denote by $x|_{A}$ the sequence with
coincides with $x$ for $n\in A$ and it is $0$ otherwise.

In the sequel, any (finite, or infinite) sequence
$\mathcal{I}=\\{I_{i}\\}_{i\in F}$ of disjoint intervals of $\mathbb{N}$ is
indexed so that $maxI_{i}<minI_{i+1}$ for $i\in F$. If $x\in J$ and
$\mathcal{I}=(I_{i})_{i\in F}$ is a sequence of disjoint intervals of
$\mathbb{N}$, we denote $\|x\|_{\mathcal{I}}=\bigl{(}\sum_{i\in
F}\bigl{|}\sum_{n\in I_{i}}x(n)\bigr{|}^{2}\bigr{)}^{1/2}.$

We write $(J^{*},\|\cdot\|_{*})$ for the dual of $J$. Since $(e_{n})$ is a
boundedly complete basis for $J$, the sequence of biorthogonal functionals
$(e_{n}^{*})_{n}\subset J^{*}$ is a shrinking basis for its closed linear span
in $J^{*}$. We denote this subspace of $J^{*}$ by $J_{*}$ because it is a
predual of $J$.

For $I$ an interval of $\mathbb{N}$ we set $I^{*}=\sum_{n\in I}e_{n}^{*}\in
J^{*}$. Moreover,

(2.2)
$\mathcal{D}=\bigl{\\{}\sum_{i=1}^{n}\alpha_{i}I_{i}^{*}:\\{I_{i}\\}_{i}\text{
disjoint finite intervals of
}\mathbb{N},\,\bigl{(}\sum_{i=1}^{n}\alpha_{i}^{2}\bigr{)}^{1/2}\leq
1\bigr{\\}}$

is a subset of $B_{J^{*}}$ that norms $J$ [14], hence
$B_{J^{*}}=\overline{\mathrm{co}}^{w^{*}}(\mathcal{D})$.

We denote by $\\#A$ the cardinal of a set $A$.

##  3\. A characterization of the extreme points of $B_{J}$

###  3.1. The existence of norming partitions

We can always extend a (finite or infinite) sequence
$\mathcal{I}=\\{I_{i}\\}_{i\in F}$ of disjoint intervals of $\mathbb{N}$ to a
partition of $\mathbb{N}$ into intervals. We denote by
$\mathcal{P}_{I}(\mathbb{N})$ the set of all partitions of $\mathbb{N}$ into
intervals.

Observe that
$\|x\|_{J}=\sup\\{\|x\|_{\mathcal{I}}:\mathcal{I}\in\mathcal{P}_{I}(\mathbb{N})\\}$
for every $x\in J$.

######  Definition 3.1.

Let $x\in J$. A sequence $\mathcal{I}=\\{I_{i}\\}_{i\in F}$ of disjoint
intervals of $\mathbb{N}$ is said to be _$x$ -norming_ if
$\|x\|_{J}=\|x\|_{\mathcal{I}}$.

Our first task is to show that every $x\in J$ admits an $x$-norming partition.

######  Definition 3.2.

We set $m_{0}=0$, and for every subset $F$ of $\mathbb{N}$ we define a
partition $\mathcal{P}_{F}\in\mathcal{P}_{I}(\mathbb{N})$ as follows:

  1. (1)

If $F=\emptyset$, then $\mathcal{P}_{F}=\\{\mathbb{N}\\}$.

  2. (2)

If $F=\\{m_{1}<\cdots<m_{k}\\}$ is non-empty and finite, then

$\mathcal{P}_{F}=\\{[m_{j}+1,m_{j+1}]:j=0,1,\ldots
k-1\\}\cup\\{[m_{k}+1,\infty)\\}.$

  3. (3)

If $F=\\{m_{1}<m_{2}<\cdots\\}$ is infinite, then

$\mathcal{P}_{F}=\\{[m_{j}+1,m_{j+1}]:j\in\\{0\\}\cup\mathbb{N}\\}.$

Moreover, we denote $\|x\|_{F}=\|x\|_{\mathcal{P}_{F}}$ for $x\in J$.

Observe that $\|x\|_{J}=\sup_{F\subset\mathbb{N}}\|x\|_{F}$ for every $x\in
J$.

In our next result, by identifying sets with their indicator functions, we
endow the powerset of $\mathbb{N}$ with the topology of pointwise convergence.

######  Proposition 3.3.

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

###### Proof.

Let $x\in J$. Let $P_{m},Q_{m}$ denote the natural projections onto the closed
subspaces generated by $\\{e_{i}:i\leq m\\}$ , $\\{e_{i}:i>m\\}$ respectively.
Then $\|x\|_{F}\leq\|x\|_{J}$ hence $\lim_{m\to\infty}\|Q_{m}x\|_{F}=0$ for
every $F\subset\mathbb{N}$.

We consider three cases:

$F=\emptyset$. In this case $\lim_{k\to\infty}min\,F_{k}=\infty$. Hence

$\lim_{k\to\infty}\|x\|_{F_{k}}=\big{|}\sum_{n=1}^{\infty}x(n)\big{|}=\|x\|_{F}.$

_$F$ finite nonempty._ Denoting $m_{0}=max\,F$, we may assume that
$F=F_{k}\cap[1,m_{0}]$ for each $k$. Then
$\|x\|_{F}^{2}=\|P_{m_{0}}(x)\|_{F}^{2}+\|Q_{m_{0}}(x)\|_{F}^{2}$ and

$\|x\|_{F_{k}}^{2}=\|P_{m_{0}}(x)\|_{F_{k}}^{2}+\|Q_{m_{0}}(x)\|_{F_{k}}^{2}=\|P_{m_{0}(}x)\|_{F}^{2}+\|Q_{m_{0}}(x)\|_{F_{k}\setminus
F}^{2}.$

Since $\lim_{k\to\infty}min\,F_{k}\setminus F=\infty$ we get
$\lim_{k\to\infty}\|Q_{m_{0}}(x)\|_{F_{k}\setminus F}=\|Q_{m_{0}}(x)\|_{F}$.
Hence

$\lim_{k\to\infty}\|x\|_{F_{k}}=\|x\|_{F}.$

_$F$ infinite._ For each $m\in\mathbb{N}$ there exists $k_{m}$ such that
$F_{k}\cap[1,m]=F\cap[1,m]$ for $k>k_{m}$. Hence
$\|P_{m}(x)\|_{F_{k}}=\|P_{m}(x)\|_{F}$ for $k>k_{m}$.

Given $\varepsilon>0$ we choose $m\in\mathbb{N}$ such that
$\|Q_{m}(x)\|_{J}<\varepsilon$. Then, for $k>k_{m}$,

$\|x\|_{F}<\|P_{m}(x)\|_{F_{k}}+\varepsilon\leq\|x\|_{F_{k}}+\varepsilon\,\text{
and }\,\|x\|_{F_{k}}<\|P_{m}(x)\|_{F}+\varepsilon\leq\|x\|_{F}+\varepsilon,$

which yields $\lim_{k\to\infty}\|x\|_{F_{k}}=\|x\|_{F}.$ ∎

######  Corollary 3.4.

For every $x\in J$ there exists $F\subset\mathbb{N}$ such that
$\|x\|_{F}=\|x\|_{J}$. Hence $\mathcal{P}_{F}$ is an $x$-norming partition.

###### Proof.

By the definition of the norm of $J$ there exists a sequence $(F_{n})$ of
finite, non-empty, subsets of $\mathbb{N}$ such that
$\lim_{n}\|x\|_{F_{n}}=\|x\|_{J}$. By compactness, there is no loss of
generality in assuming that $(F_{n})$ is pointwise convergent to some
$F\subset\mathbb{N}$. The assertion is now derived from Proposition 3.3. ∎

######  Remark 3.5.

It follows from the arguments that prove Corollary 3.4 that for each $x\in J$
the $x$-norming partitions define a closed subset of $\\{0,1\\}^{\mathbb{N}}$.
In Section 6 we will see that there exists a non-zero $x\in J$ with
uncountably many $x$-norming partitions.

###  3.2. Properties of $x$-norming partitions

######  Proposition 3.6.

Let $x\in J$ and let $\\{I_{i}\\}_{i\in F}$ be an $x$-norming partition such
that for all $i\in F$ we have $supp(x)\cap I_{i}\neq\emptyset$. Then the
following assertions hold:

  1. (1)

$\mathrm{supp}(x)\subset\cup_{i}I_{i}$.

  2. (2)

For every $G$ subinterval of $F$ we have $\sum_{i\in G}|\sum_{n\in
I_{i}}x(n)|^{2}=\|x|_{\cup_{i\in G}I_{i}}\|_{J}^{2}.$

  3. (3)

For every $i\in F$ and every $n\in I_{i}\cap\mathrm{supp}(x)$, the scalars

$\sum_{m\in I_{i}}x(m),\,\sum_{m\in I_{i},\,m\leq n}x(m),\text{ and}\sum_{m\in
I_{i},\,m\geq n}x(m)$

are all non-zero and have the same sign.

  4. (4)

For every $i\in F$, $\sum_{m\in I_{i}}x(m)$ and $\sum_{m\in I_{i+1}}x(m)$ have
alternate signs.

  5. (5)

Let $i\in F$. If $n_{1}<n_{2}$ are consecutive in $\mathrm{supp}(x)$ with
$x(n_{1})x(n_{2})>0$, then either $\\{n_{1},n_{2}\\}\subset I_{i}$, or
$\\{n_{1},n_{2}\\}\cap I_{i}=\emptyset$.

###### Proof.

(1) If not, then there exists $n\in supp(x)\backslash\cup_{i\in F}I_{i}$ which
yields that $\\{I_{i}\\}_{i\in F}$ is not $x$-norming.

(2) If not, then Corollary 3.4 allows us to choose a partition $\\{L_{m}\\}$
of $\cup_{i\in G}I_{i}$ which is $(x|_{\cup_{i\in G}I_{i}})$ norming. This
yields that the partition $\\{I_{i}\\}_{i=1}^{\infty}$ is not $x$-norming.

(3) Indeed, if $\sum_{m\in I_{i},\,m\leq n}x(m)=0$, then $|\sum_{m\in
I_{i},\,m>n}x(m)|=\|x|_{I_{i}}\|_{J}$, a contradiction, since

$x(n)\neq 0\text{ and }\|x|_{I_{i}}\|_{J}^{2}\geq\bigl{|}\sum_{m\in
I_{i},\,m>n}x(m)\bigr{|}^{2}+|x(n)|^{2}.$

Similar arguments show that all three scalars above have the same sign.

(4) Observe that since for all $i\in F$ we have $supp(x)\cap
I_{i}\neq\emptyset$, then $\sum_{m\in I_{i}}x(m)\neq 0$. If the conclusion
fails, then

$\bigl{|}\sum_{m\in I_{i}\cup I_{i+1}}x(m)\bigr{|}^{2}>\bigl{|}\sum_{m\in
I_{i}}x(m)\bigr{|}^{2}+\bigl{|}\sum_{m\in I_{i+1}}x(m)\bigr{|}^{2}$

which yields that the partition $\\{I_{i}\\}_{i\in F}$ is not $x$-norming.

(5) Indeed, otherwise we would have that $n_{1}\in I_{i}$ and $n_{2}\in
I_{i+1}$, for some interval $I_{i}$ of an $x$-norming partition. Moreover, we
may assume that $n_{1}$ is the maximum of $I_{i}$, while $n_{2}$ is the
minimum of $I_{i+1}$. Properties (3) and (4) now yield a contradiction. ∎

######  Lemma 3.7.

Let $\rho$, $\gamma$ and $\varepsilon$ be scalars with $\varepsilon\neq 0$.
Assume that

$\rho+\gamma^{2}\leq(\gamma+\varepsilon)^{2}.$

Then $\rho+(\gamma+\delta)^{2}<(\gamma+\delta+\varepsilon)^{2}$ for every
scalar $\delta$ such that $\delta\varepsilon>0$.

###### Proof.

Observe that the function $f(x)=(x+\varepsilon)^{2}-x^{2}-\rho$, defined on
$\mathbb{R}$, is strictly increasing for $\varepsilon>0$, and strictly
decreasing for $\varepsilon<0$. Therefore

$\rho+\gamma^{2}\leq(\gamma+\varepsilon)^{2}\implies\rho+x^{2}<(x+\varepsilon)^{2},\,\forall\,x>\gamma,\text{
for }\varepsilon>0$

while

$\rho+\gamma^{2}\leq(\gamma+\varepsilon)^{2}\implies\rho+x^{2}<(x+\varepsilon)^{2},\,\forall\,x<\gamma,\text{
for }\varepsilon<0,$

and the assertion of the lemma is proved. ∎

In the remaining of this subsection we assume, with no loss of generality,
that every $x$-norming partition $\\{I_{i}\\}_{i\in F}$ satisfies the
following condition:

$\forall i\in F,\,\\{minI_{i},maxI_{i}\\}\subset supp(x)\,\text{ if $I_{i}$ is
finite, and}\,minI_{i}\in supp(x)\,\text{otherwise}.$

######  Proposition 3.8.

Let $x=\sum_{n=1}^{\infty}\lambda_{n}e_{n}\in J$, $x\neq 0$, and let
$\mathcal{I}=\\{I_{i}\\}_{i\in F}$ and $\mathcal{L}=\\{L_{j}\\}_{j\in G}$ be
$x$-norming partitions. Assume that there exist $i\in F$ and $j\in G$ such
that

$I_{i}\subset L_{j},\text{ and }minI_{i}=minL_{j}.$

Then for every $i^{\prime}\in F$, $i^{\prime}\neq i$, either
$I_{i^{\prime}}\subset L_{j}$, or $I_{i^{\prime}}\cap L_{j}=\emptyset$.

Therefore, there exists a subinterval $F_{j}$ of $F$ such that

$L_{j}\cap\mathrm{supp}(x)=\cup_{i\in F_{j}}[I_{i}\cap\mathrm{supp}(x)].$

###### Proof.

Assume that the conclusion fails. We prove the case where $j=1$, since the
general case is proved with exactly the same arguments.

(i) Since $j=1$, we conclude that $i=1$.

(ii) Let $I_{k}$ be the unique element of $\\{I_{i}\\}_{i\in F}$ which is
separated by $L_{1}$. Since

$L_{1}\cap supp(x)=\bigl{[}(\cup_{m<k}I_{m})\cup(L_{1}\cap I_{k})\bigr{]}\cap
supp(x),$

we obtain

(3.1) $\sum_{n\in L_{1}}\lambda_{n}=\sum_{p<k}\sum_{n\in
I_{p}}\lambda_{n}+\sum_{n\in L_{1}\cap I_{k}}\lambda_{n}.$

(iii) Equation 3.1 combined with part (3) of Proposition 3.6 yield that the
scalars

$\sum_{m<k}\sum_{n\in I_{m}}\lambda_{n},\,\sum_{n\in L_{1}\cap
I_{k}}\lambda_{n},\,\sum_{n\in I_{k}\setminus L_{1}}\lambda_{n}\text{ and
}\sum_{n\in I_{k}}\lambda_{n}$

are all non-zero and have the same signs. Moreover, $k$ must be an odd integer
by part (4) of Proposition 3.6.

(iv) By the previous part, the scalars:

$\varepsilon=\sum_{m<k}\sum_{n\in I_{m}}\lambda_{n}\,\,\gamma=\sum_{n\in
L_{1}\cap I_{k}}\lambda_{n}\text{ and }\delta=\sum_{n\in I_{k}\setminus
L_{1}}\lambda_{n}$

are all non-zero and have the same sign.

(v) Since $L_{1}\in\mathcal{L}$, (ii) and part (2) of Proposition 3.6 yield

$\sum_{m<k}\bigl{|}\sum_{n\in
I_{m}}\lambda_{n}\bigr{|}^{2}+\gamma^{2}\leq\|x|L_{1}\|_{J}^{2}=\bigl{|}\sum_{n\in
L_{1}}\lambda_{n}\bigr{|}^{2}=(\gamma+\varepsilon)^{2}.$

Setting $\rho=\sum_{m<k}|\sum_{j\in I_{m}}\lambda_{j}|^{2}$, (iv) implies that
$\rho+\gamma^{2}\leq(\gamma+\varepsilon)^{2}$ and, by Lemma 3.7,
$\varepsilon\delta>0$. From that we infer
$\rho+(\gamma+\delta)^{2}<(\gamma+\delta+\varepsilon)^{2}.$ Hence

$\sum_{m=1}^{k}\bigl{|}\sum_{n\in
I_{m}}\lambda_{n}\bigr{|}^{2}<\bigl{|}\sum_{n\in\cup_{m=1}^{k}I_{m}}\lambda_{n}\bigr{|}^{2}.$

We get that $\\{I_{i}\\}_{i\in F}$ is not an $x$-norming partition, because
the partition

$\mathcal{Q}=\\{\cup_{m\leq k}I_{m}\\}\cup\\{I_{m}\\}_{m>k}$

gives an estimate on $x$ strictly greater than that of $\\{I_{i}\\}_{i\in F}$.
This is a contradiction that completes the proof. ∎

The $x$-norming partitions are not necessarily unique, but they have some
regularity properties that we describe in the following corollaries.

######  Corollary 3.9.

Let $x\in J$ and let $\mathcal{I}=\\{I_{i}\\}_{i\in F}$ and
$\mathcal{L}=\\{L_{j}\\}_{j\in G}$ be $x$-norming partitions. Then given $i\in
F$ and $j\in G$, either $I_{i}\subset L_{j}$ or $L_{j}\subset I_{i}$ or
$I_{i}\cap L_{j}=\emptyset$.

###### Proof.

The sets

$F_{1}=\bigl{\\{}i\in F:\,\exists j\in
G,\,minI_{i}=minL_{j}\bigr{\\}},\,\,G_{1}=\bigl{\\{}j\in G:\,\exists i\in
F,\,minI_{i}=minL_{j}\bigr{\\}}.$

are non-empty since both contain $1$. Let $i_{0}\in F_{1}$. Then there exists
a unique $j_{0}\in G_{1}$ such that $minI_{i_{0}}=minL_{j_{0}}$. Hence, by
Proposition 3.8,

$\text{either}\quad I_{i_{0}}\subset L_{j_{0}}\quad\text{or}\quad
I_{i_{0}}\cap supp(x)=\cup_{j\in Q}[L_{j}\cap supp(x)].$

In the first alternative $i_{0}$ and every $j\in G$ fulfill the conclusion,
while in the second one for every $j\in Q$ we have $L_{j}\subset I_{i_{0}}$
and any other $L_{k}$ is disjoint from $I_{i_{0}}$ (Proposition 3.8). The same
holds for $j_{0}\in G_{1}$.

Assume that $i_{0}\in G\setminus G_{1}$ and let $i_{1}=max\\{i\in
G_{1}:i<i_{0}\\}$ and $j_{1}\in G_{1}$ such that $minI_{i_{1}}=minL_{j_{1}}$.
Then Proposition 3.8 yields

$L_{j_{1}}\cap supp(x)=\cup_{i\in Q}[I_{i}\cap supp(x)]\quad\text{and}\quad
i_{0}\in Q.$

Therefore $I_{i_{0}}$ is disjoint to each $L_{j}$ with $j\neq j_{0}$, and the
result is proved. Similar arguments work in the case $j_{0}\in G\setminus
G_{1}$. ∎

Let us see that there exists a joint refinement of any two $x$-norming
partitions.

######  Corollary 3.10.

Let $x\in J$ and let $\mathcal{I}=\\{I_{i}\\}_{i\in F}$ and
$\mathcal{L}=\\{L_{j}\\}_{j\in G}$ be $x$-norming partitions. Then there
exists an $x$-norming partition $\mathcal{M}=\\{M_{p}\\}_{p\in H}$ such that

(1) $\mathcal{M}\subset\mathcal{I}\cup\mathcal{L}$.

(2) For every $i\in F$ and $j\in G$ there exist subintervals $H_{i}$ and
$H_{j}$ of $H$ such that

$I_{i}\cap\mathrm{supp}(x)=\cup_{p\in H_{i}}[M_{p}\cap\mathrm{supp}(x)]\text{
and }L_{j}\cap\mathrm{supp}(x)=\cup_{p\in H_{j}}[M_{p}\cap\mathrm{supp}(x)].$

###### Proof.

Let $\mathcal{M}=\bigl{\\{}I_{i}:\exists j\in G,\,I_{i}\subset
L_{j}\bigr{\\}}\cup\bigl{\\{}L_{j}:\exists i\in F,\,L_{j}\subset
I_{i}\bigr{\\}}$ and write $\mathcal{M}=\\{M_{p}\\}_{p\in H}$. Clearly
$\mathcal{M}$ satisfies (2). Let us see that $\mathcal{M}$ is an $x$-norming
partition.

For $i\in F$, either $I_{i}\in\mathcal{M}$ or
$I_{i}\cap\mathrm{supp}(x)=\cup_{p\in H_{i}}[M_{p}\cap\mathrm{supp}(x)]$. In
the second case, for each $p\in H_{i}$ there exists $j_{p}\in G$ such that
$M_{p}=L_{j_{p}}$. Since both $\mathcal{I}$ and $\mathcal{L}$ are $x$-norming
partitions, we have

$\bigl{|}\sum_{n\in I_{i}}x(n)\bigr{|}^{2}=\sum_{p\in H_{i}}\bigl{|}\sum_{n\in
M_{p}}x(n)\bigr{|}^{2}.$

Hence $\sum_{p\in H}\bigl{|}\sum_{n\in M_{p}}x(n)\bigr{|}^{2}=\sum_{i\in
F}\bigl{|}\sum_{n\in I_{i}}x(n)\bigr{|}^{2}$, proving that $\mathcal{M}$ is
$x$-norming. ∎

In Proposition 3.24 we will show the existence of a finest $x$-norming
partition for each non-zero $x\in J$.

######  Definition 3.11.

A vector $x\in J$ is said to be _separated by partitions_ if, given
$n_{1}<n_{2}$ consecutive elements of $\mathrm{supp}(x)$, there exists an
$x$-norming partition $\mathcal{I}=\\{I_{i}\\}_{i\in F}$ such that $n_{1}$ and
$n_{2}$ belong to different intervals of $\mathcal{I}$.

The next proposition is a consequence of Corollary 3.10 and it is the basic
ingredient for proving the characterization of the extreme points of $B_{J}$.

######  Proposition 3.12.

Let $x\in J$ with $x\neq 0$ which is separated by partitions. Then the
partition $\\{\\{n\\}\\}_{n\in supp(x)}$ is an $x$-norming partition. Hence
$\|x\|_{J}=\|x\|_{2}$.

###### Proof.

Let $supp(x)=(m_{k})_{k\in F}$. We show by induction that for every $k\in F$
there exists an $x$-norming partition $\mathcal{I}_{k}$ such that

$\mathcal{I}_{k}=\bigl{\\{}\\{n\\}:n\leq k\bigr{\\}}\cup\bigl{\\{}I_{k,i}:i\in
G_{k},\,k<i\bigr{\\}}.$

Any $x$-norming partition separating $m_{1},m_{2}$ can be taken as
$\mathcal{I}_{1}$. Assume that $k+1<supF$ and $\mathcal{I}_{k}$ exists. Then
we choose an $x$ norming partition $\mathcal{L}_{k+1}$ which separates
$m_{k+1},m_{k+2}$. Corollary 3.10 yields that there exists a joint refinement
$\mathcal{I}_{k+1}$ of $\mathcal{L}_{k+1},\mathcal{I}_{k}$ which can easily be
checked that it satisfies the inductive conclusion. If $k+1=supF$ then
$\mathcal{I}_{k}$ is an $x$-norming partition consisting from singletons. This
completes the inductive proof.

If $F$ is a finite initial interval of $\mathbb{N}$ the result has been
proved. If $F=\mathbb{N}$ then we set $\mathcal{I}$ to be the pointwise limit
of $(\mathcal{I}_{k})_{k}$, which, from Proposition 3.3, is an $x$-norming
partition and clearly satisfies the conclusion. ∎

###  3.3. On the extreme points of $B_{J}$

The main result of this subsection is the following characterization of the
extreme points of $B_{J}$, denoted $Ext(B_{J})$.

######  Theorem 3.13.

Let $x\in J$. Then $x\in Ext(B_{J})$ if and only if $\|x\|_{J}=\|x\|_{2}=1$.

We postpone the proof of Theorem 3.13. First we collect some facts about the
extreme points of $B_{J}$.

######  Proposition 3.14.

Let $x\in Ext(B_{J})$. Then any pair $n_{1}<n_{2}$ of consecutive elements of
$\mathrm{supp}(x)$ is separated by partitions.

###### Proof.

Suppose that $x=\sum_{n}a_{n}e_{n}\in S_{J}$ and $n_{1}<n_{2}$ are consecutive
elements of $\mathrm{supp}(x)$ not separated by partitions. Corollary 3.4
yields

$\bigl{\|}\sum_{n\leq n_{1}}a_{n}e_{n}\bigr{\|}_{J}^{2}+\bigl{\|}\sum_{n\geq
n_{2}}a_{n}e_{n}\bigr{\|}_{J}^{2}<1.$

Thus, by the continuity of the norm, there exists $t>0$ small enough so that

$\bigl{\|}\sum_{n<n_{1}}a_{n}e_{n}+(a_{n_{1}}+t)e_{n_{1}}\bigr{\|}_{J}^{2}+\bigl{\|}(a_{n_{2}}-t)e_{n_{2}}+\sum_{n>n_{2}}a_{n}e_{n}\bigr{\|}_{J}^{2}<1\,\text{
and }$
$\bigl{\|}\sum_{n<n_{1}}a_{n}e_{n}+(a_{n_{1}}-t)e_{n_{1}}\bigr{\|}_{J}^{2}+\bigl{\|}(a_{n_{2}}+t)e_{n_{2}}+\sum_{n>n_{2}}a_{n}e_{n}\bigr{\|}_{J}^{2}<1.$

Now it is clear that

$\bigl{\|}\sum_{n<n_{1}}a_{n}e_{n}+(a_{n_{1}}+t)e_{n_{1}}+(a_{n_{2}}-t)e_{n_{2}}+\sum_{n>n_{2}}a_{n}e_{n}\bigr{\|}_{J}^{2}\leq
1\,\text{ and }$
$\bigl{\|}\sum_{n<n_{1}}a_{n}e_{n}+(a_{n_{1}}-t)e_{n_{1}}+(a_{n_{2}}+t)e_{n_{2}}+\sum_{n>n_{2}}a_{n}e_{n}\bigr{\|}_{J}^{2}\leq
1.$

Hence $x$ is the average of two different vectors in $B_{J}$, and thus
$x\notin Ext(B_{J})$. ∎

######  Corollary 3.15.

If $x\in Ext(B_{J})$ and $n_{1}<n_{2}$ are consecutive members of
$\mathrm{supp}(x)$, then $n_{1}$ and $n_{2}$ are separated by partitions.
Moreover, $x(n_{1})x(n_{2})<0$.

###### Proof.

The separation between $n_{1}$ and $n_{2}$ by an $x$-norming partition is a
direct consequence of Proposition 3.14. We may thus choose an $x$-norming
partition $\mathcal{I}$ and successive intervals $I_{1}<I_{2}$ of
$\mathcal{I}$ so that $n_{k}\in I_{k}$ for $k=1,2$. Since $n_{1}$ and $n_{2}$
are consecutive members of the support of $x$, there is no loss of generality
in assuming that $n_{1}=\max I_{1}$, $n_{2}=\min I_{2}$ and no interval of
$\mathcal{I}$ lies between $I_{1}$ and $I_{2}$. By applying properties (3) and
(4) of Proposition 3.6, we conclude that $x(n_{1})x(n_{2})<0$. ∎

###### Proof of Theorem 3.13.

Suppose that $x=(y+z)/2$, where $\|y\|_{J}\leq 1$ and $\|z\|_{J}\leq 1$. Since
$\|u\|_{2}\leq\|u\|_{J}$ for all $u\in J$, both $y$ and $z$ lie in
$B_{\ell_{2}}$. By our hypothesis, $x$ is in the unit sphere of $\ell_{2}$.
Then $x\in Ext(B_{\ell_{2}})$, hence $x=y=z$, and thus $x\in Ext(B_{J})$.

Conversely, for $x=\sum_{j=1}^{\infty}a_{j}e_{j}\in Ext(B_{J})$, we will show
that $\sum_{j=1}^{\infty}|a_{j}|^{2}=1$. Indeed, Corollary 3.15 yields that
any $n_{1}<n_{2}$ in $supp(x)$ are separated by an $x$-norming partition.
Hence Proposition 3.12 provides the result. ∎

###  3.4. Further properties of $Ext(B_{J})$

In this subsection we point out a few direct corollaries of Theorem 3.13. We
start with the following result stated in [5].

######  Corollary 3.16.

The set $Ext(B_{J})$ is norm-closed.

###### Proof.

Since $\|x\|_{2}\leq\|x\|_{J}$, there is a continuous natural inclusion
$i:J\to\ell_{2}$, and $Ext(B_{J})=S_{J}\cap i^{-1}(S_{\ell_{2}})$. ∎

Given $x\in J$ with $supp(x)=\\{m_{1}<\dots<m_{k}<\dots\\}$ and a set
$F=\\{n_{1}<\dots<n_{k}<\dots\\}\subset\mathbb{N}$, we denote by $x^{F}$ the
element of $J$ satisfying $x(m_{k})=x^{F}(n_{k})$ for each $k\in\mathbb{N}$
and $suppx^{F}=F$. Similarly, we define $x^{F}$ when $supp(x)$ and $F$ are
finite sets.

######  Corollary 3.17.

For $x$ and $F$ as above, $x$ is in $Ext(B_{J})$ if and only if so is $x^{F}$.

###### Proof.

The definition of $\|\cdot\|_{J}$ implies that the map $e_{m_{k}}\to
e_{n_{k}}$ induces an isometry between
$\overline{\langle(e_{m_{k}})_{k}\rangle}$ and
$\overline{\langle(e_{n_{k}})_{k}\rangle}$. Hence $\|x\|_{J}=\|x^{F}\|_{J}$.
And clearly $\|x\|_{2}=\|x^{F}\|_{2}$. ∎

######  Corollary 3.18.

Let $x\in S_{J}$.

  1. (1)

If $x\in Ext(B_{J})$ then $\|x|I\|_{J}=\|x|I\|_{2}$ for every interval $I$.

  2. (2)

If $(I_{i})_{i\in F}$ is an $x$-norming partition and we set $b_{i}=\sum_{n\in
I_{i}}x(n)$ for every $i\in F$, then $\sum_{i\in F}b_{i}e_{k_{i}}\in
Ext(B_{J})$ for every strictly increasing sequence $(k_{i})_{i\in F}$
$\mathbb{N}$.

###### Proof.

(1) is a consequence of Proposition 3.6.

(2) It is enough to consider the case $k_{i}=i$ for all $n\in\mathbb{N}$. Then
the general case follows from Theorem 3.13 and the fact that $(e_{n})$ is a
spreading basis for $J$.

By part (2) of Proposition 3.6, $\sum_{i\in F}|b_{i}|^{2}=1$. Let
$(E_{i})_{i=1}^{d}$ be a finite sequence of bounded disjoint intervals of
$\mathbb{N}$. Since the $\cup_{i\in F}I_{i}\supset supp\,(x)$,

$\sum_{r=1}^{d}\biggl{|}\sum_{i\in
E_{r}}b_{i}\biggr{|}^{2}=\sum_{r=1}^{d}\biggl{|}\sum_{i\in E_{r}}\sum_{j\in
I_{i}}x(j)\biggr{|}^{2}\leq\|x\|_{J}^{2}.$

Then $y=\sum_{i\in F}b_{i}e_{k_{i}}\in J$ with $\|y\|_{J}\leq 1$, hence
$\|y\|_{J}=\|y\|_{2}=1$, and by Theorem 3.13 we conclude that $y\in
Ext(B_{J})$. ∎

######  Definition 3.19.

A vector $x\in J$ has _non-positive remainder,_ denoted NPR, provided that
$\bigl{|}\sum_{n}x(n)\bigr{|}^{2}\leq\sum_{n}|x(n)|^{2}.$

The vector $x$ has _NPR hereditarily_ if $x|_{I}$ has NPR for every interval
$I$ of $\mathbb{N}$.

Our next corollary is an immediate consequence of the first part of Corollary
3.18.

######  Corollary 3.20.

Let $x\in J$. Then $x\in Ext(B_{J})$ if and only if $x$ has NPR hereditarily
and $\|x\|_{2}=1$.

We derive a characterization of the norm $\|\cdot\|_{*}$ which does not
involve $\|\cdot\|_{J}$.

######  Corollary 3.21.

Let $x^{*}\in J^{*}$. Then,

$\|x^{*}\|_{*}=\sup\\{x^{*}(x):\,x\text{ has NPR hereditarily and
}\|x\|_{2}=1\\}.$

We close this subsection with a characterisation of the extreme points of
$B_{J_{s}}$. We recall that $J_{s}$ is the James space defined with the square
variational norm.

######  Corollary 3.22.

Let $x=\sum_{n}a_{n}e_{n}\in J_{s}$ with $\|x\|_{s}=1$. Then $x\in
Ext(B_{J_{s}})$ if and only if the sequence $(a_{n}-a_{n+1})_{n}$ is $NPR$
hereditarily.

###### Proof.

As we have mentioned the map $(c_{n})_{n}\to(c_{n}-c_{n+1})_{n}$ is an onto
isometry from $J_{s}$ to $J$. Hence $(c_{n})_{n}$ is an extreme point of
$B_{J_{s}}$ if and only if $(c_{n}-c_{n+1})_{n}$ is an extreme point of
$B_{J}$. The result follows from Corollary 3.20. ∎

###  3.5. The finest $x$-norming partition

Here we show that for every non-zero $x\in J$ there exists a finest
$x$-norming partition which corresponds to the maximal cost path in [5]. In
this subsection every $x$-norming partition $\mathcal{I}=\\{I_{i}\\}_{i\in F}$
satisfies

$\forall\,i\in F,\,\\{minI_{i},maxI_{i},\\}\subset supp(x)\,\text{ if
}\,\\#I_{i}<\infty\,\text{ and }\,minI_{i}\in supp(x)\,\text{ otherwise.}$

######  Definition 3.23.

(1) Let $x\in J$ and let $\mathcal{I}=\\{I_{i}\\}_{i\in F}$ and
$\mathcal{L}=\\{L_{j}\\}_{j\in G}$ be $x$-norming partitions. We say that
_$\mathcal{I}$ refines $\mathcal{L}$_ if for every $j\in G$ there exists a
$F_{j}$ subinterval of $F$ such that $L_{j}\cap supp(x)=\cup_{i\in
F_{j}}[I_{i}\cap supp(x)]$.

(2) Let $x\in J$. An $x$-norming partition $\mathcal{I}$ is said to be the
_finest $x$-norming partition_ if it refines any other $x$-norming partition.

If $\mathcal{I}=\\{I_{i}\\}_{i\in F}$ and $\mathcal{L}=\\{L_{j}\\}_{j\in G}$
are $x$-norming partitions and $\mathcal{I}$ refines $\mathcal{L}$ then for
every $i\in F$ and $j\in G$ either $I_{i}\subset L_{j}$ or $I_{i}\cap
L_{j}=\emptyset$.

######  Proposition 3.24.

Every non-zero $x\in J$ admits a finest $x$-norming partition.

###### Proof.

Let $x\in J$. We denote by $\mathcal{P}$ the set of all $x$-norming
partitions, and equip $\mathcal{P}$ with a partial order defined as follows:
$\mathcal{L}\prec\ \mathcal{I}$ if $\mathcal{I}$ refines $\mathcal{L}$.

As in Definition 3.2, we identify each $\mathcal{I}=\\{I_{i}\\}_{i\in
F}\in\mathcal{P}$ with $\mathcal{\hat{I}}\subset\mathbb{N}$, where

$\mathcal{\hat{I}}=\\{minI_{i},maxI_{i}:\\#I_{i}<\infty\\}\cup\\{minI_{i}:\\#I_{i}=\infty\\}.$

So we view $\mathcal{P}$ as a closed subset of $\\{0,1\\}^{\mathbb{N}}$, by
Corollary 3.4 and Remark 3.5. Also Corollary 3.10 implies that
$(\mathcal{P},\prec)$ is a directed net. Therefore there exists a subnet
$\\{\mathcal{I}_{q}\\}_{q}$ converging to $\mathcal{I}=\\{I_{i}\\}_{i\in F}$,
an $x$-norming partition, in the sense that $\\{\mathcal{\hat{I}}_{q}\\}_{q}$
pointwise converges to $\mathcal{\hat{I}}$. We claim that $\mathcal{I}$ is a
finest $x$-norming partition.

Indeed, if $\mathcal{L}=\\{L_{j}\\}_{j\in G}$ is an $x$-norming partition and
$L_{j}$ is a finite interval of $\mathbb{N}$, then there exists another
$\mathcal{M}=\\{M_{p}\\}_{p\in H}\in\mathcal{P}$ that refines $\mathcal{L}$
and $\mathcal{\hat{I}}$ coincides with $\mathcal{\hat{M}}$ on
$\\{1,\dots,maxL_{j}\\}$. Hence there exists a subinterval $F_{j}$ of $F$ such
that

$L_{j}\cap supp(x)=\cup_{i\in F_{j}}[I_{i}\cap supp(x)].$

If $L_{j}$ is an infinite interval, then any $\mathcal{M}=\\{M_{p}\\}_{p\in
H}\in\mathcal{P}$ that refines $\mathcal{L}$ contains a unique $M_{p_{0}}$
such that $minL_{j}=minM_{p_{0}}$. Therefore if $\mathcal{M}$ refines
$\mathcal{L}$ and $\mathcal{\hat{I}}$ coincides with $\mathcal{\hat{M}}$ on
$\\{1,\dots,minL_{j}\\}$, then there exists $I_{i_{0}}$ with
$minL_{J}=minI_{i_{0}}$ which yields

$L_{j}\cap supp(x)=\cup_{i_{0}\leq i}[I_{i}\cap supp(x)],$

and the proof is complete. ∎

######  Remark 3.25.

The existence of the finest $x$-norming partition and Proposition 3.14 easily
imply that for every $x\in Ext(B_{J})$ we have $\|x\|_{J}=\|x\|_{2}$. Indeed,
$x$ is an extreme point if and only if its finest $x$-norming partition
consists of singletons.

##  4\. The norm of $J^{**}$ and the extreme points of $B_{J^{**}}$

Here we give an expression for the norm of $J^{**}$ and we present a
characterisation of the extreme points of the unit ball of $J^{**}$.

###  4.1. A norming set for $J^{**}$

In the sequel, $I_{\infty}^{*}$ and $I_{[k,\infty)}^{*}$ denote the
functionals in $J^{*}$ that correspond to the set $\mathbb{N}$ and to the
final segment of $\mathbb{N}$ starting from $k\in\mathbb{N}$.

The next result will be necessary later. It is well-known ([3], [9]), and it
is also a consequence of the main result of [1].

######  Lemma 4.1.

Let $(x_{n})_{n}$ be a normalized block sequence in $J$ such that
$I_{\infty}^{*}(x_{n})\to 0$. Then $(x_{n})$ has a subsequence equivalent to
$l_{2}$ basis.

The following result is also well-known. We prove it for the sake of
completeness. Here $(e_{n}^{*})_{n}$ is the sequence of biorthogonal
functionals of basis of $J$.

######  Lemma 4.2.

  1. (1)

$\\{e_{n}^{*}:n\in\mathbb{N}\\}\cup\\{I_{\infty}^{*}\\}$ generates a norm-
dense subspace of $J^{*}$.

  2. (2)

$J^{*}=J_{*}\oplus\langle I_{\infty}^{*}\rangle$, where $J_{*}$ is the closed
subspace of $J^{*}$ generated by $(e_{n}^{*})$, which is a predual of $J$.

  3. (3)

The canonical copy of $J$ in $J^{**}$ has codimension one.

###### Proof.

(1) If not, there exists $x^{**}\in S_{J^{**}}$ such that
$\\{e_{n}^{*}:n\in\mathbb{N}\\}\cup\\{I_{\infty}^{*}\\}\subset Ker\,x^{**}$.
Since $J$ is separable and contains no copy of $\ell_{1}$ there exists a
normalized sequence $(x_{k})_{k}$ such that $x_{k}\xrightarrow{w^{*}}x^{**}$.
Observe that for each $n\in\mathbb{N}$, $x_{k}(n)\to 0$. Hence we may assume
that $(x_{k})_{k}$ is a block sequence and $I_{\infty}^{*}(x_{k})\to 0$. So
Lemma 4.1 yields that $(x_{k})_{k}$ has a subsequence equivalent to the
$l^{2}$ basis, a contradiction.

(2) Since the natural basis $(e_{n})_{n}$ is boundedly complete the sequence
$(e_{n}^{*})_{n}$ generates the canonical embedding in $J^{*}$ of the predual
$J_{*}$ of $J$. Clearly, $I_{\infty}^{*}\notin J_{*}$ and the first part of
the proof yields the result.

(3) It is clear that $J^{**}=(J_{*}\oplus\langle
I_{\infty}^{*}\rangle)^{*}=J\oplus\mathbb{R}$. Since $J_{*}$ is the canonical
copy of the predual of $J$ in $J^{*}$, it follows that in the above equality
$J$ denotes the canonical copy of $J$ in $J^{**}$. ∎

Recall from Section 2 the definition of the set $\mathcal{D}\subset J_{*}$
which norms the space $J$:

$\mathcal{D}=\Big{\\{}x^{*}=\sum_{i=1}^{n}\alpha_{i}I_{i}^{*}:\\{I_{i}\\}_{i}\quad\text{disjoint
finite intervals and}\quad(\sum_{i=1}^{n}\alpha_{i}^{2})^{1/2}\leq
1\Big{\\}}.$

The following fact is a consequence of a Theorem of Bessaga and Pelczynski
([7]) and for the shake of completeness we present a proof of it.

######  Lemma 4.3.

The set $\mathcal{D}_{1}=\overline{\mathcal{D}}^{w^{*}}\subset J^{*}$ norms
$J^{**}$.

###### Proof.

We denote by $\|\cdot\|_{**}$ the norm of $J^{**}$ and we will show that

$\|x^{**}\|_{**}=\sup_{x^{*}\in\mathcal{D}_{1}}|x^{**}(x^{*})|,\,\forall\,x^{**}\in
J^{**}.$

Let $x^{*}\in J^{*}$. Using Choquet’s theorem ([10]), we choose a probability
measure $\mu$ on the $w^{*}$-Borel subsets of $B_{X^{*}}$, supported on
$\mathcal{D}_{1}$, such that

$x^{*}(x)=\int_{\mathcal{D}_{1}}xd\mu,\,\forall\,x\in X.$

Let $x^{**}\in J^{**}$. Since $J^{*}$ is separable, Goldstine’s theorem yields
a sequence $(x_{n})$ in $X$ such that $\|x_{n}\|_{J}\leq\|x^{**}\|_{**}$ for
all $n\in\mathbb{N}$ and $x^{**}=w^{*}$-$\lim_{n}x_{n}$. By the dominated
convergence theorem

$x^{**}(x^{*})=\int_{\mathcal{D}_{1}}x^{**}d\mu,$

which yields $|x^{**}(x^{*})|\leq
sup\bigl{\\{}|x^{**}(y^{*})|:y^{*}\in\mathcal{D}_{1}\bigr{\\}}$, which proves
our assertion and the lemma clearly follows. ∎

We denote by $\omega+1$ the set $\mathbb{N}\cup\\{\omega\\}$ with the order
that extends the usual one in $\mathbb{N}$.

######  Remark 4.4.

(1) Since $I_{\infty}^{*}(e_{n})=1$ for each $n\in\mathbb{N}$, $(e_{n})$ is a
weakly Cauchy, non-weakly convergent sequence in $J$. Therefore, there exists
$e_{\omega}\in J^{**}$ such that $e_{n}\xrightarrow{w^{*}}e_{\omega}$.

(2) For $k\in\mathbb{N}$, $I_{[k,\infty)}^{*}(e_{\omega})=1$. Thus
$I_{[k,\infty)}^{*}(x^{**})=\sum_{k\geq n}x^{**}(n)+x^{**}(\omega)$ for all
$x^{**}\in J^{**}$.

###  4.2. The norm of $J^{**}$

Here we consider $\mathcal{D}_{1}$ as a subset of $J^{***}$, $w^{*}$ is the
weak∗-topology on $J^{***}$, and for an interval $I$ of $\omega+1$, $I^{*}\in
J^{***}$ is the associated functional.

######  Lemma 4.5.

  1. (1)

For every interval $I$ in of $\omega+1$ there exists a sequence of intervals
$(I_{n})_{n}\subset\mathcal{D}_{1}$ such that $I_{n}\to I$ pointwise. Hence
$I_{n}^{*}\xrightarrow{w^{*}}I^{*}$.

  2. (2)

$\overline{\mathcal{D}_{1}}^{w^{*}}=\\{\sum_{i\in
F}\alpha_{i}I_{i}^{*}:\sum_{i\in F}\alpha_{i}^{2}\leq 1,\,(I_{i})_{i\in
F}\,\text{ disjoint intervals of }\,\omega+1\\},$ where F is either a finite
initial interval of $\mathbb{N}$ or $\mathbb{N}$ or $\omega+1$.

###### Proof.

(1) As we have said before, $I_{[k,\infty)}^{*}$ acting on $J^{**}$ coincides
with $I^{*}$ where $I=[k,\infty)\cup\\{\omega+1\\}$. Therefore, the only
intervals $I$ of $\omega+1$ such that $I^{*}\notin\mathcal{D}_{1}$ are
$I_{0}=\\{\omega+1\\}$ and $I_{k}=[k,\infty)$. For the first case we set
$I_{n}=[n,\infty)\cup\\{\omega+1\\}$, $n\in\mathbb{N}$ and it easily follows
that $I_{n}^{*}\xrightarrow{w^{*}}I_{0}^{*}$. In the second case we set
$I_{n}=[k,n]$ , $k<n$ and we have that
$I_{n}^{*}\xrightarrow{w^{*}}I_{k}^{*}$.

(2) Consider the functional $x^{***}=\sum_{p\in\omega+1}\alpha_{p}I_{p}^{*}$,
where $\sum_{p\in\omega+1}\alpha_{p}^{2}\leq 1$ and $\\{I_{p}\\}_{p}$ are non-
empty disjoint finite intervals. Clearly, $I_{\omega}=\\{\omega\\}$. For
$n\in\mathbb{N}$ we set

$x_{n}^{***}=\sum_{p=1}^{n}\alpha_{p}I_{p}^{*}+\alpha_{\omega}I_{[k,\infty)}^{*}.$

It is easy to check that $x_{n}^{***}\in\mathcal{D}_{1}$ and
$x_{n}^{***}\xrightarrow{w^{*}}x^{***}$.

The remaining cases can proved in a similar manner. ∎

For $x^{**}\in J^{**}$ and a disjoint family of intervals
$\mathcal{I}=\\{I_{i}\\}_{i\in F}$ of $\omega+1$ we denote

$\|x^{**}\|_{\mathcal{I}}=(\sum_{i\in F}(\sum_{p\in
I_{i}}x^{**}(p))^{2})^{1/2}.$

The proof of the following result is easy.

######  Lemma 4.6.

Let $\mathcal{I}=\\{I_{i}\\}_{i\in F}$ be a disjoint family of intervals of
$\omega+1$. Then

$\|x^{**}\|_{\mathcal{I}}=sup\\{\sum_{i\in
F}\alpha_{i}I_{i}^{*}(x^{**}):\sum_{i\in F}\alpha_{i}^{2}\leq 1\\}\text{ for
each }x^{**}\in J^{**}.$

Since $\mathcal{D}_{1}\subset\overline{\mathcal{D}_{1}}^{w*}\subset
B_{J^{***}}$ and $\mathcal{D}_{1}$ norms $J^{**}$ (Lemma 4.3), Lemmas 4.5 and
4.6 yield the next description of the norm of $J^{**}$:

######  Proposition 4.7.

For $x^{**}\in J^{**}$

$\|x^{**}\|_{**}=sup\biggl{\\{}\bigl{(}\sum_{i\in F}|\sum_{n\in
I_{i}}x^{**}(n)|^{2}\bigr{)}^{1/2}:\\{I_{i}\\}_{i\in F}\ \text{ disjoint
intervals of}\ \omega+1\biggr{\\}},$

where $F$ is either a finite initial interval of $\mathbb{N}$, or
$\mathbb{N}$, or $\omega+1$.

######  Remark 4.8.

As in the case of $J$, for every $x^{**}\in J^{**}$ there exists an
$x^{**}$-norming partition satisfying the properties of Propositions 3.6 and
3.8.

######  Proposition 4.9.

The family $(e_{p})_{p\in\omega+1}$ is a bimonotone transfinite basis for
$J^{**}$.

###### Proof.

For $x=\sum_{i=1}^{n}\alpha_{i}e_{i}+\alpha_{\omega}e_{\omega}$ and
$y=\sum_{i=1}^{n}\alpha_{i}e_{i}$ we have $\|y\|_{**}\leq\|x\|_{**}$.

Indeed, since $y\in\langle(e_{n})_{\in\mathbb{N}}\rangle$, its norm is
determined by a finite sequence of disjoint finite intervals of $\mathbb{N}$.
This family gives the same evaluation on $x$. The remaining cases are treated
in a similar manner. We conclude that the basis $(e_{p})_{p\in\omega+1}$ is
bimonotone for $\langle(e_{p})_{p\in\omega+1}\rangle$, and this property is
extended to $J^{**}$. ∎

######  Remark 4.10.

In the case of $J$ we first defined the norm and then we found the norming set
$\mathcal{D}\subset J^{*}$. For $J^{**}$ we followed a reverse way. First we
proved that the set $\mathcal{D}_{1}$ norms $J^{**}$, and then we used
$\overline{\mathcal{D}_{1}}^{w^{*}}\subset J^{***}$ to define the norm of
$J^{**}$.

The dual spaces of quasireflexive spaces were studied by Bellenot in [4].

######  Remark 4.11.

(1) In the fourth dual of $J$, denoted by $J^{2(**)}$, the basis $(e_{n})_{n}$
of $J$ is not $w^{*}$-converging to $e_{\omega}$ since
$I_{\\{\omega\\}}^{*}\in J^{***}$ separates $(e_{n})_{n}$ from $e_{\omega}$,
but $(e_{n})_{n}$ remains being non trivial weak Cauchy. Hence $J^{2(**)}$ is
generated by $(e_{n})_{n\in\mathbb{N}}$, $e_{\omega}$ and $e_{\omega+1}$ in
the natural order, where $e_{\omega+1}$ is the image of $e_{\omega}\in J^{**}$
by the canonical embedding into $J^{2(**)}$, and the $w^{*}$-limit of
$(e_{n})_{n}$ is the new $e_{\omega}$. Therefore
$\\{e_{n}:n\in\mathbb{N}\\}\cup\\{e_{\omega+1}\\}$ generates the canonical
copy of $J^{**}$ in $J^{2(**)}$.

The norm of $J^{2(**)}$ is defined using the intervals of $\omega+2$ as we did
for $J^{**}$. This argument can repeated for all the spaces $J^{n(**)}$.

(2) The higher even duals of $J$ are more easily described if we view them as
subspaces of $J\mathbb{Q}$, where $\mathbb{Q}$ is the set of rational numbers.
Namely, $J\mathbb{Q}$ is the completion of $c_{00}(\mathbb{Q})$ endowed with a
James type norm defined as follows:

For $x=(x(p))_{p\in\mathbb{Q}}\in c_{00}(\mathbb{Q})$ we set

$\|x\|_{J\mathbb{Q}}=\sup\bigl{\\{}\bigl{(}\sum_{i=1}^{n}|\sum_{p\in
I_{i}}x(p)|^{2}\bigr{)}^{1/2}\,:\,\\{I_{i}\\}_{i}\text{ disjoint intervals of
}\mathbb{Q}\bigr{\\}}.$

For every $n\in\mathbb{N}$, the space $J^{n(**)}$ is isometric to any subspace
of $J\mathbb{Q}$ generated by
$(e_{t_{k}})_{k\in\mathbb{N}}\cup(e_{s_{m}})_{m=1}^{n}$ with
$(t_{k})_{k}\subset\mathbb{Q}$ strictly increasing and
$(s_{m})_{m=1}^{n}\subset\mathbb{Q}$ strictly decreasing with
$\lim_{k}t_{k}<s_{n}$. Moreover the subspace generated by
$(e_{t_{k}})_{k\in\mathbb{N}}\cup(e_{s_{m}})_{m=1}^{l},1\leq l<n$ is isometric
to the natural embedding of $J^{l(**)}$ into $J^{n(**)}$.

The space $J^{\omega(**)}$ is the limit of $J^{n(**)}$. It is isometric to any
subspace of $J\mathbb{Q}$ generated by two sequences
$(e_{t_{k}})_{k\in\mathbb{N}},(e_{s_{m}})_{m\in\mathbb{N}}$ with
$(t_{k})_{k}\subset\mathbb{Q}$ strictly increasing,
$(s_{m})_{m\in\mathbb{N}}\subset\mathbb{Q}$ strictly decreasing and
$\lim_{k}t_{k}<\lim_{m}s_{m}$.

These arguments can be extended inductively to $J^{\alpha(**)}$ for
$\alpha\leq\omega^{2}$.

###  4.3. The extreme points of $B_{J^{**}}$

The proof of the following result is similar to the one we did for
$Ext(B_{J})$. So it is left to the interested reader.

######  Theorem 4.12.

Let $x^{**}=\sum_{j=1}^{\infty}a_{j}e_{j}+\ a_{\omega}e_{\omega}\in
S_{J^{**}}$. Then $x^{**}\in Ext(B_{J^{**}})$ if and only if
$\|x^{**}\|_{**}=\|x^{**}\|_{2}$.

The properties of the extreme points of $B_{J}$ described in Subsection 3.3,
remain valid for the extreme points of $B_{J^{**}}$. In particular the
following analogue of Corollary 3.18 is proved with similar arguments.

######  Corollary 4.13.

Let $x^{**}\in S_{J^{**}}$.

  1. (1)

If $x^{**}\in Ext(B_{J^{**}})$ then $\|x^{**}|I\|_{**}=\|x^{**}|I\|_{2}$ for
every interval $I$.

  2. (2)

If $(I_{i})_{i\in F}$ is an $x^{**}$-norming partition and $b_{i}=\sum_{n\in
I_{i}}x^{**}(n)$ for each $i\in F$, then $\sum_{i\in F}b_{i}e_{k_{i}}\in
Ext(B_{J^{**}})$ for every strictly increasing sequence $(k_{i})_{i\in F}$ in
$\mathbb{N}$.

##  5\. The extreme points of $B_{J^{*}}$

Here we provide a complete description of the extreme points of $B_{J^{*}}$.
As in the previous section, every interval $I$ of $\mathbb{N}$ has associated
$I^{*}\in J^{*}$, we write $I_{[k,\infty)}^{*}$ when $I=[k,\infty)$, the set
$\mathcal{D}$ norms $J$, and we denote
$\mathcal{D}_{1}=\overline{\mathcal{D}}^{w^{*}}\subset J^{*}$.

Standard arguments yield the next result.

######  Lemma 5.1.

The set $\mathcal{D}_{1}$ is the union of the sets

$\biggl{\\{}\sum_{i\in F}\alpha_{i}I_{i}^{*}:\\{I_{i}\\}_{i}\text{ are
disjoint finite intervals and }(\sum_{i\in F}|\alpha_{i}|^{2})^{1/2}\leq
1\biggr{\\}}\quad\text{and}$
$\biggl{\\{}\sum_{i=1}^{n}\alpha_{i}I_{i}^{*}+\alpha_{n+1}I_{[k,\infty)}^{*}:\\{I_{i}\\}_{i}\text{
are disjoint finite intervals and
}(\sum_{i=1}^{n+1}|\alpha_{i}|^{2})^{1/2}\leq 1\biggr{\\}},$

where $F$ is either an initial segment of $\mathbb{N}$ or is equal to
$\mathbb{N}$, and in the second set we assume that $maxI_{n}<k$.

In the sequel, $F$ denotes either a finite initial segment of $\mathbb{N}$, or
$\mathbb{N}$.

######  Remark 5.2.

(1) Since for $x^{*}=\sum_{i\in_{F}}\alpha_{i}I_{i}^{*}\in\mathcal{D}_{1}$ we
have $\|x^{*}\|_{*}\leq(\sum_{i\in F}|\alpha_{i}|^{2})^{1/2}$, when all the
segments $I_{i}$ are finite we have $x^{*}=\sum_{F}\alpha_{i}I_{i}^{*}\in
J_{*}$, the predual of $J$.

(2) For every $x^{*}\in\mathcal{D}_{1}\cap J_{*}$ there exists $x\in S_{J}$
such that $x^{*}(x)=1$.

(3) The only elements of $\mathcal{D}_{1}$ that could not be normed by $B_{J}$
are those of the form
$x^{*}=\sum_{i=1}^{n}\alpha_{i}I_{i}^{*}+\alpha_{n+1}I_{[k,\infty)}^{*}$.

We do not know if there exist any $x^{*}\in\mathcal{D}_{1}$ which does not
attain its norm on $B_{J}$.

Our next result connects the set $\mathcal{D}_{1}$ with $Ext(B_{J})$.

######  Proposition 5.3.

Let $x^{*}=\sum_{F}\alpha_{i}I_{i}^{*}\in\mathcal{D}_{1}$. Then
$\|x^{*}\|_{*}=1$ if and only if $\sum_{i\in F}\alpha_{i}e_{i}$ is an extreme
point of $B_{J}$.

###### Proof.

Suppose $\|x^{*}\|_{*}=1$ and take $x^{**}\in S_{J^{**}}$ such that
$x^{**}(x^{*})=1$ and set $\beta_{i}=I_{i}^{*}(x^{**})$ for $i\in F$. Since
$\sum_{i\in F}\alpha_{i}\beta_{i}=1$, $\sum_{i\in_{F}}|\alpha_{i}|^{2}\leq 1$
and $\sum_{i\in F}|\beta_{i}|^{2}\leq 1$ we conclude $\sum_{i\in
F}|\beta_{i}|^{2}=1$. So $\\{I_{i}\\}_{i}$ is an $x^{**}$-norming partition
with $\alpha_{i}=\beta_{i}$ for all $i\in F$, and Corollary 4.13 (2) yields
that $\sum_{i\in F}\alpha_{i}e_{i}\in Ext(B_{J})$.

Conversely, assume that $x^{*}=\sum_{i\in
F}\alpha_{i}I_{i}^{*}\in\mathcal{D}_{1}$ and $x=\sum_{i\in
F}\alpha_{i}e_{i}\in Ext(B_{J})$. We define $y=\sum_{i\in
F}\alpha_{i}e_{n_{i}}$ where $n_{i}=minI_{i}$ $\forall i\in F$. Then

$\|y\|_{J}=(\sum_{i\in F}|\alpha_{i}|^{2})^{1/2}=1\quad\text{and}\quad
x^{*}(y)=1.$

Hence $\|x^{*}\|_{*}=1$. ∎

Proposition 5.3 (2) shows that the extreme points of $B_{J^{*}}$ are among the
$x^{*}\in\mathcal{D}_{1}$ whose coefficients form an extreme point of $B_{J}$.
The next result shows that not all these $x^{*}\in\mathcal{D}_{1}$ are in
$Ext(B_{J^{*}})$.

######  Proposition 5.4.

Let $x=\sum_{n\in F}\alpha_{n}e_{n}\in Ext(B_{J})$ and let $\\{I_{n}\\}_{n\in
F}$ be a nonempty disjoint family of intervals of $\mathbb{N}$. If $\cup_{n\in
supp(x)}I_{n}$ is not an interval of $\mathbb{N}$, then $x^{*}=\sum_{n\in
F}\alpha_{n}I_{n}^{*}$ is not an extreme point of $B_{J^{*}}$.

###### Proof.

By our assumption there are $k<m$ successive elements of $supp(x)$ such that
$maxI_{k}+1<minI_{m}$. Setting $I_{k,m}=(maxI_{k},minI_{m})$, we have

$I_{k,m}\cap(\cup_{n\in supp(x)}I_{n})=\emptyset.$

We set $J_{k}=I_{k}\cup I_{k,m}$ , $J_{m}=I_{m}\cup I_{k,m}$ which are
intervals of $\mathbb{N}$. Notice that if there exists $n\in F$ such that
$I_{k,m}\cap I_{n}\neq\emptyset$ then the corresponding $\alpha_{n}=0$. We
define

$y^{*}=\sum_{n\in F,\\\ n\neq
k}\alpha_{n}I_{n}^{*}+\alpha_{k}J_{k}^{*}\quad\textrm{and}\quad
z^{*}=\sum_{n\in F,\\\ n\neq m}\alpha_{n}I_{n}^{*}+\alpha_{m}J_{m}^{*}.$

From our assumptions, $\alpha_{k},\alpha_{m}\neq 0$. Since $x\in Ext(B_{J})$
and $k,m$ are successive in $supp(x)$, they must have alternate signs.
Assuming $\alpha_{m}>0$ we get $0\in(\alpha_{k},\alpha_{m})$. Hence there
exists $0<\lambda<1$ such that $\lambda\alpha_{k}+(1-\lambda)\alpha_{m}=0$. It
is easy to check that $x^{*}=\lambda y^{*}+(1-\lambda)z^{*}$ with $y^{*}\neq
z^{*}$. ∎

Propositions 5.3 and 5.4 yield the following result.

######  Corollary 5.5.

Let $x^{*}\in Ext(B_{J^{*}})$. Then $x^{*}=\sum_{i\in F}\alpha_{i}I_{i}^{*}$
with $\alpha_{i}I_{i}^{*}\neq 0$ for each $i\in F$, $x=\sum_{i\in
F}\alpha_{i}e_{i}\in Ext(B_{J})$ and $\cup_{n\in supp(x)}I_{n}$ is an interval
of $\mathbb{N}$.

It is natural to ask if the conclusions (1), (2) determine a characterization
of the extreme points of $B_{J^{*}}$. Next we give a positive answer of this
question.

######  Lemma 5.6.

Let $x^{*},y^{*}\in\mathcal{D}_{1}$ satisfying the following conditions:

$(a)$ $x^{*}=\sum_{i\in F}\alpha_{i}I_{i}^{*}\in\mathcal{D}_{1}$ with
$\alpha_{i}I_{i}^{*}\neq 0$ for every $i\in F$ and

$w=\sum_{i\in F}\alpha_{i}e_{i}\in Ext(B_{J})$.

$(b)$ $y^{*}=\sum_{j\in G}\beta_{j}L_{j}^{*}\in\mathcal{D}_{1}$ such that
$\beta_{j}L_{j}^{*}\neq 0$ for every $j\in G$ and for every $z=\sum_{i\in
F}\alpha_{i}e_{k_{i}}$ with $k_{i}\in I_{i}$ we have $y^{*}(z)=1$.

Then the following assertions hold:

  1. (1)

For $i\in F$ setting $n_{i}=minI_{i}$ and $x=\sum_{i\in
F}\alpha_{i}e_{n_{i}}$, we have that $\\{L_{j}\\}_{j\in G}$ is an $x$-norming
partition.

  2. (2)

For $j\in G$ setting $H_{j}=\\{i:n_{i}\in L_{j}\\}$ and $k_{j}=minH_{j}$, we
have that

$\beta_{j}=\text{sign}(\alpha_{k_{j}})\bigl{(}\sum_{i\in
H_{j}}|\alpha_{i}|^{2}\bigr{)}^{1/2}\,\text{ for every }j\in G.$

  3. (3)

For every $j\in G$ we have $\cup_{i\in H_{j}}I_{i}\subset L_{j}$.

  4. (4)

If, additionally, $\cup_{i\in supp(w)}I_{i}$ is an interval of $\mathbb{N}$
then $\cup_{j\in G}L_{j}$ is also an interval and for every $j\in G$,
$1<j<sup\,G$, we have $L_{j}=\cup_{i\in H_{j}}I_{i}.$

Moreover, if there exists $m\in supp(y^{*})$ with $max\,supp(x^{*})<m$, then
we have $\text{sign}(y^{*}(m))=\text{sign}(\alpha_{d})$ for $d=max\,supp(w)$.

###### Proof.

(1) For $x=\sum_{i\in F}\alpha_{i}e_{n_{i}}$ our assumptions yield
$x^{*}(x)=y^{*}(x)=1$. Therefore

$y^{*}(x)=\sum_{j\in G}\beta_{j}\sum_{i\in
H_{j}}\alpha_{i}=1,\,\sum_{j}|\beta_{j}|^{2}\leq 1,\,\textrm{ and
}\,\sum_{j}|\sum_{i\in H_{j}}\alpha_{i}|^{2}\leq 1,$

which implies $\sum_{j}|\sum_{i\in H_{j}}\alpha_{i}|^{2}=1$; hence
$\\{L_{j}\\}_{j}$ is an $x$-norming partition.

(2) From the previous we conclude that $\beta_{j}=\sum_{i\in H_{j}}\alpha_{i}$
for all $j\in G$. Furthermore, since $x$ is an extreme point and
$\\{L_{j}\\}_{j\in G}$ is an $x$-norming partition,

$|\sum_{i\in H_{j}}\alpha_{i}|^{2}=\sum_{i\in
H_{j}}|\alpha_{i}|^{2}\quad\text{and}\quad\text{sign}(\sum_{i\in
H_{j}}\alpha_{i})=\text{sign}(\alpha_{k_{j}}).$

By part (3) of Proposition 3.6,
$\beta_{j}=\text{sign}(\alpha_{k_{j}})\bigl{(}\sum_{i\in
H_{j}}|\alpha_{i}|^{2}\bigr{)}^{1/2}\,\text{ for every }j\in G.$

(3) Assume that there exists a $j_{0}\in G$ and $i_{0}\in H_{j_{0}}$ such that
$I_{i_{0}}\backslash L_{j_{0}}\neq\emptyset$, and set

$E_{j_{0}}=\\{i\in H_{j_{0}}:I_{i}\backslash L_{j_{0}}\neq\emptyset\\}\,\text{
and }\,Q_{j_{0}}=H_{j_{0}}\backslash E_{j_{0}}.$

Clearly $E_{j_{0}}\neq\emptyset$ and, for $i\in E_{j_{0}}$, we choose
$m_{i}\in I_{i}\backslash L_{j_{0}}$. We set

$z=\sum_{i\notin H_{j_{0}}}\alpha_{i}e_{n_{i}}+\sum_{i\in
Q_{j_{0}}}\alpha_{i}e_{n_{i}}+\sum_{i\in E_{j_{0}}}\alpha_{i}e_{m_{i}}.$

From our assumptions $y^{*}(z)=1$. On the other hand

$\beta_{j_{0}}L_{j_{0}}^{*}(z)=\beta_{j_{0}}\sum_{i\in
Q_{j_{0}}}\alpha_{i}\,\text{ and }\,\sum_{i\in
H_{j_{0}}}\alpha_{i}\neq\sum_{i\in Q_{j_{0}}}\alpha_{i},$

which yields a contradiction, completing the proof of (3).

(4) The first part is a direct consequence of (3) and the fact that
$\cup_{i\in supp(w)}I_{i}$ is an interval. For the second part, since
$\\{L_{j}\\}_{j\in G}$ is an $x$-norming partition, part (3) of Proposition
3.6 yields that for every $j\in G$ and $p_{j}=maxL_{j}\cap supp(x)$

$\text{ sign}\bigl{(}\sum_{i\in
L_{j}}\alpha_{i}\bigr{)}=\text{sign}\alpha_{p_{j}},$

and by (2), $\text{sign}(\beta_{j})=\text{sign}(\alpha_{p_{j}}$). To finish
the argument, observe that $m\in L_{j_{0}}$ with $j_{0}=maxG$. Hence
$\text{sign}(y^{*}(m))=\text{sign}(\beta_{j_{0}})=\text{sign}(\alpha_{d})$. ∎

######  Remark 5.7.

Assume that $x^{*}\in\mathcal{D}_{1}$ with $x^{*}=\sum_{i\in
F}\alpha_{i}I_{i}^{*}$ where $x=\sum_{i\in F}\alpha_{i}e_{i}$ is an extreme
point of $B_{J}$ and $\cup_{i\in supp(x)}I_{i}$ is an interval of
$\mathbb{N}$. Then for $n\notin supp(x)$ such that $I_{n}\neq\emptyset$ it
follows that $n<min\,supp(x)$, or $max\,supp(x)<n$. Thus, in the sequel, we
may assume that $supp(x)=F$ with $F$ an initial segment of $\mathbb{N}$ or
equal to $\mathbb{N}$.

In our next lemma we adopt the notation used in the statement of Lemma 5.6.

######  Lemma 5.8.

Let $x^{*}=\sum_{i\in F}\alpha_{i}I_{i}^{*}\in\mathcal{D}_{1}$ with
$\alpha_{i}I_{i}^{*}\neq 0$ for every $i\in F$ and satisfying

$w=\sum_{i\in F}\alpha_{i}e_{i}\in Ext(B_{J})\,\text{and}\,\cup_{i\in
supp(x)}I_{i}\,\text{ is an interval of}\,\mathbb{N}.$

Then there exists a $w^{*}$-closed face $E$ of $B_{J^{*}}$ such that $x^{*}\in
E$ and every $y^{*}=\sum_{j\in G}\beta_{j}L_{j}^{*}\in E\cap\mathcal{D}_{1}$
with $\beta_{j}L^{*}_{j}\neq 0$, $j\in G$, satisfies the following conditions:

  1. (1)

$\\{L_{j}\\}_{j\in G}$ is an $x$-norming partition for $x=\sum_{i\in
F}\alpha_{i}e_{n_{i}}$ and $n_{i}=minI_{i}$.

  2. (2)

For every $j\in G$, if $H_{j}=\\{i:n_{i}\in L_{j}\\}$ and $k_{j}=minH_{j}$
then

$\beta_{j}=\text{sign}(\alpha_{k_{j}})\bigl{(}\sum_{i\in
H_{j}}|\alpha_{i}|^{2}\bigr{)}^{1/2}.$

  3. (3)

$\cup_{j\in G}L_{j}$ is an interval of $\mathbb{N}$, $\cup_{i\in
H_{j}}I_{i}\subset L_{j}$ for each $j\in G$ and $\cup_{i\in H_{j}}I_{i}=L_{j}$
for each $1<j<sup\,G$. Moreover, if $max\,supp(x^{*})<max\,supp(y^{*})$ then
$\text{sign}(y^{*}(m))=\text{sign}(\alpha_{d})$ for $d=max\,supp(w)$.

###### Proof.

We set $S=\bigl{\\{}\sum_{i\in F}\alpha_{i}e_{m_{i}}\,:\,m_{i}\in
I_{i}\bigr{\\}}$, and for each $s\in S$ we define

$E_{s}=\bigl{\\{}x^{*}\in B_{J^{*}}:x^{*}(s)=1\bigr{\\}}.$

Then $E_{s}$ is a $w^{*}$-closed face of $B_{J^{*}}$ and $x^{*}\in E_{s}$.
Hence $E=\cap_{s\in S}E_{s}$ is a non empty $w^{*}$-closed face and $x^{*}\in
E$.

Let $y^{*}\in E\cap\mathcal{D}_{1}$ and $s\in S$ then $x^{*}(s)=1,y^{*}(s)=1$.
Therefore for every $y^{*}\in E\cap\mathcal{D}_{1}$ the pair $x^{*},y^{*}$
satisfies the assumptions of Lemma 5.6, which yields the desired properties
for $y^{*}$. ∎

######  Proposition 5.9.

Let $x^{*}=\sum_{i\in F}\alpha_{i}I_{i}^{*}\in\mathcal{D}_{1}$ such that
$x=\sum_{i=1}^{n}\alpha_{i}e_{i}\in Ext(B_{J})$ and $\cup_{i\in supp(x)}I_{i}$
is an interval of $N$. Then $x^{*}\in Ext(B_{J^{*}})$.

###### Proof.

For $i\in F$ we can assume $\alpha_{i}I^{*}_{i}\neq 0$. Let $E$ be the face of
$B_{J^{*}}$ defined in Lemma 5.8. Then $Ext(E)\subset
Ext(B_{J^{*}})\subset\mathcal{D}_{1}$, hence
$\overline{co}^{w^{*}}(\mathcal{D}_{1}\cap E)=E.$

Since $\mathcal{D}_{1}\cap E$ is $w^{*}$ compact, the easy version of
Choquet’s theorem ([10], [12]) yields that for $z^{*}\in E$ there exists a
(not necessarily unique) probability measure $\mu_{z^{*}}$ with
$supp(\mu_{z^{*}})\subset\mathcal{D}_{1}\cap E$ which represents $z^{*}$.
Namely, for every $w\in J$

$z^{*}(w)=\int_{E}y^{*}(w)d\mu_{z^{*}}(y^{*}).$

We denote by $\mu_{x^{*}}$ a probability measure with
$supp(\mu_{x^{*}})\subset\mathcal{D}_{1}\cap E$ representing $x^{*}$. We will
prove that $\mu_{x^{*}}=\delta_{x^{*}}$, hence $x^{*}\in Ext(E)$. Otherwise
$x^{*}=(z^{*}+w^{*})/2$ with $z^{*}\neq w^{*}$ and if $\mu_{z^{*}}$,
$\mu_{w^{*}}$ represent $z^{*},w^{*}$ then
$\mu_{x^{*}}=(\mu_{z^{*}}+\mu_{w^{*}})/2\neq\delta_{x^{*}}$.

Let $y^{*}=\sum_{j\in G}\beta_{j}L_{j}^{*}\in\mathcal{D}_{1}\cap E$ with
$\beta_{j}I_{j}\neq 0$ for all $j\in G$. Part (3) of Lemma 5.8 yields
$minL_{1}\leq minI_{1}$. We set

$W_{0}=\biggl{\\{}y^{*}\in\mathcal{D}_{1}\cap E:y^{*}=\sum_{j\in
G}\beta_{j}L_{j}^{*}\text{ and }minL_{1}<minI_{1}\biggr{\\}}\,\text{ and}$

$V_{0}=\biggl{\\{}y^{*}\in\mathcal{D}_{1}\cap E:y^{*}=\sum_{j\in
G}\beta_{j}L_{j}^{*}\ \text{and}\ minL_{1}=minI_{1}\biggr{\\}}.$

Since for $y^{*}=\sum_{j\in G}\beta_{j}L_{j}^{*}\in\mathcal{D}_{1}\cap E$ we
have $minL_{1}\leq minI_{1}$, we conclude

$W_{0}\cup V_{0}=\mathcal{D}_{1}\cap E.$

Let us show that $\mu_{x^{*}}(W_{0})=0$. Indeed, assume that
$W_{0}\neq\emptyset$ and set $m=minI_{1}-1$. Then for $y^{*}=\sum_{j\in
G}\beta_{j}L_{j}^{*}\in W_{0}$, Lemma 5.8 yields

$y^{*}(e_{m})=\beta_{1},\text{
sign}(\beta_{1})=\text{sign}(\alpha_{1}),\,|\beta_{1}|\geq|\alpha_{1}|.$

Assuming that $\mu_{x^{*}}(W_{0})>0$, since $y^{*}(e_{m})=0$ for $y^{*}\in
V_{0}$, we have

$\displaystyle
0<\biggl{|}\int_{W_{0}}y^{*}(e_{m})d\mu_{x^{*}}\biggr{|}=\biggl{|}\int_{W_{0}}y^{*}(e_{m})d\mu_{x^{*}}+\int_{V_{0}}y^{*}(e_{m})d\mu_{x^{*}}\biggr{|}=$
$\displaystyle\biggl{|}\int_{\mathcal{D}_{1}\cap
E}y^{*}(e_{m})d\mu_{x^{*}}\biggr{|}=\biggl{|}\int_{E}y^{*}(e_{m})d\mu_{x^{*}}\biggr{|}=|x^{*}(e_{m})|=0,$

a contradiction. Hence $\mu_{x^{*}}(W_{0})=0$, and $\mu_{x^{*}}(V_{0})=1$
because $\mathcal{D}_{1}\cap E\setminus W_{0}=V_{0}$.

For $k\in F$ with $k<supF$ we set

$V_{k}=\biggl{\\{}y^{*}\in\mathcal{D}_{1}\cap E:y^{*}=\sum_{i\leq
k}\alpha_{i}I_{i}^{*}+\sum_{\begin{subarray}{c}j\in
G,\,k<j\end{subarray}}\beta_{j}L_{j}^{*}\biggr{\\}},$

where $G$ is an initial interval of $\mathbb{N}$ that depends on $y^{*}$, and
we inductively show that $\mu_{x^{*}}(V_{k})=1$. We first prove the inductive
assumption for $k=1$ and $1<supF$.

Observe that $V_{1}\subset V_{0}$. If $y^{*}=\sum_{j\in
G}\beta_{j}L_{j}^{*}\in V_{0}$ with $|H_{1}|=1$ (see Lemma 5.6 for the
definition of $H_{1}$) then, since $1<supF$, we have $L_{1}=I_{1}$. Indeed,
$I_{2}\subset L_{2}$, $I_{1}\cup I_{2}$ is an interval and
$minL_{1}=minI_{1}$. This yields that $y^{*}\in V_{1}$, hence

$V_{0}\setminus V_{1}=W_{1}=\bigl{\\{}y^{*}\in V_{0}:y^{*}=\sum_{j\in
G}\beta_{j}L_{j}^{*}\ \text{and}\ |H_{1}|>1\bigr{\\}}.$

We will show that $\mu_{x^{*}}(W_{1})=0$. We fix $m\in I_{1}$. Then
$y^{*}=\sum_{j\in G}\beta_{j}L_{j}^{*}\in W_{1}$ satisfies

$y^{*}(e_{m})=\beta_{1},\text{
sign}(\beta_{1})=\text{sign}(\alpha_{1}),\,|\beta_{1}|\geq(|\alpha_{1}|^{2}+|\alpha_{2}|^{2})^{1/2}$

by Lemma 5.8 (2). Also for $y^{*}\in V_{1}$ we have that
$y^{*}(e_{m})=\alpha_{1}$. Therefore, assuming that $\mu_{x^{*}}(W_{1})>0$ we
have

$\int_{W_{1}}y^{*}(e_{m})d\mu_{x^{*}}=\gamma\mu_{x^{*}}(W_{1}),\
\text{sign}(\gamma)=\text{sign}(\alpha_{1})\ \text{and}\
|\gamma|>|\alpha_{1}|.$

Hence

$\displaystyle|x^{*}(e_{m})|=\biggl{|}\int_{W_{1}}y^{*}(e_{m})d\mu_{x^{*}}+\int_{V_{1}}y^{*}(e_{m})d\mu_{x^{*}}\biggr{|}=\bigl{|}\gamma\mu_{x^{*}}(W_{1})+\alpha_{1}\mu_{x^{*}}(V_{1})\bigr{|}>|\alpha_{1}|,$

which gives a contradiction since $x^{*}(e_{m})=\alpha_{1}$. Thus
$\mu_{x^{*}}(V_{1})=\mu_{x^{*}}(V_{0})=1$ and the proof for $k=1$ is complete.

Let $k\in F$, $1<k<supF$ and assume that $\mu_{x^{*}}(V_{k-1})=1$. To show
that $\mu_{x^{*}}(V_{k})=1$ we follow the same steps as in the case $k=1$ with
$V_{k-1}$ in the place of $V_{0}$. This completes the inductive proof.

If $x^{*}=\sum_{i=1}^{\infty}\alpha_{i}I_{i}^{*}$ then
$\mu_{x^{*}}=\delta_{x^{*}}$ since $\cap_{k\in\mathbb{N}}V_{k}=\\{x^{*}\\}$
and for all $k\in\mathbb{N}$, $\mu_{x^{*}}(V_{k})=1$.
For $x^{*}=\sum_{i=1}^{d}\alpha_{i}I_{i}^{*}$ with $I_{d}$ an infinite
subinterval of $\mathbb{N}$, then we have proved by induction that
$\mu_{x^{*}}(V_{d-1})=1$ and

$V_{d-1}=\biggl{\\{}y^{*}\in\mathcal{D}_{1}\cap E:y^{*}=\sum_{i\leq
d-1}\alpha_{i}I_{i}^{*}+\beta_{d}L_{d}\biggr{\\}}.$

It follows from part (3) of Lemma 5.8 that $I_{d}\subset L_{d}$. Since
$\cup_{i\in F}I_{i}$ is an interval, we have $minI_{d}=minL_{d}$,
$I_{d}=L_{d}$ and $\beta_{d}=\alpha_{d}$. Hence $V_{d-1}=\\{x^{*}\\}$, and
thus $\mu_{x^{*}}=\delta_{x^{*}}$.

It remains the case when $x^{*}=\sum_{i=1}^{d}\alpha_{i}I_{i}^{*}$ and $I_{d}$
is a finite interval. In this case $V_{d-1}=\\{x^{*}\\}\cup W_{d}$, where

$W_{d}=\bigg{\\{}y^{*}\in
V_{d-1}:y^{*}=\sum_{i=1}^{d-1}\alpha_{i}I_{i}^{*}+\alpha_{d}L_{d}\quad
maxI_{d}<maxL_{d}\biggr{\\}}.$

Inductively we have shown that $\mu_{x^{*}}(V_{d-1})=1$ . Finally, we prove
that $\mu_{x^{*}}(W_{d})=0$. Indeed, set $m=maxI_{d}+1$ and observe that for
$y^{*}\in W_{d}$ , $y^{*}(e_{m})=\alpha_{d}\neq 0$, while $x^{*}(e_{m})=0$.
Therefore,

$0=x^{*}(e_{m})=\int_{V_{d-1}}y^{*}(e_{m})d\mu_{x^{*}}=\int_{W_{d}}y^{*}(e_{m})d\mu_{x^{*}}=\alpha_{d}\mu_{x^{*}}(W_{d}),$

which yields $\mu_{x^{*}}(W_{d})=0$. Hence
$\mu_{x^{*}}(\\{x^{*}\\})=\mu_{x^{*}}(V_{d-1})=1$, implying
$\mu_{x^{*}}=\delta_{x^{*}}$. ∎

Corollary 5.5 and Proposition 5.9 imply the main characterization of
$Ext(B_{J^{*}})$:

######  Theorem 5.10.

A vector $x^{*}\in J^{*}$ is in $Ext(B_{J^{*}})$ if and only if
$x^{*}=\sum_{i\in F}\alpha_{i}I_{i}^{*}$ and the following properties are
satisfied:

  1. (1)

$x=\sum_{i\in F}\alpha_{i}e_{i}$ is an extreme point of $B_{J}$

  2. (2)

$\cup_{i\in supp(x)}I_{i}$ is an interval of $N$.

In contrast to $Ext(B_{J})$ (see Corollary 3.16), $Ext(B_{J^{*}})$ is not norm
closed as the next example shows.

######  Example 1.

We set

$x^{*}=\frac{1}{\sqrt{2}}(e_{1}^{*}-e_{4}^{*})\,\text{ and
}\,x_{m}^{*}=\frac{m}{\sqrt{2(m^{2}+1)}}(e_{1}^{*}-\frac{1}{m}e_{2}^{*}+\frac{1}{m}e_{3}^{*}-e_{4}^{*})\,\text{
for }\,m\in\mathbb{N}.$

It is easy to check that $x^{*},\,(x_{m}^{*})_{m}$ are norm-one vectors such
that

$x^{*}\notin Ext(B_{J^{*}}),\,(x_{m}^{*})_{m}\subset Ext(B_{J^{*}})\,\text{
and }\,\|x_{m}^{*}-x^{*}\|_{*}\to 0.$

This example raises the question whether
$\overline{Ext(B_{J^{*}})}^{\|\cdot\|_{*}}=\mathcal{D}_{1}\cap S_{J^{*}}$.
Next we show that this is not the case.

######  Definition 5.11.

Let $x^{*}\in\mathcal{D}_{1}$ with $x^{*}=\sum_{F}\alpha_{i}I_{i}^{*}$ where
$F$ is an initial interval of $\mathbb{N}$ and $\alpha_{i}\neq 0$ for each
$i\in F$. The _gap function $g_{x^{*}}:F\to\mathbb{N}$ of $x^{*}$_ is defined
as follows:

$g_{x^{*}}(i)=minI_{i+1}-maxI_{i}-1,\text{ if }i<supF,\text{ and
}g_{x^{*}}(i)=0\text{ if }i=supF$

######  Lemma 5.12.

Let $x^{*}\in\mathcal{D}_{1}\cap S_{J^{*}}$. Suppose that $x^{*}=\sum_{i\in
F}\alpha_{i}I_{i}^{*}$ and $\alpha_{i}I^{*}_{i}\neq 0$ for all $i\in F$. If
$1\in g_{x^{*}}[F]$ then $x^{*}\notin Ext(B_{J^{*}})$.

###### Proof.

Since $\|x^{*}\|_{*}=1$, Proposition 5.3 yields that $x=\sum_{i\in
F}\alpha_{i}e_{i}\in Ext(B_{J})$. Therefore, for every $i<supF$ we have that
$\alpha_{i}\alpha_{i+1}<0$ (Corollary 3.15).

Assume, on the contrary, that there exists a sequence
$(x_{m}^{*})_{m\in\mathbb{N}}\subset Ext(B_{J^{*}})$ such that
$x_{m}^{*}\xrightarrow{\|_{\cdot}\|_{*}}x^{*}$. Choose $i_{0}\in F$ such that
$g_{x^{*}}(i_{0})=1$ and set $n_{0}=maxI_{i_{0}}$. Then
$n_{0}+2=minI_{i_{0}+1}$ and since
$x_{m}^{*}\xrightarrow{\|_{\cdot}\|_{*}}x^{*}$ we conclude that

$x_{m}^{*}(e_{n_{0}})\to\alpha_{i_{0}},\,x_{m}^{*}(e_{n_{0}+2})\to\alpha_{i_{0}+1}\text{
and }x_{m}^{*}(e_{n_{0}+1})\to 0.$

We set $x_{m}^{*}=\sum_{i\in F_{m}}\alpha_{i,m}I_{i,m}^{*}$. Theorem 5.10
yields that $\cup_{i\in F_{m}}I_{i,m}$ is an interval and $x_{m}=\sum_{i\in
F_{m}}\alpha_{i,m}e_{i}\in Ext(B_{J})$. Thus, for large $m\in\mathbb{N}$,
$x_{m}^{*}(e_{n_{0}})x_{m}^{*}(e_{n_{0}+2})<0$ and either
$x_{m}^{*}(e_{n_{0}+1})=x_{m}^{*}(e_{n_{0}})$, or
$x_{m}^{*}(e_{n_{0}+1})=x_{m}^{*}(e_{n_{0}+2})$. We get a contradiction, since
$x_{m}^{*}(e_{n_{0}+1})\to 0$ and $\alpha_{i},\alpha_{i+1}\neq 0$. ∎

Now we complete the description of
$\overline{Ext(B_{J^{*}})}^{\|\cdot\|_{*}}$.

######  Proposition 5.13.

Let $x^{*}=\sum_{i\in F}\alpha_{i}I_{i}^{*}\in\mathcal{D}_{1}\cap S_{J^{*}}$
with $\alpha_{i}I^{*}_{i}\neq 0$ for all $i\in F$. Then
$x^{*}\in\overline{Ext(B_{J^{*}})}^{\|\cdot\|_{*}}$ if and only if $1\notin
g_{x^{*}}[F]$.

###### Proof.

The direct implication is a consequence of Lemma 5.12.

For the converse, assume that $1\notin g_{x^{*}}[F]$, and set $G=\\{i\in
F:g_{x^{*}}(i)>0\\}$. If $G=\emptyset$ then $x^{*}\in Ext(B_{J^{*}})$
(Proposition 5.3, Theorem 5.10). Thus we assume that $G\neq\emptyset$.

We define a sequence $(x_{m}^{*})_{m\in\mathbb{N}}\subset Ext(B_{J^{*}})$ such
that $x_{m}^{*}\xrightarrow{\|_{\cdot}\|_{*}}x^{*}$, and we proceed as in
Example 1. For $m\in\mathbb{N}$, we choose positive scalars $(\beta_{i})_{i\in
G}$ with $\sum_{i\in G}\beta_{i}^{2}<1/(2m^{2})$, we decompose the interval
$(maxI_{i},minI_{i+1})$ into two nonempty subintervals $L_{i}<M_{i}$ for each
$i\in G$, and we define $z_{m}^{*}=\sum_{i\in
G}\bigl{(}-sign(\alpha_{i})\beta_{i}L_{i}^{*}+sign(\alpha_{i})\beta_{i}M_{i}^{*}\bigr{)}.$

Note that $\|z_{m}^{*}\|_{*}<1/(\sqrt{2}m)$. So, setting
$y_{m}^{*}=x^{*}+z_{m}^{*}$ and
$x_{m}^{*}=\frac{1}{\|y_{m}^{*}\|_{*}}y_{m}^{*}$, it is easy to check that
$(x_{m}^{*})_{m\in\mathbb{N}}\subset Ext(B_{J^{*}})$ and
$\|x_{m}^{*}-x^{*}\|_{*}\to 0$, completing the proof. ∎

##  6\. Vectors with multiple $x$-norming partitions

This is an appendix to Section 3. We provide examples of $x\in Ext(B_{J})$
admitting multiple $x$-norming partitions, as it is stated in the following
result.

######  Proposition 6.1.

Let $\alpha$ be either $\mathrm{k}\in\mathbb{N}$ or $\aleph_{0}$ or
$\mathfrak{c}$. Then there exists $x\in Ext(B_{J})$ admitting exactly $\alpha$
$x$-norming partitions. These values of $\alpha$ are the only possible
cardinalities of a set of $x$-norming partitions.

We will give the proof at the end of this section.

######  Lemma 6.2.

Let $x=a_{1}e_{1}+a_{2}e_{2}+a_{3}\in\mathbb{R}^{3}$ such that $a_{1}>0$,
$a_{2}<0$ and $a_{3}>0$. Then the only possible $x$-norming partitions are
$\mathcal{I}=\\{1,2,3\\}$ and $\mathcal{L}=\\{\\{1\\},\\{2\\},\\{3\\}\\}$.

###### Proof.

Since $a_{1}a_{2}<0$, $\\{1,2\\}$ is not an interval of an $x$-norming
partition and the same happens for $\\{a_{2},a_{3}\\}$. ∎

######  Lemma 6.3.

The set

$E=\bigl{\\{}\sum_{i=1}^{3}a_{i}e_{i}\in\mathbb{R}^{3}:\,a_{1},a_{3}>0,\,a_{2}<0,\,\sum_{i=1}^{3}a_{i}=1\text{
and }\,\sum_{i=1}^{3}a_{i}^{2}=1\bigr{\\}}$

satisfies the following properties:

  1. (1)

$E$ is uncountable.

  2. (2)

For every $0<a_{1}<1$ there exists a unique pair $a_{2},a_{3}$ in $\mathbb{R}$
such that

$a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}\in E.$

  3. (3)

For every $a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}\in E$ we have
$|a_{2}|<min\\{a_{1},a_{2}\\}$. If additionally $a_{1}\geq 1/\sqrt{2}$ then
$a_{1}>a_{3}$.

###### Proof.

(1) We consider the hyperplane
$H=\\{a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}\in\mathbb{R}^{3}:a_{1}+a_{2}+a_{3}=1\\}$
and we set $C=H\cap B_{\mathbb{R}^{3}}$. Then $C$ is the circumcircle of the
triangle with vertices $\\{e_{1},e_{2},e_{3}\\}$. We claim that $E$ is the set
of points of the circumference of $C$ lying strictly between $e_{1}$ and
$e_{3}$.

Indeed, the center of $C$ is $x_{0}=\sum_{i=1}^{3}\frac{1}{3}e_{i}$, and for
$x\in C$ lying strictly between $e_{1}$ and $e_{2}$, the segments $[x_{0},x]$
and $[e_{1},e_{2}]$ satisfy $[x_{0},x]\cap[e_{1},e_{2}]=\\{z\\}$. Then

$z=sx_{0}+(1-s)x=te_{1}+(1-t)e_{2},$

hence $z(2)=s\frac{1}{3}+(1-s)x(2)=0$. This yields that $x(2)<0$. Moreover
$x(1)$ and $x(3)$ are positive because $x(1)+x(2)+x(3)=1\,\text{ and
}\,\|x\|_{\infty}\leq 1.$

The same arguments show that any other vector in $C$ does not belong to $E$.

(2) Observe that $E$ is connected and, if $P_{1}$ is the projection onto the
first coordinate, then $P_{1}(E)=(0,1)$, showing the existence of
$a_{2},a_{3}$. Moreover the hyperplane $P_{1}^{-1}(a_{1})$ meets $E$ in
exactly one point. This fact implies the uniqueness of the pair $a_{2},a_{3}$.

(3) It follows from the properties of $E$. Namely, $\sum_{i=1}^{3}a_{i}=1$ and
$\sum_{i=1}^{3}a_{i}^{2}=1$. ∎

Clearly, if $x\in E$ then $x\in Ext(B_{J})$ and admits two different
$x$-norming partitions.

######  Lemma 6.4.

Let $G$ be an initial interval of $\mathbb{N}$ and let $(a_{i})_{i\in
G}\subset\mathbb{R}$ satisfying

$\quad a_{1}>0,\quad\sum_{i\in
G\setminus\\{1\\}}a_{i}>0\quad\text{and}\quad|\sum_{i\in
G}a_{i}|^{2}=\sum_{i\in G}a_{i}^{2}.$

Then for $0<b_{1}<a_{1}$ we have $|b_{1}+\sum_{i\in
G\setminus\\{1\\}}a_{i}|^{2}<b_{1}^{2}+\sum_{i\in
G\setminus\\{1\\}}a_{i}^{2}$.

###### Proof.

From $|\sum_{i\in G}a_{i}|^{2}=\sum_{i\in G}a_{i}^{2}$ we obtain

$a_{1}\bigl{(}\sum_{i\in
G\setminus\\{1\\}}a_{i}\bigr{)}+\sum_{\begin{subarray}{c}i,j\neq 1,i\neq
j\end{subarray}}a_{i}a_{j}=0\,\text{ and }\,a_{1}\bigl{(}\sum_{i\in
G\setminus\\{1\\}}a_{i}\bigr{)}>0.$

These facts and $0<b_{1}<a_{1}$ yield

$b_{1}\bigl{(}\sum_{i\in
G\setminus\\{1\\}}a_{i}\bigr{)}+\sum_{\begin{subarray}{c}i,j\neq 1,i\neq
j\end{subarray}}a_{i}a_{j}<0,$

which implies $|b_{1}+\sum_{i\in
G\setminus\\{1\\}}a_{i}|^{2}<b_{1}^{2}+\sum_{i\in
G\setminus\\{1\\}}a_{i}^{2}.$ ∎

Our next result is the key to construct vectors $x\in J$ with multiple
$x$-norming partitions.

######  Proposition 6.5.

$(a)$ Let $(r_{i})_{i=1}^{k}$ be a finite strictly increasing sequence with
$r_{1}\geq 1/\sqrt{2}$ and $r_{k}=1$. Then there exists a unique finite
sequence $(a_{j})_{j=1}^{2k+1}$ satisfying

  1. (1)

$a_{1}=r_{1}$, and for $1<m<k$,

$\sum_{j=1}^{2m+1}a_{j}=r_{m+1}\,\text{ and
}\,\bigl{|}\sum_{j=1}^{2m+1}a_{j}\bigr{|}^{2}=\sum_{j=1}^{2m+1}a_{j}^{2}.$

  2. (2)

For every subinterval $I$ of $\\{1,\dots,2k+1\\}$ with $\\#I>1$ and either
$1<minI$ or $minI=1$ and $maxI=2n$, we have $|\sum_{j\in
I}a_{j}|^{2}<\sum_{j\in I}a_{j}^{2}$.

$(b)$ If $(r_{k})_{k\in\mathbb{N}}$ is strictly increasing with $r_{1}\geq
1/\sqrt{2}$ and $\lim r_{k}=1$ then there exists a sequence
$(a_{j})_{j\in\mathbb{N}}$ satisfying the analogous to the previous conditions
(1), (2) in $(a)$.

###### Proof.

$(a)$ We define $(a_{j})_{j=1}^{2k+1}$ as follows. For $j=1$ we set
$a_{1}=r_{1}$. For $m=2,\dots,k$, by part $(2)$ of Lemma 6.3, there are unique
numbers $a_{2m},a_{2m+1}$ satisfying

$r_{m}+a_{2m}+a_{2m+1}=r_{m+1}\quad\text{and}\quad
r_{m}^{2}+a_{2m}^{2}+a_{2m+1}^{2}=r_{m+1}^{2}.$

This completes the definition of $(a_{j})_{j=1}^{2k+1}$. It is easy to check
that

$\sum_{j=1}^{2m+1}a_{j}=r_{m+1}\,\text{ and
}\,|\sum_{j=1}^{2m+1}a_{j}|^{2}=\sum_{j=1}^{2m+1}a_{j}^{2},$

and part $(2)$ of Lemma 6.3 implies that the sequence $(a_{j})_{j=1}^{2k+1}$
is unique. So the proof of $(1)$ is complete.

To prove $(2)$, note that for $1\leq m\leq k-1$ the vector
$x_{m}=r_{m}e_{1}+a_{2m-2}e_{2}+a_{2m-1}e_{3}$ belongs to the set

$E_{m}=\bigl{\\{}\sum_{i=1}^{3}a_{i}e_{i}\in\mathbb{R}^{3}:\,a_{1},a_{3}>0,\,a_{2}<0,\,\sum_{i=1}^{3}a_{i}=r_{m+1}\,\text{
and }\,\sum_{i=1}^{3}a_{i}^{2}=r_{m+1}^{2}\bigr{\\}}.$

Hence $a_{2m}<0$ and $a_{2m+1}>0$. Since $r_{m}\geq\frac{1}{\sqrt{2}}$, fact
(3) in Lemma 6.3 yields

$0<a_{2m+1}<r_{m},\quad a_{2m}<0\quad\text{and}\quad|a_{2m}|<a_{2m+1}.$

Let $I$ be a subinterval of $\\{1,\dots,2k-1\\}$ with $\\#I>1$. First we
assume that $1<2m+1=minI$ and $maxI=2l+1$. Then

$\sum_{j=2m+1}^{2l+1}a_{j}=\bigl{(}\sum_{j=1}^{2l+1}a_{j}-\sum_{j=1}^{2m+1}a_{j}\bigr{)}+a_{2m+1}=(r_{l+1}-r_{m+1})+a_{2m+1}>0$

and $\sum_{j=2m+2}^{2l+1}a_{j}=r_{l+1}-r_{m+1}>0$. Therefore

$|r_{m+1}+\sum_{j=2m+2}^{2l-1}a_{j}|^{2}=r_{m+1}^{2}+\sum_{j=2m+2}^{2l+1}a_{j}^{2}\quad\text{and}\quad
r_{m+1}>a_{2m+1}.$

The last inequality holds since $r_{m+1}>r_{m}>a_{2m+1}$. Thus the assumptions
of Lemma 6.4 are fulfilled, and we get $|\sum_{j\in I}a_{j}|^{2}<\sum_{j\in
I}a_{j}^{2}$.

This completes the proof for subintervals of $\\{1,\dots,2k-1\\}$ with odd
ends. The other cases can be reduced to this one.

Indeed, assume first that $minI=2m+1$ and $maxI=2l$. Then from the previous
case we get $\sum_{j=2m+1}^{2l-1}a_{j}>0$ and $a_{2l}<0$. Hence

$|\sum_{j=2m+1}^{2l}a_{j}|^{2}<|\sum_{j=2m+1}^{2l-1}a_{j}|^{2}+a_{2l}^{2}\leq\sum_{j=2m+1}^{2l}a_{j}^{2}.$

With the same arguments we treat the case $minI=2m$ and $maxI=2l+1$. The
remaining case is $minI=2m$ and $maxI=2l$.

Again, as in the previous cases, we have $\sum_{j=2m}^{2l-1}a_{j}>0$ and
$a_{2l}<0$. And similarly we conclude that

$|\sum_{j=2m}^{2l}a_{j}|^{2}<|\sum_{j=2m}^{2l-1}a_{j}|^{2}+a_{2l}^{2}<\sum_{j=2m}^{2l}a_{j}^{2}.$

To finish the proof of $(2)$ of part $(a)$ we examine the case $min\,I=1$ and
$max\,I=2n$, in which we have

$\sum_{j=1}^{2n}a_{j}=\sum_{j=1}^{2n-1}a_{j}+a_{2n}=r_{n}+a_{2n}$

with $r_{n}>0$ , $a_{2n}<0$ and $|a_{2n}|<r_{n}$. Hence

$|\sum_{j=1}^{2n}a_{j}|^{2}=|r_{n}+a_{2n}|^{2}<|r_{n}|^{2}<\sum_{j=1}^{2n}a_{j}^{2}.$

To prove $(b)$, let $(r_{k})_{k\in\mathbb{N}}$ as in the statement. We define
the sequence $(a_{j})_{j\in\mathbb{N}}$ in the same way as in the finite case,
and it satisfies

$\forall k\in\mathbb{N}\,\sum_{j=1}^{2k+1}a_{j}=r_{k+1}\,\text{ and
}\,|\sum_{j=1}^{2k+1}a_{j}|^{2}=\sum_{j=1}^{2k+1}a_{j}^{2}=r_{k+1}^{2}.$

Clearly $a_{j}\to 0$. Hence $\sum_{j=1}^{\infty}a_{j}=1$ and
$|\sum_{j=1}^{\infty}a_{j}|^{2}=\sum_{j=1}^{\infty}a_{j}^{2}=1.$

The uniqueness of $(a_{j})_{j\in\mathbb{N}}$ can be proved as in the finite
case. Thus part $(1)$ of $(b)$ is proved. To prove $(2)$, we shall show that
for every interval $I$ of $\mathbb{N}$ with $\\#I>1$ and either $minI>1$ or
$minI=1$ and $maxI=2n$, we have $|\sum_{j\in I}a_{j}|^{2}<\sum_{j\in
I}a_{j}^{2}$.

If $I$ is finite and satisfies one of the above two conditions then the proof
is identical with the corresponding one in part $(a)$. Thus we assume that $I$
is infinite and $minI>1$.

If $minI=2m+1$, then

$\sum_{j=2m+1}^{\infty}a_{j}=\sum_{j=2m+2}^{\infty}a_{j}+a_{2m+1}=1-r_{m+1}+a_{2m+1}>0\,\text{
and }$

$\sum_{j=2m}^{\infty}a_{j}=1-r_{m}>0\,\text{ and }\,r_{m}>a_{2m+1}.$

Hence Lemma 6.4 yields the result.

If $minI=2m$ then $a_{2m}<0$ and $\sum_{j=2m+1}^{\infty}a_{j}>0$. Hence

$|\sum_{j=2m}^{\infty}a_{j}|^{2}<a_{2m}^{2}+|\sum_{j=2m+1}^{\infty}a_{j}|^{2}<\sum_{j=2m}^{\infty}a_{j}^{2},$

part $(2)$ in $(b)$ is proved and the proof is complete. ∎

For $x\in J$, we denote
$\mathcal{N}(x)=\\{\mathcal{I}:\mathcal{I}\,\,x\text{-norming partition}\\}$.

######  Corollary 6.6.

$(a)$ For every $k\in\mathbb{N}$ there exists $x_{k}\in Ext(B_{J})$ such that
$\\#\mathcal{N}(x_{k})=k$.

$(b)$ There exists $x_{\infty}\in Ext(B_{J})$ such that
$\\#\mathcal{N}(x_{\infty})=\aleph_{0}.$

###### Proof.

(a) Let $k\in\mathbb{N}$. If $k=1$ then every $e_{n}$ satisfies the
conclusion. If $k>1$ we choose $(r_{i})_{i=1}^{k}$ and set
$x_{k}=\sum_{j=1}^{2k-1}a_{j}e_{j}$, where $(a_{j})_{j=1}^{2k-1}$ is the
sequence associated to $(r_{i})_{i=1}^{k}$ in Proposition 6.5.

Let $\\{I_{p}\\}_{p\in G}$ be an $x_{k}$-norming partition. Then Proposition
6.5 yields that there is at most one $I_{p}$ with $\\#I_{p}>1$ and, if it
exists, then $minI_{p}=1,maxI_{p}=2l-1$. Therefore $x_{k}\in Ext(B_{J})$ and
$\\#\mathcal{N}(x_{k})=k$.

$(b)$ The vector $x_{\infty}$ is defined in a similar way. ∎

######  Proposition 6.7.

The set of all $x\in Ext(B_{J})$ for which $\\#\mathcal{N}(x_{k})>\aleph_{0}$
is uncountable.

###### Proof.

We take $x=a_{1}e_{1}+a_{2}e_{2}+a_{3}\in E$ and we set
$y=-a_{3}e_{4}-a_{2}e_{5}-a_{1}e_{6}$. Then we decompose $\mathbb{N}$ into
$(F_{n})_{n\in\mathbb{N}}$ disjoint intervals with $\\#F_{n}=6$ $\forall
n\in\mathbb{N}$. We also denote $x_{n}+y_{n}$ the shift of $x+y$ into the
interval $F_{n}$, and $z=\sum_{n=1}^{\infty}2^{-n}(x_{n}+y_{n}).$

Let $\\{I_{i}\\}_{i\in G}$ be a $z$-norming partition. Then

  1. (1)

For every $i\in G$ there exists $n\in\mathbb{N}$ such that

$\text{either }I_{i}\subset supp(x_{n})\,\text{ or }\,I_{i}\subset
supp(y_{n}).$

  2. (2)

For every $i\in G$ there exists $n\in\mathbb{N}$ such that

$\text{either }I_{i}=supp(x_{n})\,\text{ or }\,I_{i}=supp(y_{n})\,\text{ or
}\,\exists k\in F_{n}\,\text{ with }\,I_{i}=\\{k\\}.$

Both assertions are easily established. Therefore, if we denote

$\mathcal{I}_{x_{n}}=\\{\\{k\\}:k\in
supp(x_{n})\\},\quad\mathcal{L}_{x_{n}}=supp(x_{n})$

and we define

$P=\prod_{n=1}^{\infty}\bigl{\\{}\mathcal{I}_{x_{n}},\mathcal{L}_{x_{n}}\bigr{\\}}\times\bigl{\\{}\mathcal{I}_{y_{n}},\mathcal{L}_{y_{n}}\bigr{\\}},$

then every $\mathcal{I}\in P$ defines a $z$-norming partition, showing that
$(\|z\|_{J})^{-1}z\in Ext(B_{J})$ and admits uncountable many $z$-norming
partitions. Since $z$ is different for different vectors in $E$, the result is
proved. ∎

######  Remark 6.8.

If for some $x\in J$ the set of $x$-norming partitions is uncountable, then
this set has the cardinality of the continuum. Indeed, we showed in Remark 3.6
that this set is in correspondence one-to-one with a closed subset of
$\\{0,1\\}^{N}$.

###### Proof of Proposition 6.1.

It is a direct consequence of Corollary 6.6, Proposition 6.7 and Remark 6.8. ∎

## References

  * [1] I. Amemiya and I. Ito, Weakly null sequences in James spaces on trees, Kodai Math. J. 4 (1981), 418–425.
  * [2] A. Andrew, James’s quasi-reflexive space is not isomorphic to any subspace of its dual, Israel J. Math. 38 (1981), 276–282.
  * [3] A. Andrew, Spreading basic sequences and subspaces of James’ quasi-reflexive space, Math. Scand. 48 (1981), 109–118.
  * [4] S. Bellenot, Transfinite duals of quasireflexive Banach spaces, Trans. Amer. Math. Soc. 273 (1982), 551–577.
  * [5] S. Bellenot, The maximum path theorem and extreme points of James’s space, Studia Math. 94 (1989), 1–15.
  * [6] S. Bellenot Isometries of James spaces, Contemp. Math. A.M.S. 85 (1989), 1–18.
  * [7] C. Bessaga and A. Pełczyński, On extreme points in separable conjugate space, Israel J. Math. 4 (1966), 262–264.
  * [8] P.G. Casazza, James’ quasi-reflexive space is primary, Israel J. Math 26 (1977), 294–305.
  * [9] P.G. Casazza, B.L. Lin, and R. H. Lohman, _On James’ quasi-reflexive Banach space_ , Proc. Amer. Math. Soc. 67 (1977), 265–271.
  * [10] G. Choquet, Theory of Capacities, Annales de l’Institut Fourier, 5 (1954), 131–295.
  * [11] H. Fetter and B. Gamboa de Buen, The James Forest  Lecture Note Series London Math. Soc. 236, 1997.
  * [12] V.P. Fonf, J. Lindenstrauss and R.R. Phelps Infinite Dimensional Convexity, Handbook of the Geometry of Banach spaces. North Holland vol. I (2001), 599–670.
  * [13] R.C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174–177.
  * [14] R.C. James, Banach spaces quasi-reflexive of order one, Studia Math. 60 (1977), 157–177.
  * [15] J. Lindenstrauss, On James’s paper ”Separable conjugate spaces”, Israel J. Math. 9 (1971), 279–284.
  * [16] G. Pisier, The dual $J^{*}$ of the James space has cotype $2$ and the Gordon-Lewis property, Math. Proc. Cambridge Phil. Soc. 103 (1988), 323–331.