# The HI extension of the standard HI spaces

Spiros A. Argyros  National Technical University of Athens, Faculty of Applied
Sciences, Department of Mathematics, Zografou Campus, 157 80, Athens, Greece.
<EMAIL_ADDRESS>,  Antonis Manoussakis  School of Environmental
Engineering, Technical University of Crete, University Campus, 73100 Chania,
Greece<EMAIL_ADDRESS>and  Pavlos Motakis  Department of Mathematics
and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3,
Canada<EMAIL_ADDRESS>
###### Abstract.

A Hereditarily Indecomposable (HI) Banach space $X$ admits an HI extension if
there exists an HI space $Z$ such that $X$ is isomorphic to a subspace $Y$ of
$Z$ and $Z/Y$ is of infinite dimension. The problem whether or not every HI
space admits an HI extension is attributed to A. Pelczynski. In this paper we
present a method to define HI-extensions of the standard HI spaces, a class
which includes the Gowers-Maurey space, asymptotic $\ell_{p}$-HI spaces and
others.

2010 Mathematics Subject Classification: Primary 46B03, 46B06.

##  1\. Introduction

The problem of the HI extension of an HI Banach space can be considered as
part of a general problem which concerns the embedding of Banach spaces into a
space with tight structure. There are several results related to this problem
with the more representative being the following: The first one stated and
proved in [7] asserts that every separable reflexive Banach space is
isomorphic to a subspace of a reflexive indecomposable Banach space. The
second one is similar to the previous result where the superspace admits very
few operators. In particular every separable reflexive Banach space is
isomorphic to a subspace of a $\mathcal{L}_{\infty}$ space which satisfies the
“scalar-plus-compact” property. This result is presented in [8] for uniformly
convex spaces and the general case is proved in an unpublished paper of the
same authors. In this direction it remains open whether every separable Banach
space not containing $c_{0}$ is embedded into a $\mathcal{L}_{\infty}$ space
with very few operators.

Returning to the problem of the HI extension of an HI space, attributed to A.
Pelczynski, we point out that this is a delicate and hard problem. This is
mainly due to the quasi-minimality of HI Banach spaces [14]. To explain the
obstacle a little further, assume that $X$ is an HI space and $Z$ is also HI
containing isomorphically the space $X$. Then every subspace $Y$ of $X$ and
every subspace $E$ of $Z$ contain further subspaces $D_{1}$ of $Y$ and $D_{2}$
of $E$ which are isomorphic. This means that for a possible construction of an
HI superspace $Z$ of $X$ we ought to extract the structure of the space $X$ in
the space $Z$ and this is hard, in particular when we do not have a precise
description of the structure of the space $X$.

As it stands, there exists a specific type of HI space for which the answer to
Pelczynski’s problem is known to be affirmative. In [11], J. M. F. Castillo,
V. Ferenczi and M. Gonzalez proved that Ferenczi’s uniformly convex HI space
$\mathcal{F}$ from [13] admits an HI extension $\mathcal{F}_{2}$. The space
$\mathcal{F}$ is the outcome of a complex interpolation scheme, and this is an
unavoidable component of the definition of $\mathcal{F}_{2}$. As such, the
method from [11] is not applicable to the more standard HI spaces defined via
the method of norming sets. The aim of this paper is to exhibit a method that
permits us to define HI extensions of the latter class of HI spaces, which
includes the first generation of HI spaces such as the Gowers-Maurey HI space,
[15], the Argyros-Deliyanni asymptotic $\ell_{1}$ HI space, [3], and the
Deliyanni-Manoussakis asymptotic $\ell_{p}$-HI spaces, [12]. We have chosen to
apply the method on a variant of Gowers-Maurey space, denoted by $X_{hi}$,
with its norming set $W_{hi}$ being a subset of the norming set of the mixed
Tsirelson space $T[(\mathcal{A}_{n_{j}},m_{j}^{-1})_{j\in\mathbb{N}}]$.

The norming set of any HI space which has been defined in the literature has
two inevitable characteristics. The first one is some unconditional structure
which is usually imposed by a mixed Tsirelson space. For example in Gowers-
Maurey space, Schlumprecht space [20] plays that role. The second one is the
conditional structure which is responsible for the tight structure of the HI
space and it is traced in the earlier work of B. Maurey and H. P. Rosenthal
[19]. In the case of $X_{hi}$ the norming set $W_{hi}$ is defined as follows:
The set $W_{hi}$ is a subset of $c_{00}(\mathbb{N})$ which is minimal with
respect to the following properties.

  1. (i)

It is rationally convex, symmetric, closed on the projections on intervals of
$\mathbb{N}$ and contains $\pm e_{n}^{*}$, $n\in\mathbb{N}$.

  2. (ii)

It is closed in all $(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$ operations. That
means that for every family $(f_{i})_{i=1}^{n_{2j}}$ of elements of $W_{hi}$
with successive supports the functional
$f=m_{2j}^{-1}\sum_{i=1}^{n_{2j}}f_{i}\in W_{hi}$. In this case we say that
$f$ is a weighted functional with weight $w(f)$ of $f$ equal to $m_{2j}$.(
Note that this is the unconditional structure mentioned above).

  3. (iii)

It includes all $f=m_{2j-1}^{-1}\sum_{i=1}^{n_{2j-1}}f_{i}$ where
$(f_{i})_{i=1}^{n_{2j-1}}$ is a $(2j-1)$-special sequence. For such a
functional we set $w(f)=m_{2j-1}$.

$(2j-1)$-special sequences are sequences $(f_{i})_{i=1}^{n_{2j-1}}$ of
successive weighted functionals which are defined through a coding function.
Their important property is a tree-like property, namely for every $i>1$ the
weight $w(f_{i})=m_{2j_{i}}$ of $f_{i}$ uniquely determines the previous
elements of the sequence. For a precise definition of the $(2j-1)$-special
sequences see Definition 4.1. This type of functionals imposes the HI
structure of the space $X_{hi}$.

We now pass to describe the HI extension of the space $X_{hi}$ mentioned
above. Our approach is the natural one. Namely, we shall define a new norming
set $W_{ex}$ such that the induced norm on $c_{00}(\mathbb{N})$ is an HI one
and also the completion of the subspace generated by the even coordinates
$(e_{2n})_{n\in\mathbb{N}}$ is isometric to the initial space $X_{hi}$. For
this, we let $\widehat{W}_{hi}=\\{\hat{g}\mathrel{\mathop{\mathchar
58\relax}}g\in W_{hi}\\}$ where for $g\in W_{hi}$ and $n\in\mathbb{N}$,
$\hat{g}(2n)=g(n)$ and $\hat{g}(2n-1)=0$. Then, $W_{ex}$ is the norming set
that satisfies the following two conditions:

  1. (a)

The norm induced by $W_{ex}$ is an HI one.

  2. (b)

$W_{ex}|_{\mathbb{N}_{ev}}=\widehat{W}_{hi}$, where $\mathbb{N}_{ev}$ is the
set of even positive integers and $g|_{\mathbb{N}_{ev}}=g\cdot
1_{\mathbb{N}_{ev}}$.

We set $X_{ex}$ the completion of $(c_{00}(\mathbb{N}),\|\cdot\|_{W_{ex}})$.
Condition (b) yields that the subspace $X_{ev}=\overline{\langle
e_{2n}\mathrel{\mathop{\mathchar 58\relax}}n\in\mathbb{N}\rangle}$ of $X_{ex}$
is isometric to $X_{hi}$. In the sequel we denote the subspace $X_{ev}$ by
$\widehat{X}_{hi}$. Let us also observe that since the space $X_{ex}$ is an HI
space, the set $\widehat{W}_{hi}$ cannot be a subset of $W_{ex}$. Otherwise,
the subspace $\widehat{X}_{hi}$ would be complemented in $X_{ex}$. Moreover,
since $\|\cdot\|_{W_{ex}}$ is HI, the norming set $W_{ex}$ must include
special functionals which are entirely responsible for the HI structure. At
the same time, for every special functional $f\in W_{ex}$,
$f|_{\mathbb{N}_{ev}}=\hat{g}$ with $g\in W_{hi}$. This means that the added
special functional in $W_{ex}$ does not contribute to $\widehat{X}_{hi}$ norm
beyond the elements of $\widehat{W}_{hi}$. To satisfy the previously explained
two necessary requirements, we introduce paired special functionals that will
be described below. The norming set $W_{ex}$ is defined to satisfy the
following properties.

  1. (i)

It contains $(e_{n}^{*})_{n\in\mathbb{N}}$, it is symmetric, closed under
rational convex combinations, and the restrictions on intervals of
$\mathbb{N}$.

  2. (ii)

It is closed under the even operations
$(\mathcal{A}_{n_{2j}},m_{2j}^{-1})_{j}$.

  3. (iii)

$W_{ex}|_{\mathbb{N}_{ev}}=\widehat{W}_{hi}$.

  4. (iv)

It contains the paired special functionals.

The paired special functionals are of the form
$f=m_{2j+1}^{-1}\sum_{k=1}^{n_{2j+1}}(f_{k}+\hat{g}_{k})$ such that
$\operatorname{supp}f_{k}\subset\mathbb{N}_{od}$, the set of the odd positive
integers, $w(f_{k})=w(g_{k})$, and $m_{2j+1}^{-1}\sum_{k=1}^{n_{2j+1}}g_{k}$
is a special functional in the norming set $W_{hi}$. The definition of the
paired special sequence
$f_{1},\hat{g}_{1},\ldots,f_{n_{2j+1}},\hat{g}_{n_{2j+1}}$ uses a new coding
function so that the tree-like property is also satisfied. Notice that the
paired special functionals satisfy the requirement that
$f|_{\mathbb{N}_{ev}}=\hat{g}$ with $g\in W_{hi}$. The HI property of the
space $X_{ex}$ follows from the following criterion from [11, Lemma 7.1] (see
Proposition 6.1): the subspace $\widehat{X}_{hi}$ is HI and the quotient map
$Q\mathrel{\mathop{\mathchar 58\relax}}X_{ex}\to X_{ex}/\widehat{X}_{hi}$ is a
strictly singular operator. The latter is a consequence of the next
proposition (Proposition 7.7).

######  Proposition 1.1.

For every infinite dimensional subspace $Y$ of $X_{ex}$ we have that
$\mathrm{dist}(S_{Y},S_{\widehat{X}_{hi}})=0$.

The key ingredients for proving this result are paired special functionals.

The paper is organized into nine sections.The first three sections, after the
introduction, are devoted to a brief presentation of basic ingredients and
results for defining the norming set $W_{hi}$ and proving its properties. The
next section contains the definition of the norming set $W_{ex}$ of the space
$X_{ex}$. In the following two sections the proof that the space $X_{ex}$ is
HI is given. In the last two sections we present two further HI extensions.
The first one concerns a norming set $W$ which is not closed in rational
convex combinations and the second one refers to HI extensions of asymptotic
$\ell_{1}$ HI spaces.

##  2\. Mixed Tsirelson spaces

In this section we recall the definition of a class of mixed Tsirelson spaces
and some of their properties that we shall use. We refer to [9] for a detailed
presentation. Also we follow standard terminology as in [1],[18].

Let $(m_{j})_{j=0}^{+\infty}$, $(n_{j})_{j=0}^{+\infty}$ be two strictly
increasing sequences satisfying the following:

  1. (1)

$m_{0}=2$ and $m_{j+1}=m_{j}^{5}$.

  2. (2)

$n_{0}=4$ and $n_{j+1}=(12n_{j})^{s_{j}}$ where $2^{s_{j}}\geq m_{j+1}^{3}$.

By $\mathcal{A}_{n}$ we denote the family of the finite subsets of
$\mathbb{N}$ with at most $n$ elements, i.e.,
$\mathcal{A}_{n}=\\{F\subset\mathbb{N}\mathrel{\mathop{\mathchar
58\relax}}\\#F\leq n\\}$.

######  Definition 2.1.

1) Let $n\in\mathbb{N}$. A sequence $(f_{i})_{i=1}^{k}$, $k\leq n$, of
elements of $c_{00}(\mathbb{N})$ is said to be $\mathcal{A}_{n}$-admissible if
$\operatorname{supp}(f_{1})<\operatorname{supp}(f_{2})<\dots<\operatorname{supp}(f_{k})$.

2) The $(\mathcal{A}_{n_{j}},m_{j}^{-1})$-operation on $c_{00}(\mathbb{N})$ is
the operation which assigns to each $n_{j}$-admissible sequence
$(f_{i})_{i=1}^{k}$ the vector $m_{j}^{-1}(f_{1}+f_{2}+\dots+f_{k})$.

For $f\in c_{00}(\mathbb{N})$ we shall write $w(f)=m_{j}$ if $f$ is the result
of an $(n_{j},m_{j}^{-1})$\- operation.

We present the norming set of the mixed Tsirelson space defined by the
families $\mathcal{A}_{n_{j}}$ and the “weights” $(m_{j})_{j}$ which is a
discretization of the norming set of Schlumprecht space $\mathcal{S}$, [20].

######  Definition 2.2.

Let $W_{mT}$ be the smallest subset of $c_{00}(\mathbb{N})$ with the following
properties

  1. (1)

$\pm e^{*}_{n}\in W_{mT}$ for every $n\in\mathbb{N}$.

  2. (2)

$W_{mT}$ is closed in the $(\mathcal{A}_{n_{j}},m_{j}^{-1})$-operations, i.e.,
for every $\mathcal{A}_{n_{j}}$-admissible sequence $(f_{i})_{i=1}^{k}$ of
elements of $W_{mT}$ we have that $m_{j}^{-1}\sum_{i=1}^{k}f_{i}\in W_{mT}$.

  3. (3)

$W_{mT}$ is symmetric and closed under the restriction of its elements to
intervals of $\mathbb{N}$.

  4. (4)

$W_{mT}$ is closed in rational convex combinations.

The set $W_{mT}$ can also be defined inductively as
$W_{mT}=\cup_{n=0}^{\infty}W_{mT}^{n}$ where $W_{mT}^{0}=\\{\pm
e^{*}_{n}\mathrel{\mathop{\mathchar 58\relax}}n\in\mathbb{N}\\}$ and
$W_{mT}^{n}$ is the smallest subset of $c_{00}(\mathbb{N})$ that contains
$W_{mT}^{n-1}$ and satisfies properties (2)-(3) of the above definition with
respect to the functionals of $W_{mT}^{n-1}$.

We denote by $X_{mT}=T[(\mathcal{A}_{n_{j}},m_{j}^{-1})_{j}]$ the completion
of $c_{00}(\mathbb{N})$ under the norm $\lVert\cdot\rVert_{W_{mT}}$ induced by
the set $W_{mT}$, i.e.,

$\lVert x\rVert_{W_{mT}}=\sup\\{f(x)\mathrel{\mathop{\mathchar 58\relax}}f\in
W_{mT}\\}$

######  Definition 2.3.

Let $f\in W_{mT}$. For a finite tree $\mathcal{A}$, a family
$(f_{\alpha})_{\alpha\in\mathcal{A}}$ is said to be a tree analysis of $f$ if
the following are satisfied.

  1. (1)

$\mathcal{A}$ has a unique root denoted by 0 and $f_{0}=f$.

  2. (2)

For every maximal node $\alpha\in\mathcal{A}$ we have that $f_{\alpha}=\pm
e^{*}_{n_{\alpha}}$.

  3. (3)

Let $\alpha$ be a non-maximal node of $\mathcal{A}$ and denote by $S_{\alpha}$
the set of the immediate successors of $\alpha$. Then either there exists
$j\in\mathbb{N}$ such that $(f_{\beta})_{\beta\in S_{\alpha}}$ is a
$n_{j}$-admissible subset of $W_{mT}$ and $f_{\alpha}=m_{j}^{-1}\sum_{\beta\in
S_{\alpha}}f_{\beta}$ or there exists a rational convex combination
$(r_{\beta})_{\beta\in S_{a}}$ such that $f_{\alpha}=\sum_{\beta\in
S_{\alpha}}r_{\beta}f_{\beta}$.

######  Proposition 2.4.

Every $f\in W_{mT}$ admits a tree analysis.

The proof is easy. We simple consider all $f\in W_{mT}$ admitting a tree
analysis and then we show that this set satisfies (1),(2) and (3) of the above
definition. The result follows from the minimality of the set $W_{mT}$.

######  Definition 2.5.

Let $f\in W_{mT}$ and $(f_{\alpha})_{\alpha\in\mathcal{A}}$ be a tree analysis
of $f$. For $\alpha\in\mathcal{A}$ the functional $f_{\alpha}$ is said to be
weighted with weight $w(f_{\alpha})=m_{j}$ if $f_{\alpha}$ is the result of an
$(\mathcal{A}_{n_{j}},m_{j}^{-1})$-operation of its successors.

It was proved in [10], Lemma 3.15, that every $f\in W_{mT}$ admits a tree
analysis $(f_{\alpha})_{\alpha\in\mathcal{A}}$ such that whenever $f_{\alpha}$
is rational convex combination of its successors, i.e.,
$f_{\alpha}=\sum_{\beta\in S_{\alpha}}r_{\beta}f_{\beta}$, then every
$f_{\beta}$ is a weighted functional and
$\operatorname{ran}(f_{\beta})\subset\operatorname{ran}(f_{\alpha})$.

##  3\. Even operation complete subsets of $W_{mT}$

######  Definition 3.1.

A subset $W$ of $W_{mT}$ is said to be even operation complete (EO-complete)
if

  1. (1)

$\pm e^{*}_{n}\in W$ for every $n\in\mathbb{N}$.

  2. (2)

$W$ is closed in the $(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$-operation for every
$j\in\mathbb{N}$.

  3. (3)

$W$ is symmetric and closed under the restriction of its elements to intervals
of $\mathbb{N}$.

  4. (4)

Every $f\in W$ admits a tree analysis $(f_{\alpha})_{\alpha\in\mathcal{A}}$
such that $f_{\alpha}\in W$ for every $\alpha\in\mathcal{A}$.

In the rest of this section by $W$ we will denote any EO-complete subset of
$W_{mT}$. We will denote by $X_{W}$ the completion of $c_{00}(\mathbb{N})$
under the norm $\lVert\cdot\rVert_{W}$ induced by $W$.

######  Definition 3.2.

A vector $x\in X_{W}$ is said to be a $C-\ell_{1}^{k}$\- average if there
exists $x_{1}<x_{2}<\dots<x_{k}$ such that $x=\frac{1}{k}\sum_{i=1}^{k}x_{i}$,
and $\lVert x_{i}\rVert\leq C\lVert x\rVert$ for each $i$.

Moreover if $\lVert x\rVert=1$ then $x$ is called a normalized
$C-\ell_{1}^{k}$-average.

######  Lemma 3.3.

Let $Y$ be a block subspace of $X_{W}$. Then for every $j\in\mathbb{N}$ there
exists a block sequence $(y_{i})_{i=1}^{n_{2j}}$ such that $y_{i}\in Y$ and
$n_{2j}^{-1}\sum_{i=1}^{n_{2j}}y_{i}$ is a normalized
$2-\ell_{1}^{n_{2j}}$-average

The existence of normalized $\ell_{1}^{k}$-averages was first established by
Th. Schlumprecht in his space $S$ ([20]), with the use of Krivine theorem,
since $\ell_{1}$ is finitely block representable in $S$. Later W. T. Gowers
and B. Maurey, ([15] Lemma 3), provided a James type argument to construct
$2-\ell_{1}^{k}$-averages in $S$. For a proof in the setting of the space
$X_{W}$ we refer the reader to [9] Lemma II.22, [4] Lemma 4.6

The following lemma concerning $\ell_{1}^{n{j}}$-averages is an essential tool
for providing upper bounds on the norm of certain vectors.

######  Lemma 3.4.

Let $x$ be an $C-\ell_{1}^{n_{j}}$-average in $X_{W}$ and $f\in W$ with
$w(f)=m_{i}$, $i<j$. Then

$\mathinner{\\!\left\lvert f(x)\right\rvert}\leq\frac{3C}{2m_{i}}.$

We refer to [15] Lemma 4, or [9] Lemma II.23, for a proof.

######  Definition 3.5.

A block sequence $(x_{k})_{k\in\mathbb{N}}$ in $X_{W}$ is said to be a
$(C,\epsilon)$-rapidly increasing sequence (RIS), if there exists a strictly
increasing sequence $(j_{k})_{k}$ of positive integers such that

  1. (1)

$\lVert x_{k}\rVert\leq C$ for all $k$.

  2. (2)

$\frac{1}{m_{j_{k+1}}}\cdot\\#\operatorname{supp}(x_{k})\leq\epsilon$ for
every $k$.

  3. (3)

For every $k=1,2,\dots$ and every $f\in W$ with $w(f)=m_{i},i<j_{k}$ we have
that $|f(x_{k})|\leq\nicefrac{{C}}{{m_{i}}}$.

######  Proposition 3.6.

Let $Y$ be a block subspace of $X_{W}$. Then for $\epsilon>0$ there exists a
normalized $(3,\epsilon)$-RIS sequence $(x_{k})_{k}$ with $x_{k}\in Y$ for
every $k$.

Indeed, Lemma 3.3 yields that we can get a block sequence $(x_{j})_{j}$ such
that $x_{j}$ is a normalized 2-$\ell_{1}^{n_{2j}}$-average and $x_{j}\in Y$
for every $k$. Choose a subsequence $(x_{j_{k}})_{k}$ such that
$\\#\operatorname{supp}(x_{j_{k}})\leq\epsilon m_{2j_{k+1}}$. Lemma 3.4 yields
that $(x_{j_{k}})_{k}$ is a $(3,\epsilon)$-RIS.

###  3.1. The auxiliary space

In order to get upper bounds on the action of weighted functionals on certain
vectors we shall use an auxiliary space to reduce the high complexity of the
calculations (see subsection 3.2). The auxiliary space $X_{aux}$ is the mixed
Tsirelson space $T[(\mathcal{A}_{4n_{j}},m_{j}^{-1})_{j\in\mathbb{N}}]$.

The result we shall use from the auxiliary space are the upper estimations of
the functionals of its norming set $W_{aux}$ on the averages of its basis, and
in particular the following lemma.

######  Lemma 3.7.

Let $x=\frac{m_{j}}{n_{j}}\sum_{k=1}^{n_{j}}e_{k}\in X_{aux}$. Then for every
$f\in W_{aux}$ with $w(f)=m_{i}$,

$\mathinner{\\!\left\lvert
f(x)\right\rvert}\leq\begin{cases}\frac{2}{m_{i}}\,\,\text{if $i<j$}\\\
\frac{m_{j}}{m_{i}}\,\,\text{if $i\geq j$}.\end{cases}$

Moreover if $f$ has a tree analysis $(f_{\alpha})_{\alpha\in\mathcal{A}}$ such
that for every weighted $f_{\alpha}$, $w(f_{\alpha})\neq m_{j}$ then
$\mathinner{\\!\left\lvert f(x)\right\rvert}\leq\frac{2}{m^{2}_{j}}$.

For a proof of this result see [9],Lemma II.9, (or [4], Lemma 4.2).

######  Remark 3.8.

Let us note that the above estimations hold also for $n_{j}$-averages of the
basis of the space $X_{W}$.

###  3.2. The basic inequality

The basic inequality is the fundamental tool which permits us to show that the
action of a weighted functional $f\in W$ on a $n_{j}$-average of a RIS admits
similar upper bound as the action of a functional $g$ in $W_{aux}$ with the
same weight with $f$, acting on a $n_{j}$-average of the basis of $X_{aux}$.
In particular the following holds.

######  Proposition 3.9.

Let $(x_{k})_{k}$ be a $(C,\epsilon)$-RIS in $X_{W}$ and $(\lambda_{k})_{k}\in
c_{00}(\mathbb{N})$ be a sequence of scalars. Then for every $f\in W$ of type
$I$ we can find $g_{1}$ such that $g_{1}=h_{1}$ or $g_{1}=e^{*}_{t}+h_{1}$
with $t\notin\operatorname{supp}(h_{1})$ where $h_{1}\in W_{aux}$ with
$w(h_{1})=w(f)$ and $g_{2}\in c_{00}(\mathbb{N})$ with $\lVert
g_{2}\rVert_{\infty}\leq\epsilon$ with $g_{1},g_{2}$ having nonnegative
coordinates and such that

$\mathinner{\\!\left\lvert f(\sum_{k}\lambda_{k}x_{k})\right\rvert}\leq
C(g_{1}+g_{2})(\sum_{k}\mathinner{\\!\left\lvert\lambda_{k}\right\rvert}e_{k}).$
(3.1)

If we additionally assume that there exists $j_{0}\in\mathbb{N}$ such that for
every $h\in X_{W}$ with $w(h)=m_{j_{0}}$ and every interval $E$ of
$\mathbb{N}$ we have that

$\mathinner{\\!\left\lvert h(\sum_{k\in E}\lambda_{k}x_{k})\right\rvert}\leq
C(\max_{k\in
E}\mathinner{\\!\left\lvert\lambda_{k}\right\rvert}+\epsilon\sum_{k\in
E}\mathinner{\\!\left\lvert\lambda_{k}\right\rvert}),$ (3.2)

then if $w(f)\neq m_{j_{0}}$, we may select $h_{1}$ to have a tree analysis
$(h_{t})_{t\in\mathcal{T}}$ with $w(h_{t})\neq m_{j_{0}}$ for all
$t\in\mathcal{T}$ with $h_{t}$ a weighted functional.

We refer the reader to [9] Proposition II.14 or [4] Proposition 4.3 for a
proof. As a first application of the basic inequality we get the following
result concerning the action of weighted functionals on RIS.

######  Corollary 3.10.

Let $(x_{k})_{k=1}^{n_{j}}$ be a $(C,m_{j}^{-2})$-RIS and
$x=n_{j}^{-1}\sum_{k=1}^{n_{j}}x_{k}$. Then for $f\in W$ with $w(f)=m_{i}$ we
have that

$\mathinner{\\!\left\lvert
f(x)\right\rvert}\leq\begin{cases}\frac{3C}{m_{i}m_{j}}\,\,&\text{if}\,\,i<j\\\
\frac{C}{m_{i}}+\frac{2C}{n_{j}}\,\,&\text{if}\,\,i\geq j\\\ \end{cases}.$
(3.3)

In particular for every $j\in\mathbb{N}$, $\displaystyle m^{-1}_{2j}\leq\lVert
n_{2j}^{-1}\sum_{k=1}^{n_{2j}}x_{k}\rVert\leq 2Cm_{2j}^{-1}$.

######  Definition 3.11.

A pair $(x,f)$ with $x\in X_{W}$ and $f\in W$ is said to be a $(C,j)$-exact
pair if the following conditions are satisfied:

  1. (1)

$1\leq\lVert x\rVert\leq C$, for every weighted functional $g\in W$ with
$w(g)=m_{i}$, $i\neq j$ we have $\mathinner{\\!\left\lvert
g(x)\right\rvert}\leq\frac{3C}{m_{i}}$ if $i<j$ , while
$\mathinner{\\!\left\lvert g(x)\right\rvert}\leq\frac{C}{m_{j}^{2}}$ if $i>j$.

  2. (2)

$f$ is weighted and $w(f)=m_{j}$.

  3. (3)

$f(x)=1$ and $\operatorname{ran}(f)\subset\operatorname{ran}(x)$.

The primitive example of a $(2,2j)$-exact pair is

$(\frac{m_{2j}}{n_{2j}}\sum_{k=1}^{n_{2j}}e_{k},m_{2j}^{-1}\sum_{k=1}^{n_{2j}}e_{k}^{*}).$
(3.4)

Indeed using Lemma 3.7 and Remark 3.8 it readily follows that the pair given
by (3.4) is a $(2,2j)$-exact pair. Also for every block subspace $Y$ of
$X_{W}$ we can find a $(6,2j)$-exact pair $(y,f)$ with $y\in Y$. Indeed let
$(x_{k})_{k=1}^{n_{2j}}$ be a $(3,m_{2j}^{-3})$-RIS where every $x_{k}\in Y$
is a $2-\ell_{1}^{n_{2j_{k}}}$-average with $\lVert x_{k}\rVert>1$. Choose for
every $k$ a $f_{k}\in W$, such that $f_{k}(x_{k})=1$ and
$\operatorname{ran}(f_{k})\subset\operatorname{ran}(x_{k})$. Then from
Proposition 3.10 we get that that pair $(y,f)$ where
$y=\frac{m_{2j}}{n_{2j}}\sum_{k=1}^{n_{2j}}x_{k}$ and
$f=m_{2j}^{-1}\sum_{k=1}^{n_{2j}}f_{k}$, is a $(6,2j)$-exact pair.

######  Definition 3.12.

Let $C\geq 1$ and $j_{0}\in\mathbb{N},j_{0}\geq 2$. We call a pair $(x,f)$ for
$x\in X_{W}$ and $f\in W$ a $(C,j_{0})$-standard exact pair (SEP) if there
exists a $(C,m_{j_{0}}^{-2})$-RIS $(x_{k})_{k=1}^{n_{j_{0}}}$ with

  1. (1)

$x=\frac{m_{j_{0}}}{n_{j_{0}}}\sum_{k=1}^{n_{j_{0}}}x_{k}$.

  2. (2)

$f=m_{j_{0}}^{-1}\sum_{k=1}^{n_{j_{0}}}f_{k}$ where $f_{k}\in W$ with
$f_{k}(x_{k})=1$ and $\operatorname{ran}(f_{k})=\operatorname{ran}(x_{k})$.

  3. (3)

$x_{k}$ is a $\ell_{1}^{n_{k}}$-average and $n_{j_{0}}^{2}<m_{n_{1}}$.

######  Proposition 3.13.

Let $j\in\mathbb{N}$ and $(x_{i},f_{i})_{i=1}^{n_{2j-1}}\subset X_{W}\times W$
such that $(x_{i})_{i=1}^{n_{2j-1}}$ is a RIS and $(x_{i},f_{i})$ is standard
exact pair for every $i$. Let $x=n_{2j-1}^{-1}\sum_{i=1}^{n_{2j-1}}x_{i}\in
X_{W}$ and assume that for every $g\in W$ with $w(g)=m_{2j-1}$ we have
$\mathinner{\\!\left\lvert g(x)\right\rvert}\leq\frac{C}{m^{2}_{2j-1}}$. Then

$\lVert x\rVert_{W}\leq\frac{4C}{m_{2j-1}^{2}}.$ (3.5)

This result follows from the basic inequality and especially the moreover
part. We refer the reader to [4] Proposition 4.9 or [9] Proposition II.19 for
a proof.

A consequence of the basic inequality is the reflexivity of the space $X_{W}$.

######  Proposition 3.14.

The space $X_{W}$ is reflexive.

We refer to [9] Proposition II.28, for a proof. As a consequence of the
reflexivity of the space $X_{W}$ and the rational convexity of the set $W$ we
get the following corollary.

######  Corollary 3.15.

For the space $X_{W}$ we have that
$B_{X^{*}_{W}}=\overline{W}^{\lVert\cdot\rVert}$.

##  4\. Standard HI spaces

In this section we show how we can refine $W$ in order to get a norming set
$W_{hi}$ which will be the norming set of an HI-space. We start with the
definition of the special sequences.

Let $\mathbb{Q}_{s}=\\{(f_{1},f_{2},\dots,f_{l})\mathrel{\mathop{\mathchar
58\relax}}f_{i}\in c_{00}(\mathbb{N}),f_{i}\neq
0,f_{i}(n)\in\mathbb{Q}\,\forall n\in\mathbb{N},\forall i\leq
l,f_{1}<f_{2}<\dots<f_{l}\\}$. We fix a pair $\Omega_{1},\Omega_{2}$ of
disjoint infinite subsets of $\mathbb{N}$. From the fact that $\mathbb{Q}_{s}$
is countable there exists an injective coding function
$\sigma\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Q}_{s}\to\\{2j\mathrel{\mathop{\mathchar
58\relax}}j\in\Omega_{2}\\}$ such that

$m_{\sigma(f_{1},f_{2}\dots,f_{l})}>\max\\{\frac{1}{\mathinner{\\!\left\lvert
f_{i}(e_{k})\right\rvert}}\mathrel{\mathop{\mathchar
58\relax}}k\in\operatorname{supp}f_{i},i\leq
l\\}\cdot\operatorname{maxsupp}f_{l}.$ (4.1)

######  Definition 4.1.

For $j\in\mathbb{N}$ and $k\leq n_{2j-1}$ a sequence $(f_{1},\dots,f_{k})$ of
elements of $W$ is said to be a $(2j-1)$ special sequence if
$w(f_{1})=m_{2j_{1}}$ with $j_{1}\in\Omega_{1}$, $m_{2j_{1}}\geq n_{2j-1}^{2}$
and $w(f_{i+1})=m_{\sigma(f_{1},\dots,f_{i})}$ for every $i\geq 1$.

######  Remark 4.2.

The significant property of special sequences is the tree like behavior, i.e.,
if $j_{1},j_{2}\in\mathbb{N}$, $(f_{i})_{i=1}^{n_{2j_{1}}-1}$ is an
$(2j_{1}-1)$-special sequence, $(g_{i})_{i=1}^{n_{2j_{2}}-1}$ is an
$(2j_{2}-1)$-special sequence and if for some $k,l$ we have
$w(f_{k})=w(g_{l})$, then $k=l$ and $f_{i}=g_{i}$ for every $i<k$. Moreover if
$f_{k}\neq g_{k}$ then $w(f_{k+l})\neq w(g_{k+i})$ for every $i,l\geq 1$.

###  4.1. The norming set of a standard HI-space

We consider the minimal subset $W_{hi}$ of $c_{00}(\mathbb{N})$ satisfying the
following

  1. (1)

$\pm e^{*}_{n}\in W_{hi}$ for every $n\in\mathbb{N}$.

  2. (2)

$W_{hi}$ is closed in the $(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$-operations.

  3. (3)

$W_{hi}$ is closed in the $(\mathcal{A}_{n_{2j-1}},m_{2j-1}^{-1})$-operations
on special sequences, i.e., for every $(2j-1)$-special sequence
$(f_{1},\dots,f_{k})$ with $f_{i}\in W_{hi}$ for every $i$, the functional
$m_{2j-1}^{-1}\sum_{i=1}^{k}f_{i}\in W_{hi}$.

  4. (4)

$W_{hi}$ is symmetric and closed under the restriction of its elements to
intervals of $\mathbb{N}$.

  5. (5)

$W_{hi}$ is closed under rational convex combinations.

The set $W_{hi}$ is defined inductively as follows: Set $W_{hi}^{0}=\\{\pm
e^{*}_{n}\mathrel{\mathop{\mathchar 58\relax}}n\in\mathbb{N}\\}$ and assume
that $W_{hi}^{n}$ has been defined. To get $W_{hi}^{n+1}$ , first we close the
set of the functionals $\cup_{k=1}^{n}W_{hi}^{k}$ in the
$(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$-operations, and then we close the above
set of functionals in the $(\mathcal{A}_{n_{2j-1}},m_{2j-1}^{-1})$ operations
on special sequences. The set $W_{hi}^{n+1}$ is the set of the rational convex
combinations of restrictions to intervals of the functionals we defined in the
above two steps.

We denote by $X_{hi}$ the completion of $c_{00}(\mathbb{N})$ under the norm
induced by $W_{hi}$.

Note that by the definition of $W_{hi}$ we get that it is an EO-complete
subset of $W_{mT}$ and hence all the results of the previous section hold for
the space $X_{hi}$.

######  Definition 4.3.

Let $j\in\mathbb{N}$. A sequence
$(x_{1},f_{1},x_{2},f_{2},\dots,x_{n_{2j-1}},f_{n_{2j-1}})$ with $x_{i}\in
X_{hi}$, $f_{i}\in W_{hi}$ is said to be $(C,2j-1)$-dependent if

  1. (1)

$(f_{1},\dots,f_{n_{2j-1}})$ is a $W_{hi}-(2j-1)$ special sequence

  2. (2)

each pair $(x_{i},f_{i})$ is a $(C,2j_{i})$-SEP where $m_{2j_{i}}=w(f_{i})$.

It follows from the definition of a $(C,2j-1)$-dependent sequence
$(x_{i},f_{i})_{i=1}^{n_{2j-1}}$ that the block sequence
$(x_{i})_{i=1}^{n_{2j-1}}$ is a $3C$-RIS.

######  Remark 4.4.

From the results of the previous section we get that in every block subspace
$Y$ of $X_{hi}$ we can find for every $j\in\mathbb{N}$ an $(3,2j)$-SEP $(x,f)$
with $x\in Y$. Hence if $Y,Z$ are any two block subspaces of $X_{hi}$ we can
chooce a a $(3,2j-1)$-dependent sequence $(x_{i},f_{i})_{i=1}^{n_{2j-1}}$ with
$x_{2i-1}\in Y$ and $x_{2i}\in Z$ for every $i$.

Regarding the norm of the average $n_{2j-1}^{-1}\sum_{i=1}^{n_{2j-1}}x_{i}$ of
the vectors of a $(C,2j-1)$-dependent sequence we have the following result.

######  Proposition 4.5.

Let $(x_{1},f_{1},x_{2},f_{2},\dots,x_{n_{2j-1}},f_{n_{2j-1}})$ be
$(C,2j-1)$-dependent sequence. Then

$m_{2j-1}^{-1}\leq\lVert\sum_{i=1}^{n_{2j-1}}\frac{x_{i}}{n_{2j-1}}\rVert\,\,\,\,\,\,\text{and}\,\,\,\,\lVert\sum_{i=1}^{n_{2j-1}}(-1)^{i+1}\frac{x_{i}}{n_{2j-1}}\rVert\leq
4Cm^{-2}_{2j-1}.$ (4.2)

The first inequality in (4.2) follows easily. Indeed if
$(x_{1},f_{1},x_{2},f_{2},\dots,x_{n_{2j-1}},f_{n_{2j-1}})$ is a
$(C,2j-1)$-dependent sequence, the properties of the set $W_{hi}$ yield that
$F=m_{2j-1}^{-1}\sum_{i=1}^{n_{2j-1}}f_{i}\in W_{hi}$. Using this functional
the first inequality in (4.2) readily follows. Note also
$F(\sum_{i=1}^{n_{2j-1}}(-1)^{i+1})x_{i})=0$. It is proved also that

$\mathinner{\\!\left\lvert
g(\sum_{i=1}^{n_{2j-1}}(-1)^{i+1}x_{i})\right\rvert}\leq
2C(1+2m_{2j-1}^{-2})\quad\text{ for every $g\in W_{hi}$ with
$w(g)=m_{2j-1}$}.$

This inequality is a consequence of the following two facts:

The first one is the tree structure of the special sequences (see Remark 4.2).
The second one is that every pair $(x_{k},f_{k})$ of a dependent sequence
$(x_{i},f_{i})_{i=1}^{n_{2j-1}}$ is a $(C,2j_{k})$-SEP. This yields that for
every $g\in W_{hi}$ with $w(g)\neq m_{2j_{k}}$ we have
$\mathinner{\\!\left\lvert g(x_{k})\right\rvert}\leq
3C\max\\{w(g)^{-1},m_{2j_{k}}^{-2}\\}$ Combining the above two facts we get
that the assumption of Proposition 3.13 holds. This gives us the second
inequality in (4.2).

As a corollary of the previous proposition we get the following corollary.

######  Corollary 4.6.

The space $X_{hi}$ is an HI space.

Indeed let $Y,Z$ be any two block subspaces of $X_{0}$ and $j\in\mathbb{N}$.
From Remark 4.4 we get a $(3,2j-1)$-dependent sequence
$(x_{i},f_{i})_{i=1}^{n_{2j-1}}$ with $x_{2i-1}\in Y$ and $x_{2i}\in Z$ for
every $i$. From (4.2) we get that

$\lVert\sum_{i=1}^{n_{2j-1}}(-1)^{i+1}x_{i}\rVert\leq\frac{12}{m_{2j-1}}\lVert\sum_{i=1}^{n_{2j-1}}x_{i}\rVert.$
(4.3)

Since $j\in\mathbb{N}$ the above inequality yields that
$\operatorname{dist}(S_{Y},S_{Z})=0$ and this implies that $X_{0}$ is an HI
space.

##  5\. The defintion of the space $X_{ex}$.

In this section we define the HI space $X_{ex}$ which contains as an isometric
subspace the initial HI space $X_{hi}$. For this we will define a norming set
$W_{ex}$, a subset of $c_{00}(\mathbb{N})$, that induces the norm of the
space. Actually the set $W_{ex}$ is a subset of the set $W_{mT}$ which is the
norming set of the space
$X_{mT}=T[(\mathcal{A}_{n_{2j}},m_{2j}^{-1})\cup(\mathcal{A}_{2n_{2j-1}},m_{2j-1}^{-1})]$.
Note that $X_{mT}$ uses in the odd operations the families
$\mathcal{A}_{2n_{2j-1}}$ instead of $\mathcal{A}_{n_{2j-1}}$ which were used
in the definition of $X_{hi}$.

###  5.1. The definition of $W_{ex}$.

Let $\mathbb{N}=\mathbb{N}_{od}\cup\mathbb{N}_{ev}$, $\mathbb{N}_{od}$ the odd
and $\mathbb{N}_{ev}$ the even positive integers.

We shall denote also by $K=\cup_{j}K^{2j}$ the subset of $W_{mT}$, where

$K^{2j}=\\{m_{2j}^{-1}\sum_{i\in F}e_{i}^{*}\mathrel{\mathop{\mathchar
58\relax}}\\#F=n_{2j}\\}.$

For every $j$, let $(K_{k}^{2j})_{k\in\mathbb{N}}$, be disjoint subsets of
$K^{2j}$ such that $\sup\\{\min(f)\mathrel{\mathop{\mathchar 58\relax}}f\in
K_{k}^{2j}\\}=+\infty$ for every $k$. For
$g=\sum_{i=1}^{+\infty}a_{i}e^{*}_{i}\in c_{00}(\mathbb{N})$ we denote by
$\widehat{g}$ the vector $\widehat{g}=\sum_{i}a_{i}e_{2i}^{*}\in
c_{00}(\mathbb{N})$. We shall denote by $\widehat{K_{k}^{2j}}$ the set
$\\{\widehat{f}\mathrel{\mathop{\mathchar 58\relax}}f\in K^{2j}_{k}\\}$ and by
$\widehat{W}_{hi}$ the set $\\{\widehat{f}\mathrel{\mathop{\mathchar
58\relax}}f\in W_{hi}\\}$.

######  Definition 5.1 (The coding functions
$\varrho_{k}^{2i},k\in\mathbb{N},i=0,1,\dots$).

Let $(D_{k})_{k\in\mathbb{N}}$ be a sequence of disjoint subsets of $W_{mT}$,
to be determined later. For every $i,k\in\mathbb{N}$ we shall denote by
$L_{k}^{2i}$ the following subset of $D_{k}$,

$L_{k}^{2i}=\\{f\in D_{k}\mathrel{\mathop{\mathchar
58\relax}}w(f)=m_{2i}\,\,\,\,\,\text{and}\,\,\operatorname{supp}(f)\subset\mathbb{N}_{od}\\}.$

For every $k,i\in\mathbb{N}$ let $\varrho_{k}^{2i}\mathrel{\mathop{\mathchar
58\relax}}L_{k}^{2i}\to K_{k}^{2i}$ be an injection such that

$\displaystyle\text{for every $f\in L_{k}^{2i}$ we have}\quad
f<\widehat{\varrho_{k}^{2i}(f)}.\quad$ (5.1)

######  Remark 5.2.

From the above definitions it follows that if $g\in K_{k}^{2i}$ for some
$i,k\in\mathbb{N}$ then there is no $n\neq k$ and $f\in L_{n}^{2i}$ such that
$g=\varrho_{n}^{2i}(f)\in K_{n}^{2i}$.

This follows from the fact that the domains of
$\varrho_{k}^{2i},\varrho_{n}^{2i}$ are disjoint as well as their ranges
$K_{k}^{2i},K_{n}^{2i}$.

Our goal in this subsection is to define an EO-complete subset $W_{ex}$ of
$W_{mT}$ such that the restriction of every element $f\in W_{ex}$ to the set
$\mathbb{N}_{ev}$ belongs to $\widehat{W}_{hi}$ and in particular
$\widehat{W}_{hi}=\\{f_{|\mathbb{N}_{ev}}\mathrel{\mathop{\mathchar
58\relax}}f\in W_{ex}\\}$. Denoting by $X_{ex}$ the space defined by the set
$W_{ex}$, the aforementioned property yields that its subspace
$\widehat{X}_{hi}$ generated by the subsequence $(e_{2n})_{n\in\mathbb{N}}$ of
the basis is isometric to the initial HI space $X_{hi}$. We shall also include
in $W_{ex}$ some new special functionals, named paired special functionals
which will permit us to export the HI structure of the space
$\widehat{X}_{hi}$ to the whole space $X_{ex}$.

We proceed to give three definitions of special functionals which will be used
in the definition of the norming $W_{ex}$. We assume the existence of a family
$(D_{k})_{k\in\mathbb{N}}$ of disjoint subsets of $W_{mT}$ that will be
specified later and we consider the corresponding functions
$\varrho_{k}^{2i}\mathrel{\mathop{\mathchar 58\relax}}L_{k}^{2i}\to
K_{k}^{2i}$, $k\in\mathbb{N}$,$i\geq 0$ as they were defined in Definition 5.1

######  Definition 5.3.

Let $j,n\in\mathbb{N}$ and $V\subset W_{mT}$. We call a
$(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n},i\geq 0)$\- paired special sequence any
block sequence $(f_{1},\widehat{g}_{1},\dots,f_{p},\widehat{g}_{p})$
satisfying the following:

  1. (1)

$p\leq n_{2j-1}$ and for every $k\leq p$ there exist $j_{k}\in\mathbb{N}$ and
$i_{k}\leq n$ such that $f_{k}\in V\cap L_{i_{k}}^{2j_{k}}$ and
$g_{k}=\varrho^{{2j_{k}}}_{i_{k}}(f_{k})$.

  2. (2)

The finite sequence $(g_{1},g_{2},\dots,g_{p})$ is a $(2j-1)$-special sequence
of $W_{hi}$.

######  Definition 5.4.

Let $j,n\in\mathbb{N}$ and $V\subset W_{mT}$. We call a
$(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n}),i\geq 0)$-semi paired special sequence
any block sequence of the form
$(f_{i},\widehat{g}_{i})_{i=1}^{l}\,{}^{\frown}(\widehat{g}_{i})_{i=l+1}^{p}$
satisfying the following:

  1. (1)

$p\leq n_{2j-1}$, $l\geq 0$ and
$(f_{1},\widehat{g}_{1}f_{2},\widehat{g}_{2},\dots,f_{l},\widehat{g}_{l})$ is
a $(2j-1,V,(\varrho_{i}^{2k})_{i=1}^{n},k=0,1,\dots)$-paired special sequence.

  2. (2)

For every $m=l+1,\dots,p$ we have that $g_{m}\in K^{2j_{m}}_{i_{m}}$ for some
$i_{m}\leq n$, and $(g_{1},g_{2},\dots,g_{p})$ is a $(2j-1)$\- special
sequence of $W_{hi}^{1}$.

  3. (3)

There is no $\phi\in V$ such that
$(f_{1},\widehat{g}_{1}f_{2},\widehat{g}_{2}\dots,f_{l},\widehat{g}_{l},\phi,\widehat{g}_{l+1})$
is a $(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$ paired special
sequence.

The third definition is the following.

######  Definition 5.5.

Let $j,n\in\mathbb{N}$ and $V\subset W_{mT}$. We call a
$(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n},i\geq 0)$-special sequence any block
sequence
$(f_{1},\widehat{g}_{1},\dots,f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m},h_{m+1},h_{m+2},\dots,h_{p})$
satisfying the following:

  1. (1)

$p\leq n_{2j-1},$ $0\leq l\leq m$ and
$(f_{1},\widehat{g}_{1},\dots,f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m})$
is a $(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$-semi paired special
sequence.

  2. (2)

$h_{m+1|\mathbb{N}_{ev}}\in
V\setminus\cup_{j=0}^{+\infty}\cup_{i=1}^{n}\widehat{K}_{i}^{2j}$ and for
every $q\geq 1$,
$h_{m+q|\mathbb{N}_{ev}}=\widehat{\phi}_{m+q}\in\widehat{W}_{hi}$ and
$(g_{1},\dots,g_{m},\phi_{m+1},\phi_{m+2},\dots,\phi_{p})$ is a
$(2j-1)$-special sequence of $W_{hi}$.

######  Remark 5.6.

It follows from the above definitions that the paired special sequences are
semi paired special seqeunces and the later are special sequences.

###  5.2. Definition of the norming set $W_{ex}$

We proceed by induction to define an increasing family of sets $W_{n}$,
$n=0,1,2,\dots$ which will be subsets of $W_{mT}$ such that
$W_{ex}=\cup_{n=0}^{\infty}W_{n}$. For every $n\geq 1$ we shall choose a set
$D_{n}\subset W_{n}$ such that the sets $(D_{n})_{n\in\mathbb{N}}$ are
disjoint and for every $n\in\mathbb{N}$, $i\geq 0$, we will consider the
injective function $\varrho_{n}^{2i}\mathrel{\mathop{\mathchar
58\relax}}L_{n}^{2i}\to K_{n}^{2i}$ as it was defined in Definition 5.1.

Let $W_{0}=\\{\pm e^{*}_{n}\mathrel{\mathop{\mathchar
58\relax}}n\in\mathbb{N}\\}$ . Let $n\in\mathbb{N}$ and assume that for every
$k<n$ the sets $D_{k}\subset W_{k}$ and the functions $\varrho_{k}^{2i}$,
$i\geq 0$ have been defined.

We set

$U_{n}=\cup_{j}\\{m_{2j}^{-1}\sum_{i=1}^{k}f_{i}\mathrel{\mathop{\mathchar
58\relax}}(f_{i})_{i=1}^{k}\,\,\text{is an $\mathcal{A}_{n_{2j}}$-admissible
sequence of}\,\,W_{n-1}\\},$

and
$V_{n}=(U_{n}\setminus\cup_{j=0}^{\infty}\cup_{i=n+1}^{+\infty}\widehat{K_{i}^{2j}})\cup
W_{n-1}$.

We set $D_{n}=V_{n}\setminus W_{n-1}$ and for $i\geq 0$ we define

$L_{n}^{2i}=\\{f\in D_{n}\mathrel{\mathop{\mathchar
58\relax}}w(f)=m_{2i}\,\,\text{and}\,\,\operatorname{supp}(f)\subset\mathbb{N}_{od}\\}.$

For every $i\in\mathbb{N}$ we a choose an injective function
$\varrho_{n}^{2i}\mathrel{\mathop{\mathchar 58\relax}}L_{n}^{2i}\to
K_{n}^{2i}$ satisfying $f<\widehat{\varrho_{n}^{2i}(f)}$ for every $f\in
L_{n}^{2i}$.

It is clear that $D_{n}\cap W_{n-1}=\emptyset$ and hence $D_{n}$ is disjoint
from every $W_{k}$, $k<n$. Also we have that $L_{n}^{2i}\subset V_{n}$ for
every $i\geq 0$.

We define now for every $j\in\mathbb{N}$ the
$(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i\geq 0)$ special functionals.

For $j\in\mathbb{N}$ we define

$\displaystyle W_{n}^{2j-1}$
$\displaystyle=\\{\frac{1}{m_{2j-1}}\left(\sum_{i=1}^{l}(f_{i}+\widehat{g}_{i})+\sum_{i=l+1}^{m}\widehat{g}_{i}+\sum_{i=m+1}^{p}h_{i}\right)\mathrel{\mathop{\mathchar
58\relax}}$ (5.2) $\displaystyle\hskip 199.16928pt\text{special sequence}.\\}$

We set

$W_{n}=\operatorname{conv}_{\mathbb{Q}}\\{Ef\mathrel{\mathop{\mathchar
58\relax}}f\in
V_{n}\cup\cup_{j=1}^{\infty}W_{n}^{2j-1}\mathrel{\mathop{\mathchar
58\relax}}E\,\text{interval of}\,\mathbb{N}\\}.$

The norming set $W_{ex}$ is the set $W_{ex}=\cup_{n=0}^{+\infty}W_{n}$.

From the inductive assumption we have that $W_{n-1}$ is a subset of set
$W_{mT}$. It follows that the set $U_{n}$ is also subset of $W_{mT}$. Hence
for every $j\in\mathbb{N}$ the set $W_{n}^{2j-1}$ is a subset of $W_{mT}$ and
finally the set $W_{n}$ is subset of $W_{mT}$. It is clear also that the sets
$W_{n-1}$ and $D_{n}$ are subsets of $W_{n}$.

######  Remarks 5.7.

  1. a)

If
$(f_{1},\widehat{g}_{1},f_{2},\widehat{g}_{2},f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\widehat{g}_{l+2},\dots,\widehat{g}_{p})$
is a $(2j-1,V_{n},(\varrho^{2i}_{k})_{k=1}^{n},i=0,1,\dots)$-semi-paired
special and $l\geq 1$, then for the functional $\widehat{g}_{l+1}$ we have
that

    1. i)

either $g_{l+1}\notin\cup_{k=1}^{n}\varrho_{k}^{2j_{l+1}}(L_{k}^{2j_{l+1}})$,
$w(g_{l+1})=m_{2j_{l+1}}$,

    2. ii)

or $g_{l+1}=\widehat{\varrho^{2j_{l+1}}_{k_{l+1}}(f_{l+1})}$ for an
$f_{l+1}\in L_{k_{l+1}}^{2j_{l+1}}$, $k_{l+1}\leq n$ and
$\operatorname{minsupp}f_{l+1}\leq\operatorname{maxsupp}\widehat{g}_{l}$.

Note that in both cases the definition of the coding functions
$\varrho_{k}^{2i}$ and the disjointness of the sets $K_{k}^{2i}$ implies that
there is no $m>n$ and a $(2j-1,V_{m},(\varrho_{k}^{2i})_{k=0}^{m},i\geq 0)$\-
paired special sequence having as an initial segment the sequence
$(f_{i},\widehat{g}_{i})_{i=1}^{l+1}$.

  2. b)

For every $g_{1}\in\varrho^{2j_{1}}_{i_{1}}(L_{i_{1}}^{2j_{1}})$ with
$w(g_{1})=m_{2j_{1}}\geq n^{2}_{2j-1}$ for some $j\in\mathbb{N}$ and
$i_{1}\leq n$, we have that
$((\varrho^{2j_{1}}_{i_{1}})^{-1}(g_{1}),\widehat{g}_{1})$ is a
$(2j-1,V_{n},(\varrho^{2i}_{k})_{k=1}^{n},i=0,1,\dots)$-paired sequence.

  3. c)

If $(g_{1},\dots,g_{p})$ is a $(2j-1)$-special sequence in $W_{hi}$, such that
for every $l\leq p$, $g_{l}\in K^{2j_{l}}$ for some $4j_{l}\in\mathbb{N}$ then
if $\widehat{g}_{1}\notin\cup_{k=1}^{\infty}K_{k}^{2j_{1}}$ it follows that
$(\widehat{g}_{1},\dots,\widehat{g}_{p})$ is a
$(2j-1,V_{n},(\varrho^{2i}_{k})_{k=1}^{n},i=0,1,\dots)$ special sequence where
$n=\max\\{1,k\mathrel{\mathop{\mathchar 58\relax}}\exists l\leq
p\,\,\text{with}\,g_{l}\in K_{k}^{2j_{l}}\\}$.

If $g_{1}\in
K_{k_{1}}^{2j_{1}}\setminus\varrho_{k_{1}}^{2j_{1}}(L_{k_{1}}^{2j_{1}})$ then
$(\widehat{g}_{1},\dots,\widehat{g}_{p})$ is a
$(2j-1,V_{n},(\varrho^{2i}_{k})_{k=1}^{n},i=0,1,\dots)$-semi paired special
sequence where $n=\max\\{k\mathrel{\mathop{\mathchar 58\relax}}\exists l\leq
p\,\,\text{with}\,g_{l}\in K_{k}^{2j_{l}}\\}$.

It follows form the definition of $W_{ex}$ that it is an EO-complete subset of
$W_{mt}$. Hence from the results of Section 3 we have the following.

######  Proposition 5.8.

The space $X_{ex}$ is reflexive. Moreover
$B_{X_{ex}^{*}}=\overline{W_{ex}}^{\lVert\rVert}$.

We show now that the semi paired special sequences have the tree property, a
result that will be essential in the next sections.

######  Notation 5.9.

If
$(f_{1},\widehat{g}_{1},f_{2},\widehat{g}_{2},f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m})$
is a $(2j-1,V_{q},(\varrho^{2i}_{k})_{k=1}^{q},i=0,1,\dots)$-semi paired
special sequence, the pair defined by $g_{i}$, $i\leq m$ is defined to be the
pair $\\{f_{i},g_{i}\\}$ if $i\leq l$ and the singleton $\\{g_{i}\\}$ if
$i=l+1,\dots,m$. The weight of the pair is defined to be the weight of
$g_{i}$.

######  Lemma 5.10.

The set of the semi paired special sequences has a tree structure, i.e., if
$(f_{i}^{1},\hat{g_{i}^{1}})_{i=1}^{p_{1}}{}^{\frown}(\hat{g_{i}^{1}})_{i=p_{1}+1}^{m_{1}}$
and
$(f_{i}^{2},\hat{g_{i}^{2}})_{i=1}^{p_{2}}{}^{\frown}(\hat{g_{i}^{2}})_{i=p_{2}+1}^{m_{2}}$
are two semi-paired special sequences and the couples determined by some
$g_{i}^{1},g_{j}^{2}$ have the same weight then $i=j$ and then couples
determined by $g_{l}^{1}$ and $g_{l}^{2}$ are equal for all $l\leq i$.
Moreover if $g_{i}^{1}\neq g_{i}^{2}$ then all the couples determined by
$g_{k}^{1},g_{p}^{2}$, $k,p>i$ have different weights.

###### Proof.

By the definition of the semi-paired special sequences it follows that
$(g_{l}^{1})_{l=1}^{i}$ and $(g_{k}^{2})_{k=1}^{j}$ are special sequences of
$W_{hi}^{1}$. Since the special sequences of $W_{hi}$ have the tree property
and $w(g_{i}^{1})=w(g_{j}^{2})$ it follows that $i=j$ and
$g_{k}^{1}=g_{k}^{2}$ for all $k<i=j$. Also from the definition of the
(semi)-paired special sequences and the injectivity of the funtions
$\varrho_{n}^{2j}$,$n\in\mathbb{N},j\geq 0$ it follows that the couples
determined by $g_{k}^{1},g_{k}^{2}$, $k<i$, are equal and in particular they
have the same cardinality.

The same holds of $g_{i}^{1}=g_{i}^{2}$. In the case $g_{i}^{1}\neq g_{i}^{2}$
again from the tree property of the special sequences of $W_{hi}$ it follows
that $w(g_{k}^{1})\neq w(g_{l}^{2})$ for all $k,l>i$.

In the case $i=j=l_{1}$ (or $i=j=l_{2}$) we have that $l_{1}-1\leq l_{2}$ (
$l_{2}-1\leq l_{1}$).

∎

We proceed to show that
$\widehat{W}_{hi}=\\{f_{|\mathbb{N}_{ev}}\mathrel{\mathop{\mathchar
58\relax}}f\in W_{ex}\\}$. First we show that $W_{ex|\mathbb{N}_{ev}}$ is a
subset of $\widehat{W}_{hi}$. This will be an immediate consequence of the
following lemma.

######  Lemma 5.11.

For every $n\in\mathbb{N}$ and every $f\in W_{n}$ we have that
$f_{|\mathbb{N}_{ev}}\in\widehat{W}_{hi}$.

###### Proof.

We prove the lemma by induction. For every $n\in\mathbb{N}$,
$e^{*}_{n|\mathbb{N}_{ev}}=0$ if $n$ is odd otherwise if $n=2k$,
$e^{*}_{n|\mathbb{N}_{ev}}=e^{*}_{n}=\widehat{e^{*}_{k}}\in\widehat{W}_{hi}$.
Assume that the result holds for all $k\leq n$ and $f\in W_{n+1}$.

Let $f$ be the result of an $(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$\- operation
of elements of $W_{n-1}$, i.e., $f=m_{2j}^{-1}\sum_{i=1}^{d}f_{i}$. Then from
the inductive hypothesis we have that
$f_{i|\mathbb{N}_{ev}}=\widehat{\phi_{i}}\in\widehat{W}_{hi}$ for every $i$.
Since $W_{hi}$ is closed in the
$(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$-operations it follows that
$\phi=m_{2j}^{-1}\sum_{i=1}^{d}\phi_{i}\in W_{hi}$ and
$f_{|\mathbb{N}_{ev}}=\widehat{\phi}\in\widehat{W}_{hi}$.

Assume now that there exists a
$(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$-special sequence
$(f_{1},\widehat{g}_{1},f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m},h_{m+1},\dots,h_{p})$
such that
$f=m_{2j-1}^{-1}(\sum_{i=1}^{l}(f_{i}+\widehat{g}_{i})+\sum_{i=l+1}^{m}\widehat{g}_{i}+\sum_{i=m+1}^{p}h_{i})$.
By the definition of the special sequences we have that
$f_{i|\mathbb{N}_{ev}}=0$ for every $i\leq l$ and
$h_{i|\mathbb{N}_{ev}}=\widehat{\phi}_{i}\in\widehat{W}_{hi}$ for every $i\geq
m+1$. Moreover $(g_{1},\dots,g_{m},\phi_{m+1},\dotsm\phi_{p})$ is a
$(2j-1)$-special sequence of $W_{hi}$. It follows that

$\phi=m_{2j-1}^{-1}(\sum_{i=1}^{m}g_{i}+\sum_{i=m+1}^{p}\phi_{i})\in
W_{hi}\,\,\,and\,\,f_{|\mathbb{N}_{ev}}=m_{2j-1}^{-1}(\sum_{i=1}^{m}\widehat{g}_{i}+\sum_{i=m+1}^{p}h_{i|\mathbb{N}_{ev}})=\widehat{\phi}.$

The case $f$ is rational convex combination of functionals of the first two
cases is immediate consequence of the fact that $W_{hi}$ is closed in rational
convex combinations. This completes the proof. ∎

The next lemma will be used to prove the converse inclusive relation.

######  Lemma 5.12.

Let $g_{i}\in K_{n_{i}}^{2j_{i}}$, $i\leq m$ be such that
$(g_{1},\dots,g_{m})$ is a $(2j-1)$-special sequence of $W_{hi}$. Then there
exist $0\leq l\leq m$ and $f_{1},\dots,f_{l}\in W_{ex}$ such that
$(f_{1},\widehat{g}_{1},\dots,f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m})$
is a $(2j-1,V_{n},(\varrho_{k}^{n})_{k=1}^{n},i=0,\dots)$ semi-paired special
sequence where $n=\max\\{n_{i}\mathrel{\mathop{\mathchar 58\relax}}i\leq
m\\}$.

###### Proof.

Let $l\leq m$ be maximal with the properties

  1. (1)

$g_{i}\in\varrho_{n_{i}}^{2j_{i}}(L_{n_{i}}^{2j_{i}})$ for every $i\leq l$.

  2. (2)

$F=((\varrho^{2j_{i}}_{n_{i}})^{-1}(g_{i}),\widehat{g}_{i})_{i=1}^{l}$ is a
$(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1\dots)$ -paired special
sequence, where $n=\max\\{n_{i}\mathrel{\mathop{\mathchar 58\relax}}i\leq
m\\}$.

If $l=0$, Remark 5.7b) yields that
$g_{1}\notin\varrho_{n_{1}}^{2j_{1}}(L_{n_{1}}^{2j_{1}})$ and from Remark
5.7c) we get that $(\widehat{g}_{1},\widehat{g}_{2},\dots,\widehat{g}_{m})$ is
$(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots,)$-semi-paired special
sequence.

If $l=m$ we have that
$((\varrho^{2j_{i}}_{n_{i}})^{-1}(g_{i}),\widehat{g}_{i})_{i=1}^{m}$ is
$(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1\dots)$ paired special
sequence.

Let $l<m$. Then for the functional $g_{l+1}$ we have that either does not
belong to $\varrho_{n_{l+1}}^{2j_{l+1}}(L_{n_{l+1}}^{2j_{l+1}})$ or belongs to
$\varrho_{n_{l+1}}^{2j_{l+1}}(L_{n_{l+1}}^{2j_{l+1}})$ and
$\min(\varrho_{n_{l+1}}^{2i_{l+1}})^{-1}(g_{l+1})\leq\max\widehat{g}_{l}$. In
both cases we get that

$((\varrho^{2j_{1}}_{1})^{-1}(g_{1}),\widehat{g}_{1},\dots,(\varrho^{2j_{1}}_{l})^{-1}(g_{l}),\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m})$

is $(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1\dots)$ semi-paired special
sequence. ∎

######  Lemma 5.13.

For every $g\in W_{hi}$ there exists $\phi\in W_{ex}$ such that
$\phi_{|\mathbb{N}_{ev}}=\widehat{g}$.

###### Proof.

We shall prove by induction on $n$ that for every $g\in W_{hi}^{n}$ there
exists $\phi\in W_{ex}^{n_{0}}$,$n_{0}=n_{0}(g)\geq n$, such that
$\phi_{|\mathbb{N}_{ev}}=\widehat{g}$.

The case $g=\pm e^{*}_{k}\in W_{hi}^{0}$ it trivial. Let $g\in W_{hi}^{1}$. We
shall consider the following two cases.

Case 1. $g=m^{-1}_{2j}\sum_{i\in F}\pm e_{i}^{*}$ with $\\#F\leq n_{2j}$.

We have to consider the following two subcases.

Subcase a). $g\notin\cup_{n=2}^{\infty}K_{n}^{2j}$.

In this subcase we get that $\phi=m_{2j}^{-1}\sum_{i\in F}\pm e^{*}_{2i}\in
V_{1}$ and $\phi=\phi_{|\mathbb{N}_{ev}}=\widehat{g}$.

Subcase b). $g\in K_{n}^{2j}$ for some $n\geq 2$.

In this subcase we get that $\phi=m_{2j}^{-1}\sum_{i\in F}e^{*}_{2i}\in
V_{n}\subset W_{n}$ and $\phi=\widehat{g}$.

We pass now to the next case.

Case 2. Let $g=m_{2j-1}^{-1}\sum_{i=1}^{p}g_{i}$ where $g_{i}\in W_{hi}^{1}$
and $(g_{1},\dots,g_{p})$ is a $(2j-1)$-special sequence.

Let $m\leq p$ be maximal with the property $g_{k}\in K_{i_{k}}^{2j_{k}}$ for
every $k\leq m$. If $m=0$ from 5.7c) we get that
$(\widehat{g}_{1},\dots,\widehat{g_{m}})$ is
$(2j-1,V_{n_{0}},(\varrho_{k}^{2i})_{k=1}^{n_{0}},i=0,\dots)$ special sequence
for some $n$.

If $m\geq 1,$ Lemma 5.12 yields that there exist $l\leq m$,
$f_{1},\dots,f_{l}\in W_{ex}$ and $n_{0}\in\mathbb{N}$ such that
$(f_{1},\widehat{g}_{1},\dots,f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g_{m}})$
is a $(2j-1,V_{n_{0}},(\varrho_{k}^{2i})_{k=1}^{n_{0}},i=0,\dots)$ semi-paired
special sequence. Clearly by the definition of the
$(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,\dots)$ special sequences, we
get that
$(f_{1},\widehat{g}_{1},\dots,f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g_{p}})$
is $(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,\dots)$ special sequence
where $n=\max\\{r_{i}\mathrel{\mathop{\mathchar 58\relax}}g_{i}\in
K_{r_{i}}^{2j_{i}},i\in\\{1,\dots,p\\}\\}$. It follows

$\phi=m_{2j-1}^{-1}\left(\sum_{i=1}^{l}(f_{i}+\widehat{g}_{i})+\sum_{i=l+1}^{p}\widehat{g}_{i}\right)\in
W_{ex}\,\,\,\,\text{and}\,\,\,\,\phi_{|\mathbb{N}_{ev}}=m_{2j-1}^{-1}\sum_{i=1}^{d}\widehat{g}_{i}=\widehat{g}.$

The case $g$ is rational convex combination of functional of the above two
cases follows readily. This complete the first inductive step.

Assume that we have prove the result for every $k\leq n$ and let $g\in
W_{hi}^{n+1}$.

The case $g$ is that result of an
$(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$-operation of elements of $W_{n}^{hi}$
follows easily from the inductive hypothesis. Indeed if
$g=m_{2j}^{-1}\sum_{i=1}^{d}g_{i}$ then for every $i$ we have that there
exists $\phi_{i}\in W_{ex}$ such that
$\phi_{i|\mathbb{N}_{ev}}=\widehat{g}_{i}$. Since $W_{ex}$ is closed in
projections on intervals we may assume that
$\operatorname{ran}(\phi_{i})=\operatorname{ran}(\widehat{g_{i}})$. Also using
that $W_{ex}$ is closed in $(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$-operations we
get

$\phi=m_{2j}^{-1}\sum_{i=1}^{d}\phi_{i}\in
W_{ex}\,\,\,\,\,\text{and}\,\,\phi_{|\mathbb{N}_{ev}}=\widehat{g}.$

Assume now that $g=m_{2j-1}^{-1}\sum_{i=1}^{p}g_{i}$ where
$(g_{1},\dots,g_{p})$ is a $(2j-1)$-special sequence of $W_{hi}^{n+1}$

Let $m\leq p$ be maximal with respect the property $g_{k}\in
K_{i_{k}}^{2j_{k}}$ for some $i_{k}$, for every $k\leq m$. Then from Lemma
5.12 we get that there exist $l\leq m$, $f_{1},\dots,f_{l}\in W_{ex}$ and
$n_{0}\in\mathbb{N}$ such that
$(f_{1},\widehat{g}_{1},\dots,f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g_{m}})$
is a $(2j-1,V_{n_{0}},(\varrho_{k}^{2i})_{k=1}^{n_{0}},i=0,\dots)$ semi-paired
special sequence. For every $i>m$ from the inductive hypothesis we have that
there exists $h_{i}\in W_{ex}^{n_{i}}$ such that
$h_{i|\mathbb{N}_{ev}}=\widehat{g_{i}}$. Since $W_{ex}$ is closed in
projections on intervals we may assume that
$\operatorname{ran}(h_{i})=\operatorname{ran}(\widehat{g_{i}})$ for every $i$.
Moreover from the choice of $m$ we have also that
$h_{m+1|\mathbb{N}_{ev}}=\widehat{g}_{m+1}\notin\cup_{k=1}^{\infty}\widehat{K}_{k}^{2j_{m+1}}$.

It follows that
$(f_{1},\widehat{g}_{1},\dots,f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g_{m}},h_{m+1},\dotsm
h_{p})$ is a
$(2j-1,V_{\bar{n}_{0}},(\varrho_{k}^{2i})_{k=1}^{\bar{n}_{0}},i=0,\dots)$
special sequence for some $\bar{n}_{0}\geq n$. Hence

$\phi=m_{2j-1}^{-1}\left(\sum_{i=1}^{l}(f_{i}+\widehat{g}_{i})+\sum_{i=l+1}^{m}\widehat{g}_{i}+\sum_{i=m+1}^{p}h_{i}\right)\in
W_{ex}\,\,\,\text{and}\,\,\phi_{|\mathbb{N}_{ev}}=\widehat{g}.$

∎

As a corollary we get the following.

######  Proposition 5.14.

For the norming sets $W_{hi}$ and $W_{ex}$ of the spaces $X_{hi}$ and $X_{ex}$
we have that

$W_{ex|\mathbb{N}_{ev}}=\widehat{W}_{hi}.$

Morever the space $X_{hi}$ embeds isometrically into $X_{ex}$.

##  6\. A criretion for the HI property

In this section we provide a criterion for a Banach space to be an HI space.
It is equivalent to [11, Lemma 7.1], but we provide a proof for completeness.

######  Proposition 6.1.

Let $X$ be a Banach space and $X_{0}$ be an HI subspace of $X$. If for every
subspace $Z$ of $X$ we have $\operatorname{dist}(S_{X_{0}},S_{Z})=0$ then the
space $X$ is HI.

###### Proof.

We show that for every two infinite dimensional closed subspaces $Y_{1},Y_{2}$
of $X$ we have $\operatorname{dist}(S_{Y_{1}},S_{Y_{2}})=0$. Let
$\epsilon\in(0,1/10)$. We start with the following observation.

In every subspace $Y$ of $X$ we can find a Schauder basic sequence
$(y_{n})_{n\in\mathbb{N}}\subset Y$ and a normalized sequence
$(x_{n})_{n\in\mathbb{N}}\subset S_{X_{0}}$ such that $\lVert
y_{n}-x_{n}\rVert<\frac{\epsilon}{2^{n+3}}$. Indeed using Mazur Theorem [18]
we get a normalized Schauder basic sequence $(u_{n})_{n}$ in $Y$ with basis
constant $K\leq 2$. Using the hypothesis by a standard inductive argument we
get a block sequence $(y_{n})_{n\in\mathbb{N}}$ of $(u_{n})_{n}$ and a
normalized sequence $(x_{n})_{n\in\mathbb{N}}\subset S_{X_{0}}$ such that
$\lVert y_{n}-x_{n}\rVert<\frac{\epsilon}{2^{n+3}}$. This implies that
$(x_{n})_{n}$ is also Schauder basic sequence.

By the above observation we find Schauder basic sequences
$(y_{n}^{1})_{n}\subset Y_{1}$, $(y_{n}^{2})\subset Y_{2}$ and normalized
Schauder basic sequences $(x_{n}^{1})_{n},(x_{n}^{2})_{n}$ in $X_{0}$ such
that $\lVert y_{n}^{i}-x_{n}^{i}\rVert<\epsilon/2^{n+3}$ for every $n$ and
$i=1,2$. Since the subspace $X_{0}$ is HI, there exist normalized vectors
$x_{1}\in[x_{n}^{1}\mathrel{\mathop{\mathchar 58\relax}}n\in\mathbb{N}]$ and
$x_{2}\in[x_{n}^{2}\mathrel{\mathop{\mathchar 58\relax}}n\in\mathbb{N}]$ such
that $\lVert x_{1}-x_{2}\rVert<\epsilon/3$. Since $(x_{n}^{1})_{n}$ and
$(x_{n}^{2})_{n}$ are Schauder basic sequences by $\epsilon/3$ perturbation we
may assume that there exists $n_{1}\in\mathbb{N}$ such that
$x_{1}\in[x_{n}^{1}\mathrel{\mathop{\mathchar 58\relax}}n\leq n_{1}]$ and
$x_{2}\in[x_{n}^{2}\mathrel{\mathop{\mathchar 58\relax}}n\leq n_{1}]$. Let
$x_{1}=\sum_{i=1}^{n_{1}}a_{n}x_{n}^{1}$ and
$x_{2}=\sum_{i=1}^{n_{1}}b_{n}x_{n}^{2}$. Then

$\displaystyle\lVert\sum_{i=1}^{n_{1}}a_{n}y_{n}^{1}-\sum_{i=1}^{n_{1}}b_{n}y_{n}^{2}\rVert$
$\displaystyle\leq\lVert
x_{1}-x_{2}\rVert+\lVert\sum_{i=1}^{n_{1}}a_{n}y_{n}^{1}-\sum_{i=1}^{n_{1}}a_{n}x_{n}^{1}\rVert+\lVert\sum_{i=1}^{n_{1}}b_{n}x_{n}^{2}-\sum_{i=1}^{n_{1}}b_{n}y_{n}^{2}\rVert$
$\displaystyle\leq 3\frac{\epsilon}{3}+\max_{n}\mathinner{\\!\left\lvert
a_{n}\right\rvert}\frac{\epsilon}{2^{3}}+\max_{n}\mathinner{\\!\left\lvert
b_{n}\right\rvert}\frac{\epsilon}{2^{3}}<3\epsilon.$

Using once more that $\lVert
x_{n}^{i}-y_{n}^{i}\rVert<\nicefrac{{\epsilon}}{{2^{n+3}}}$ we get that
$\operatorname{dist}(S_{Y_{1}},S_{Y_{2}})<10\epsilon$. This completes the
proof that $\operatorname{dist}(S_{Y_{1}},S_{Y_{2}})=0$ and therefore $X$ is
HI. ∎

##  7\. The space $X_{ex}$ is HI

To show that the space $X_{ex}$ is HI we will use Proposition 6.1 and as
$X_{0}$ the space $\widehat{X}_{hi}$. In order to show that the assumption of
the lemma holds we will use the RISs and the standard exact pairs as we did
for the space $X_{hi}$.

###  7.1. Dependent paired sequences

In this subsection we shall define the dependent paired sequences which is the
analogue of the dependent sequences defined for the original space $X_{ex}$
and we will provide upper estimations for the vectors defined by them. The
definition of the $(C,\epsilon)$-RIS is the same as in Definition 3.5 with the
obvious modifications. Similar the definition of the $(C,j_{0})$-standard
exact pair (SEP) is as in Definition 3.12.

To get upper bounds for the estimation of a weighted functional on a
$(C,\epsilon)$-RIS we shall use the basic inequality and a slight modification
of the auxiliary space. The new auxiliary space is the mixed Tsirelson space
$X_{aux2}=T[(\mathcal{A}_{12n_{j}},m_{j}^{-1})]_{j\in\mathbb{N}}$. We denote
by $W_{aux2}$ its norming set. The choice of the sequences
$(m_{j})_{j\in\mathbb{N}},(n_{j})_{j\in\mathbb{N}}$ gives us that on the
averages of the basis we have similar estimations as in Lemma 3.7. Namely the
following holds.

######  Lemma 7.1.

Let $x=\frac{m_{j}}{n_{j}}\sum_{k=1}^{n_{j}}e_{k}\in X_{aux2}$. Then for every
$f\in W_{aux2}$ with $w(f)=m_{i}$.

$\mathinner{\\!\left\lvert
f(x)\right\rvert}\leq\begin{cases}\frac{2}{m_{i}}\,\,\text{if $i<j$}\\\
\frac{m_{j}}{m_{i}}\,\,\text{if $i\geq j$}.\end{cases}$

Moreover if $f$ has a tree analysis $(f_{\alpha})_{\alpha\in\mathcal{A}}$ such
that for every weighted $f_{\alpha}$ we have $w(f_{\alpha})\neq m_{j}$ then
$\mathinner{\\!\left\lvert f(x)\right\rvert}\leq\frac{2}{m^{2}_{j}}$.

For a proof of this result see [9],Lemma II.9, (or [4], Lemma 4.2).

The upper bounds of the action of a weighted functional on averages of a
$(C,\epsilon)$-RIS in the space $X_{ex}$ are the same as in Proposition 3.10.
For sake of completeness we state it again.

######  Proposition 7.2.

Let $(x_{k})_{k=1}^{n_{j}}$ be a $(C,m_{j}^{-2})$-RIS in $X_{ex}$ and
$x=n_{j}^{-1}\sum_{k=1}^{n_{j}}x_{k}$. Then for every $f\in W_{ex}$ with
$w(f)=m_{i}$ we have

$\mathinner{\\!\left\lvert
f(x)\right\rvert}\leq\begin{cases}\frac{3C}{m_{i}m_{j}}\,\,\,&\text{if
$i<j$}\\\ \frac{C}{m_{i}}+\frac{2C}{n_{j}}\,\,\,&\text{if $i\geq
j$}\end{cases}.$ (7.1)

To show that the assumptions of the Lemma 6.1 hold we shall use the notion of
dependent paired sequences.

######  Definition 7.3.

A double sequence $(x_{k},f_{k})_{k=1}^{2n_{2j-1}}$ is said to be a
$(C,2j-1)$-dependent paired sequence if there exist $j_{k},k\leq n_{2j-1}$
such that

  1. (1)

$x_{2k}=\frac{m_{2j_{k}}}{n_{2j_{k}}}\sum_{i\in F_{k}}e_{i}$,
$\\#F_{k}=n_{2j_{k}}$ and $F_{k}\subset\mathbb{N}_{ev}$, and
$f_{2k}=m_{2j_{k}}^{-1}\sum_{i\in F_{k}}e_{i}^{*}$.

  2. (2)

$(x_{2k-1},f_{2k-1})$ is a $(C,2j_{k})$-standard exact pair with
$\operatorname{supp}(f_{2k-1})\subset\mathbb{N}_{od}$ for every $k$.

  3. (3)

$(f_{i})_{i=1}^{2n_{2j-1}}$ is a $(2j-1)$-special sequence of $W_{ex}$ with
$w(f_{2k-1})=m_{2j_{k}}=w(f_{2k})$ for every $k$.

Let us note that if $(x_{k},f_{k})_{k=1}^{2n_{2j-1}}$ is a
$(C,2j-1)$-dependent paired sequence then the functional
$F=m_{2j-1}^{-1}\sum_{k=1}^{n_{2n-1}}(f_{2k-1}+f_{2k})\in W_{ex}$ and

$\lVert\frac{m_{2j-1}}{n_{2j-1}}\sum_{k=1}(x_{2k-1}+x_{2k})\rVert\geq
F(\frac{m_{2j-1}}{n_{2j-1}}\sum_{k=1}(x_{2k-1}+x_{2k}))=2$

while $F(n_{2j-1}^{-1}\sum_{k=1}(x_{2k-1}-x_{2k}))=0$. Our aim is to show that
there exists a $K$ such that for every $j$ and $(C,2j-1)$-dependent paired
sequence we have

$\lVert\frac{m_{2j-1}}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}}(x_{2k-1}-x_{2k})\rVert\leq
KCm_{2j-1}^{-1}.$

For this we employ a variant of the basic inequality.

######  Proposition 7.4 (A variant of the basic inequality).

Let $(x_{k},f_{k})_{k=1}^{2n_{2j-1}}$ be a $(C,2j-1)$-dependent paired
sequence and $\lambda_{k},k\leq 2n_{2j-1}$ be scalars. Then for every weighted
functional $f\in W_{ex}$ we can find $g_{1},g_{2}\in c_{00}(\mathbb{N})$
having non-negative coordinates such that $\lVert
g_{2}\rVert_{\infty}<m_{2j-1}^{-3}$ and $g_{1}=h_{1}$ or
$g_{1}=e^{*}_{k}+h_{1}$ or $g_{1}=e^{*}_{k}+e^{*}_{k+1}+h_{1}$ for some
$h_{1}\in W_{aux2}$ with $w(f)=w(h_{1})$ and
$k\notin\operatorname{supp}(h_{1})$ if $g_{1}=e^{*}_{k}+h_{1}$ or
$k,k+1\notin\operatorname{supp}(h_{1})$ if
$g_{1}=e^{*}_{k}+e^{*}_{k+1}+h_{1}$, satisfying

$\mathinner{\\!\left\lvert
f(\sum_{k=1}^{2n_{2j-1}}\lambda_{k}x_{k})\right\rvert}\leq
C(g_{1}+g_{2})\left(\sum_{k=1}^{2n_{2j-1}}\lambda_{k}e_{k}\right).$ (7.2)

If we additionally assume that there exist $j_{0}\in\mathbb{N}$ such that for
all weighted functionals $g\in W_{ex}$ with $w(g)=m_{j_{0}}$ and every
interval $E$ of $\mathbb{N}$ we have that

$\mathinner{\\!\left\lvert g(\sum_{k\in E}\lambda_{k}x_{k})\right\rvert}\leq
C(\max_{k}\mathinner{\\!\left\lvert\lambda_{k}\right\rvert}+m_{2j-1}^{-3}\sum_{k\in
E}\mathinner{\\!\left\lvert\lambda_{k}\right\rvert})$ (7.3)

then if $w(f)\neq m_{j_{0}}$ we may select $h_{1}$ to have a tree analysis
$(h_{t})_{t\in\mathcal{T}}$ with $w(h_{t})\neq m_{j_{0}}$ for all
$t\in\mathcal{T}$.

The proof is identical to the original one with the only exception that a
functional with weight in the interval $[m_{2j_{k}},m_{2j_{k+1}})$ “can norm”
at most two vectors, namely $x_{2k-1},x_{2k}$, instead of one in the case of
the original basic inequality.

As a corollary of the basic inequality we get the following.

######  Corollary 7.5.

Let $(x_{k},f_{k})_{k=1}^{2n_{2j-1}}$ be a $(C,2j-1)$-dependent paired
sequence. Then

$\lVert\frac{m_{2j-1}}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}}(x_{2k-1}-x_{2k})\rVert\leq\frac{12C}{m_{2j-1}}.$
(7.4)

The proof of the corollary follows from Proposition 7.4 and Lemma 7.1 once it
is proved that for every $(2j-1)$-special sequence $(g_{k})_{k=1}^{d}$ of
$W_{ex}$ we have

$m_{2j-1}^{-1}\sum_{k=1}^{d}g_{k}(\sum_{k\in E}(-1)^{k+1}x_{k})\leq
6C(1+(\\#E)m_{2j-1}^{-2}).$ (7.5)

The proof of this inequality is based on two key ingredients which are the
same as in the proof of Proposition 4.5. The first one is that for every $k$
the estimation of a weighted functional on $x_{2k-1}$ or $x_{2k}$ satisfies
upper bounds which depend on the weight of the functional (see Lemma 7.1 and
Proposition 7.2) and the second one is that the family of paired special
sequences has a tree structure (see Remark 5.6b).

Using these two ingredients it is shown that for every $(C,2j-1)$-dependent
paired sequence $(x_{k},f_{k})_{k=1}^{2n_{2j-1}}$ and for every
$(2j-1)$-special sequence $(g_{k})_{k=1}^{d}$ of $W_{ex}$ (7.5) holds. This
implies that the additional part of the variant of the basic inequality holds
and using Lemma 7.1 we get (7.4).

###  7.2. The HI property of $X_{ex}$

We start with the folowing lemma.

######  Lemma 7.6.

Let $\epsilon>0$ and $x\in X_{ex}$ be a normalized block vector such that
$\operatorname{dist}(x,\widehat{X}_{hi})>\epsilon$. Then for every
$\delta\in(0,\epsilon)$ there exists $f\in W_{ex}$ such that

$a)\,\operatorname{supp}(f)\subset\mathbb{N}_{od},\hskip
85.35826ptb)\,f(x)>\epsilon-\delta.$

###### Proof.

Since $\operatorname{dist}(x,\widehat{X}_{hi})>\epsilon$ the Hahn-Banach
theorem yields that there exists $g\in S_{X^{*}_{ex}}$ such that
$g(x)>\epsilon$ and $g_{|\widehat{X}_{hi}}=0$. Denoting by $E$ the range of
$x$ we get that the functional $F=Eg$ satisfies also that $F(x)>\epsilon$ and
$F(e_{2n})=0$ for every $n\in\mathbb{N}$. Using that
$B_{X^{*}_{ex}}=\overline{W_{ex}}^{\lVert\,\rVert}$ we get $x^{*}\in W_{ex}$
such that

$\lVert F-x^{*}\rVert<\delta_{1}\,\,\,\ \text{where $\delta_{1}$
satisfies}\,\,\,\delta_{1}<\frac{\delta(1-\epsilon)}{(1-\delta)(1+\\#E)}.$
(7.6)

The basis of $X_{ex}$ is bimonotone hence $\mathinner{\\!\left\lvert
x^{*}(e_{2n})\right\rvert}=\mathinner{\\!\left\lvert
x^{*}(e_{2n})-F(e_{2n})\right\rvert}<\delta_{1}$ and also we may assume that
$x^{*}=Ex^{*}$. Assuming by a small perturbation if necessary, that $\delta$
is rational we set

$f=(1-\delta)x^{*}+(1-\delta)\sum_{2n\in E}\mathinner{\\!\left\lvert
x^{*}(e_{2n})\right\rvert}\epsilon_{2n}e^{*}_{2n}$ (7.7)

where $\epsilon_{2n}=-sign(x^{*}(e_{2n}))$. Note that $f(e_{2n})=0$ for every
$n$. Indeed for every $n$ such that $2n\notin E$ we have
$x^{*}(e_{2n})=Ex^{*}(e_{2n})=0$ while if $2n\in E$,

$x^{*}(e_{2n})+\mathinner{\\!\left\lvert
x^{*}(e_{2n})\right\rvert}\epsilon_{n}e^{*}_{2n}(e_{2n})=x^{*}(e_{2n})-x^{*}(e_{2n})=0.$

Also $f$ belongs to the rational convex hull of $W_{ex}$. This follows from
the fact that $x^{*}(e_{n})\in\mathbb{Q}$ for every $n\in\mathbb{N}$ and that

$\sum_{2n\in E}\mathinner{\\!\left\lvert
Ex^{*}(e_{2n})\right\rvert}\leq\sum_{2n\in E}\lVert
Ex^{*}-F\rVert\leq\\#E\,\delta_{1}<\delta.$ (7.8)

We conclude that $f\in W_{ex}$ since $W_{ex}$ is rationally convex. To show
that $f$ is the functional we are seeking it remains to show that
$f(x)>\epsilon-\delta$. Using (7.8) we get

$\displaystyle f(x)$ $\displaystyle=(1-\delta)x^{*}(x)+(1-\delta)\sum_{2n\in
E}\mathinner{\\!\left\lvert
x^{*}(e_{2n})\right\rvert}\epsilon_{2n}e^{*}_{2n}(x)$
$\displaystyle\geq(1-\delta)F(x)-(1-\delta)\lVert F-x^{*}\rVert\cdot\lVert
x\rVert-(1-\delta)\delta_{1}\\#E$
$\displaystyle\geq(1-\delta)\epsilon-(1-\delta)\cdot\delta_{1}(1+\\#E)>\epsilon-\delta.$

∎

######  Proposition 7.7.

For every closed infinite dimensional subspace $Y$ of $X_{ex}$ we have that
$\operatorname{dist}(S_{Y},S_{\widehat{X}_{hi}})=0$.

###### Proof.

On the contrary assume that
$\operatorname{dist}(S_{Y},S_{\widehat{X}_{hi}})>\epsilon$ for an infinite
dimensional subspace $Y$. We may assume that $Y$ is a block subspace.

We first show that for every $j,p\in\mathbb{N}$ there exists a
$(6/\epsilon,2j)$\- SEP $(y,f)$ with $y\in Y$, $\min y>p$ and
$\operatorname{supp}(f)\subset\mathbb{N}_{od}$.

Since $\operatorname{dist}(S_{Y},S_{\widehat{X}_{hi}})>\epsilon$ we may choose
a RIS $(y_{k})_{k\in\mathbb{N}}$ of normalized
$2-\ell_{1}^{n_{2j_{k}}}$-averages in $Y$ such that
$\operatorname{dist}(y_{k},S_{\widehat{X}_{hi}})>\epsilon$ for every $k$. Form
Lemma 7.6 we get a sequence of functionals $(f_{k})_{k}\subset W_{ex}$ such
that

$f_{k}(y_{k})>\frac{\epsilon}{2},\quad\,\,\operatorname{ran}(f_{k})\subset\operatorname{ran}(y_{k})\,\,\,\text{and}\,\,\,f_{k|\widehat{X}_{hi}}=0.$
(7.9)

For every $k$ choose $c_{k}\in[1,2/\epsilon)$ such that $f_{k}(c_{k}y_{k})=1$.
Then for every $j\in\mathbb{N}$ and $F_{j}\subset\mathbb{N}$ with
$\\#F_{j}=n_{2j}$ and $\min F_{j}>\max\\{n_{2j},p\\}$ it follows that the pair
$(y,f)$ where

$y=\frac{m_{2j}}{n_{2j}}\sum_{k\in
F_{j}}c_{k}y_{k}\quad\text{and}\,\,\,f=m_{2j}^{-1}\sum_{k\in F_{j}}f_{k}\quad$
(7.10)

is a $(6/\epsilon,2j)$\- SEP (see Definition 3.12) with $y\in Y$,
$\min\operatorname{supp}(y)>p$ and
$\operatorname{supp}(f)\subset\mathbb{N}_{od}$.

Next for every $j\in\mathbb{N}$ we defined a $(6\epsilon^{-1},2j-1)$-dependent
paired sequence $(x_{k},f_{k})_{k=1}^{2n_{2j-1}}$ as follows:

Choose $j_{1}$ such that $n_{2j-1}^{2}<m_{2j_{1}}$. Let $(x_{1},f_{1})$ be a
$(6\epsilon^{-1},2j_{1})$-SEP as in (7.10). Since $f_{1}\in W_{ex}$ we have
that $f_{1}\in V_{n_{1}}$ for some $n_{1}$. Using the coding function
$\varrho_{n_{1}}^{2j_{1}}$ we get the functional
$g_{2}=\varrho^{2j_{1}}_{n_{1}}(f_{1})=m_{2j_{1}}^{-1}\sum_{k\in
B_{1}}e_{k}^{*}\in K_{n_{1}}^{2j_{1}}$. We set

$f_{2}=\widehat{g_{2}}\quad\text{and}\quad
x_{2}=\frac{m_{2j_{1}}}{n_{2j_{1}}}\sum_{k\in B_{1}}e_{2k}.$

Let $2j_{2}=\sigma(g_{2})$ where $\sigma$ is the coding function used to
define the special sequences in the space $X_{hi}$. Choose $(x_{3},f_{3})$ a
$(6\epsilon^{-1},2j_{2})$-SEP as in (7.10) satisfying
$\operatorname{supp}(x_{2})<\operatorname{supp}(x_{3})$. This yields that also
$\operatorname{supp}(f_{2})<\operatorname{supp}(f_{3})$. Let $n_{3}$ be such
that $f_{3}\in V_{n_{3}}$. Using the coding function
$\varrho_{n_{3}}^{2j_{2}}$ we get the functional
$g_{4}=\varrho^{2j_{2}}_{n_{3}}(f_{3})=m_{2j_{2}}^{-1}\sum_{k\in
B_{3}}e_{k}^{*}\in K_{n_{3}}^{2j_{2}}$. We set

$f_{4}=\widehat{g_{4}}\quad\text{and}\quad
x_{4}=\frac{m_{2j_{2}}}{n_{2j_{2}}}\sum_{k\in B_{3}}e_{2k}.$

It is clear now how we proceed to get a $(6\epsilon^{-1},2j-1)$-dependent
paired sequence.

Set $y=\frac{m_{2j-1}}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}}x_{2k-1}$ and
$x=\frac{m_{2j-1}}{n_{2j-1}}\sum_{k=1}^{n_{2j-1}}x_{2k}$. From the choice of
the sequence we have that $y\in Y$ and $x\in\widehat{X}_{hi}$. Corollary 7.5
yields that $\lVert y-x\rVert<72\epsilon^{-1}m_{2j-1}^{-1}$. Also since
$(x_{2k-1})_{k=1}^{n_{2j-1}}$ is a $(2,m_{2j-1}^{-2})$-RIS it follows that
$1\leq\lVert x\rVert\leq 4$. This yields
$\operatorname{dist}(S_{Y},S_{\widehat{X}_{hi}})<2\cdot
72C\epsilon^{-1}m_{2j-1}^{-1}$. It follows that

$\frac{\epsilon}{2}<\frac{72}{\epsilon\cdot m_{2j-1}}\Leftrightarrow
m_{2j-1}\epsilon^{2}<2\cdot 72$

a contradiction for large enouge $j$. Therefore
$\operatorname{dist}(S_{Y},S_{\widehat{X}_{hi}})=0$ for every infinite
dimensional subspace $Y$ of $X_{ex}$. ∎

The above yields the following.

######  Theorem 7.8.

The HI space $X_{hi}$ is isometric to an infinite codimensional subspace of
the HI space $X_{ex}$.

###### Proof.

From Proposition 5.14 we have that $W_{ex|\mathbb{N}_{ev}}=\widehat{W}_{hi}$
and this shows that the space
$\widehat{X_{hi}}=[e_{2n}\mathrel{\mathop{\mathchar 58\relax}}n\in\mathbb{N}]$
is isometric to $X_{hi}$. Propositions 6.1 and 7.7 yield that the space
$X_{ex}$ is HI and this completes the proof. ∎

##  8\. The HI extension of a variant of $X_{hi}$.

There are several examples of HI spaces where the norming set W does not
include convex combinations and this is necessary for the restricted
saturation method applied for the definition of $W$, see [5],[6]. Also the
norming set in the Argyros-Deliyanni asymptotic $\ell_{1}$-HI space is defined
by a norming set $W$ which does not include rational convex combinations.
Since the closedness in rational convex combinations is essential to show that
the extended space is HI, naturally this raises the question if the spaces
defined by a norming set not closed in rational convex combinations admit an
HI extension. In this section we provide a positive answer to this problem for
a variant of the space $X_{hi}$ used before, which is defined by a norming set
$W_{vhi}$ which is similar to $W_{hi}$ but not closed in rational convex
combinations. More precisely we consider the following norming set. Let
$\sigma\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Q}_{s}\to\\{2j\mathrel{\mathop{\mathchar
58\relax}}j\in\Omega_{2}\\}$ be a coding function satisfying (4.1).

######  Definition 8.1.

The set $W_{vhi}$ is the minimal subset of $c_{00}$ satisfying the following
conditions:

  1. (1)

$\\{\pm e_{n}\mathrel{\mathop{\mathchar 58\relax}}n\in\mathbb{N}\\}\subset
W_{vhi}$.

  2. (2)

$W_{vhi}$ is symmetric and is closed under the restriction of its elements on
intervals of $\mathbb{N}$.

  3. (3)

$W_{vhi}$ is closed under the $(\mathcal{A}_{n_{2j}},m_{2j}^{-1})$-operations,
i.e., $m_{2j}^{-1}\sum_{i=1}^{d}f_{i}\in W_{vhi}$ for all $n_{2j}$-admissible
sequences $(f_{i})_{i=1}^{d}$ of $W_{vhi}$.

  4. (4)

$W_{vhi}$ is closed under the
$(\mathcal{A}_{n_{2j-1}},m_{2j-1}^{-1})$-operations on special sequences,
i.e., $m_{2j-1}^{-1}\sum_{i=1}^{d}f_{i}\in W_{vhi}$ for all
$n_{2j-1}$-admissible sequences $(f_{i})_{i=1}^{d}$ of $W_{vhi}$ such that
$w(f_{1})=m_{2j_{1}}>n_{2j-1}^{2}$ and
$w(f_{i+1})=m_{\sigma(f_{1},\dots,f_{i})}$ for $1\leq i\leq d-1$.

The difference between $W_{vhi}$ and $W_{hi}$ is the rational convexity. It
follows by the same arguments as in section 4 that the space $X_{vhi}$ with
the set $W_{vhi}$ as its norming set is also HI.

We define a variant $W_{vex}$ of the extended norming set $W_{ex}$. The
definition of $W_{vex}$ is very similar to the previous one $W_{ex}$ and goes
as follows:

Let $\widetilde{W}_{0}=\\{\pm e_{n}^{*}\mathrel{\mathop{\mathchar
58\relax}}n\in\mathbb{N}\\}$ and assume that $\widetilde{W}_{n-1}$ has been
defined. We set

$\widetilde{U}_{n}=\cup_{j}\\{m_{2j}^{-1}\sum_{i=1}^{k}f_{i}\mathrel{\mathop{\mathchar
58\relax}}(f_{i})_{i=1}^{k}\;\text{ is an
}\mathcal{A}_{n_{2j}}\text{-admissible sequence
of}\,\,\widetilde{W}_{n-1}\\},$

and
$\widetilde{V}_{n}=(\widetilde{U}_{n}\setminus\cup_{j=0}^{\infty}\cup_{k=n+1}^{+\infty}K_{k}^{2j})\cup\widetilde{W}_{n-1}$.

For $j\in\mathbb{N}$ we define

$\displaystyle\widetilde{W}_{n}^{2j-1}$
$\displaystyle=\\{\frac{1}{m_{2j-1}}\left(\sum_{i=1}^{l}(f_{i}+\widehat{g}_{i})+\sum_{i=l+1}^{m}\widehat{g}_{i}+\sum_{i=m+1}^{p}h_{i}\right)\mathrel{\mathop{\mathchar
58\relax}}$ (8.1) $\displaystyle\hskip 199.16928pt\text{special sequence}\\}.$

where the definition of a
$(2j-1,\widetilde{V}_{n},(\varrho^{2k}_{i})_{i=1}^{n},k=0,1,\dots)$ special
sequence is identical to the corresponding definition in $W_{ex}$. We set

$\begin{split}\widetilde{W}_{n}=\cup_{j}\widetilde{W}_{n}^{2j-1}\cup\widetilde{V}_{n}\cup\\{\sum_{i=1}^{d}r_{i}f_{i}\mathrel{\mathop{\mathchar
58\relax}}&(r_{i})_{i=1}^{d}\text{is a rational convex combination},\\\
&f_{i}\in\cup_{j}\widetilde{W}_{n}^{2j-1}\cup\widetilde{V}_{n}\,\,\,\text{and}\,\,\,\operatorname{supp}(\sum_{i=1}^{d}r_{i}f_{i})\subset\mathbb{N}_{od}\\}.\end{split}$

The norming set $W_{vex}$ is the set
$W_{vex}=\cup_{n=0}^{+\infty}\widetilde{W}_{n}$.

Lemmas 5.11 and 5.13 remain valid for the norming sets $W_{vhi}$ and $W_{vex}$
and the proofs are identical. These yield that $X_{vhi}$ is isometric to the
subspace $\widehat{X}_{vhi}=[e_{2n}\mathrel{\mathop{\mathchar
58\relax}}n\in\mathbb{N}]$ of $X_{vex}$. We have to prove that $X_{vex}$ is
HI. To prove this first we note that the variant of the basic inequality,
Proposition 7.4 and Corollary 7.5 hold also for the space $X_{vex}$. Moreover
the following holds.

######  Proposition 8.2.

The space $X_{vex}$ is reflexive and
$B_{X_{vex}^{*}}=\overline{\mathrm{conv}}^{\lVert\,\rVert}(W_{vex})$

To prove that $X_{vex}$ is HI we need a slight modification in the proof of
Lemma 7.6 since in that proof we use that
$B_{X^{*}_{ex}}=\overline{W_{ex}}^{\lVert\cdot\rVert}$ which does not hold for
the space $X_{vex}$.

######  Lemma 8.3.

Let $\epsilon>0$ and $x\in X_{vex}$ be a normalized block vector such that
$\operatorname{dist}(x,\widehat{X}_{vhi})>\epsilon$. Then for every
$\delta\in(0,\epsilon)$ there exists $f\in W_{vex}$ such that

$a)\,\operatorname{supp}(f)\subset\mathbb{N}_{od},\hskip
85.35826ptb)\,f(x)>\epsilon-\delta.$

###### Proof.

Since $\operatorname{dist}(x,\widehat{X}_{vhi})>\epsilon$ there exists $f\in
S_{X^{*}_{vex}}$ such that $g(x)>\epsilon$ and $g_{|\widehat{X}_{vhi}}=0$.
Denoting by $E$ the range of $x$ we get that the functional $F=Eg$ satisfies
also that $F(x)>\epsilon$ and $F(e_{2n})=0$ for every $n\in\mathbb{N}$. Using
that $B_{X^{*}_{vex}}=\overline{\text{conv}}(W_{vex})^{\lVert\,\rVert}$ we get
$x^{*}\in\operatorname{conv}_{\mathbb{Q}}(W_{vex})$ such that

$\lVert F-x^{*}\rVert<\delta_{1}\,\,\,\ \text{where $\delta_{1}$
satisfies}\,\,\,\delta_{1}<\frac{\delta(1-\epsilon)}{(1-\delta)(1+\\#E)}.$
(8.2)

The basis of $X_{vex}$ is bimonotone hence $\mathinner{\\!\left\lvert
x^{*}(e_{2n})\right\rvert}=\mathinner{\\!\left\lvert
x^{*}(e_{2n})-F(e_{2n})\right\rvert}<\delta_{1}$ and also we may assume that
$x^{*}$ is supported by the set $E$, i.e., $x^{*}=Ex^{*}$.

Assuming by a small perturbation if necessary that $\delta$ is rational we set

$f=(1-\delta)x^{*}+(1-\delta)\sum_{2n\in E}\mathinner{\\!\left\lvert
x^{*}(e_{2n})\right\rvert}\epsilon_{2n}e^{*}_{2n}\,\,\text{where}\,\,\epsilon_{2n}=-sign(x^{*}(e_{2n})).$
(8.3)

As in Lemma 7.6 we get that that
$f\in\operatorname{conv}_{\mathbb{Q}}(W_{vex})$ and
$\operatorname{supp}(f)\subset\mathbb{N}_{od}$.

Since $W_{vex}$ is closed in the rational convex combinations whose support is
subset of $\mathbb{N}_{od}$ it follows that $f\in W_{vex}$. The rest of the
proof is identical with the proof of Lemma 7.6. ∎

Using the above lemma we get the following.

######  Proposition 8.4.

The space $X_{vex}$ is HI.

The proof of the proposition is identical to the proof of the Proposition 7.7.

######  Remark 8.5.

The extension method applied for the HI spaces defined by the families
$\mathcal{A}_{n}$ can also be applied with identical proofs for the Gowers-
Maurey space [15].

##  9\. The case of asymptotic $\ell_{1}$-HI Banach spaces

In this section we show that the extension method can be applied also to
asymptotic $\ell_{1}$-HI Banach spaces, see [3]. The method is identical to
the one for the spaces defined by the families $\mathcal{A}_{n_{j}}$ with some
slight modifications due to the usage of the Schreier families.

Let us recall the definition of the norming set of a standard asymptotic
$\ell_{1}$-HI space.

Let $(n_{j})_{j=0}^{\infty},(m_{j})_{j=0}^{\infty}$ be two sequences of
positive integers such that $m_{0}=2$, $m_{j+1}=m_{j}^{m^{j}}$, $n_{0}=1$ and
$n_{j+1}=m_{1}^{2m_{j+1}}n_{j}$. Let also $\sigma$ be a coding function
defined as in Section 4.

The norming set $W_{hi}$ of a standard asymptotic $\ell_{1}$-HI space is the
minimal subset of $c_{00}(\mathbb{N})$ satisfying the following properties:

  1. i)

It is rationally convex, symmetric, closed on the projections on intervals of
$\mathbb{N}$ and contains $\pm e_{n}^{*}$, $n\in\mathbb{N}$.

  2. ii)

It is closed in $(\mathcal{S}_{n_{2j}},m_{2j}^{-1})$-operations, i.e., if
$(f_{i})_{i\in F}$ is an $\mathcal{S}_{n_{2j}}$-admissible sequence of
$W_{hi}$ then $m_{2j}^{-1}\sum_{i\in F}f_{i}\in W_{hi}$.

  3. iii)

It is closed in $(\mathcal{S}_{n_{2j-1}},m_{2j-1}^{-1})$-operations on special
sequences, i.e., if $(f_{i})_{i\in F}$ is an $\mathcal{S}_{n_{2j}}$-admissible
sequence of $W_{hi}$ $w(f_{i})=m_{2j_{i}}=\sigma(f_{1},\dots,f_{i-1})$ and
$m_{2j_{1}}\geq n_{2j-1}^{2}$ then $m_{2j-1}^{-1}\sum_{i\in F}f_{i}\in
W_{hi}$.

To define the extended HI space we shall use a variation of the Schreier
families $(\mathcal{S}_{n})_{n\in\mathbb{N}}$, (see [10] for some of their
properties).

######  Definition 9.1.

We set
$\mathcal{S}_{1}^{f}=\\{\\{1\\}\\}\cup\\{F\subset\mathbb{N}\mathrel{\mathop{\mathchar
58\relax}}2\\#F\leq minF\\}$ and

$\mathcal{S}_{n}^{f}=\mathcal{S}_{1}^{f}[\mathcal{S}_{n-1}^{f}]=\\{\cup_{i=1}^{k}F_{i}\mathrel{\mathop{\mathchar
58\relax}}F_{1}<\dots<F_{k}\in\mathcal{S}_{n-1}^{f},\\{\min
F_{i}\mathrel{\mathop{\mathchar 58\relax}}i\leq
k\\}\in\mathcal{S}_{1}^{f}\\}.$

We also the define the families

$\mathcal{S}_{n}^{f}\odot\mathcal{A}_{2}=\\{\cup_{i=1}^{d}F_{i}\mathrel{\mathop{\mathchar
58\relax}}F_{1}<\dots<F_{d},\,\\#F_{i}\leq 2\,\,\text{and}\,\\{\max
F_{i}\\}_{i=1}^{d}\in\mathcal{S}_{n}^{f}\\}.$

It is easy to see that for every $n$ the families
$\mathcal{S}_{m}^{f},{S}_{n}^{f}\odot\mathcal{A}_{2}$ are regular.

For a finite subset $G=\\{n_{1}<\dots<n_{d}\\}$ of $\mathbb{N}$ we denote by
$\widehat{G}$ the set $\widehat{G}=\\{2n_{1}<\dots<\dots<2n_{d}\\}$. We also
denote by $\widehat{\mathcal{S}_{n}}$ the family
$\\{\widehat{F}\mathrel{\mathop{\mathchar 58\relax}}F\in\mathcal{S}_{n}\\}.$

######  Lemma 9.2.

For every $n\in\mathbb{N}$ we have

$\widehat{\mathcal{S}_{n}}=\mathcal{S}_{n}^{f}\cap[\mathbb{N}_{ev}]^{<\omega}.$

###### Proof.

First we show by induction on $n$ that

$\mathcal{S}_{n}^{f}\cap[\mathbb{N}_{ev}]^{<\omega}\subset\widehat{\mathcal{S}_{n}}.$

Let $F=\\{2n_{1}<2n_{2}<\dots<2n_{d}\\}\in\mathcal{S}_{1}^{f}$. Then $2d\leq
2n_{1}\Rightarrow d\leq n_{1}$. It follows that
$G=\\{n_{1}<n_{2}<\dots<n_{d}\\}\in\mathcal{S}_{1}$ and $F=\widehat{G}$.

Assume that the result holds for every $k\leq n$ and
$F\in\mathcal{S}_{n+1}^{f}\cap[\mathbb{N}_{ev}]^{<\omega}$. Then
$F=\cup_{i=1}^{d}F_{i}$ with $F_{i}\in\mathcal{S}_{n}^{f}$ and $\\{\min
F_{i}\mathrel{\mathop{\mathchar 58\relax}}i\leq d\\}\in\mathcal{S}_{1}^{f}$.
By the inductive hypothesis we have for every $i\leq d$ there exits
$G_{i}\in\mathcal{S}_{n}$ such that $F_{i}=\widehat{G}_{i}$. From the case
$n=1$ we get that $\\{\min G_{i}\mathrel{\mathop{\mathchar 58\relax}}i\leq
d\\}\in\mathcal{S}_{1}$ and therefore
$G=\cup_{i=1}^{d}G_{i}\in\mathcal{S}_{n+1}$. Also $\widehat{G}=F$ and this
completes the proof of the induction.

The inclusion
$\widehat{\mathcal{S}_{n}}\subset\mathcal{S}_{n}^{f}\cap[\mathbb{N}_{ev}]^{<\omega}$
is proved similarly. ∎

We give the definition of the admissibility with respect the families
$\mathcal{S}_{n}$ and $\mathcal{S}_{n}^{f}\odot\mathcal{A}_{2}$.

######  Definition 9.3.

We call a block sequence $(x_{i})_{i=1}^{d}$ in $c_{00}(\mathbb{N})$

  1. (1)

$\mathcal{S}_{n}$-admissible if
$\min\\{\operatorname{supp}(x_{i})\mathrel{\mathop{\mathchar 58\relax}}i\leq
d\\}\in\mathcal{S}_{n}$.

  2. (2)

$\mathcal{S}^{f}_{n}$-admissible if
$\min\\{\operatorname{supp}(x_{i})\mathrel{\mathop{\mathchar 58\relax}}i\leq
d\\}\in\mathcal{S}^{f}_{n}$.

  3. (3)

$\mathcal{S}^{f}_{n}\odot\mathcal{A}_{2}$-admissible if
$\\{\min\operatorname{supp}(x_{i})_{i=1}^{d}\\}=\cup_{i=1}^{m}F_{i}$,
$F_{i}<F_{i+1}$ , $\\#F_{i}\leq 2$ for every $i$ and $\\{\max
F_{i}\mathrel{\mathop{\mathchar 58\relax}}i\leq m\\}\in\mathcal{S}_{n}^{f}$.

Let us mention that admissibility with respect the families
$\mathcal{S}_{n},\mathcal{S}_{n}^{f},n\in\mathbb{N}$ is the standard one, see
[9], [10], while the admissibility with respect the families
$\mathcal{S}_{n}^{f}\odot\mathcal{A},n\in\mathbb{N}$ is a variation of the
standard one, suitable for our applications.

######  Remark 9.4.

Let $G=\\{m_{1}<m_{2}<\dots<m_{d}\\}\in\mathcal{S}_{n}$. For every $0\leq
j\leq d$ and every $k_{1}<2m_{1}<k_{2}<2m_{2}<\dots<k_{j}<2m_{j}$ we have that
the set $\\{k_{1},2m_{1},\dots,k_{j},2m_{j},2m_{j+1},2m_{j+2},\dots,2m_{d}\\}$
is $\mathcal{S}_{n}^{f}\odot\mathcal{A}_{2}$-admissible.

Indeed if $F_{i}=\\{k_{i},2m_{i}\\}$ for $i\leq j$ and $F_{i}=\\{2m_{i}\\}$
for $i>j$, then Lemma 9.2 yields that $\\{\max F_{i}\mathrel{\mathop{\mathchar
58\relax}}i\leq d\\}$ is $\mathcal{S}_{n}^{f}$-admissible and hence
$\cup_{i=1}^{d}F_{i}$ is $\mathcal{S}_{n}^{f}\odot\mathcal{A}_{2}$-admissible
(see Definition 9.3).

The norming set of the extended space will be an EO-complete subset of the
norming set $W_{mT}$ of the mixed Tsirelson space
$T[(\mathcal{S}_{n_{2j}}^{f},m_{2j}^{-1})_{j\in\mathbb{N}},(\mathcal{S}_{n_{2j-1}}^{f}\odot\mathcal{A}_{2},m_{2j-1}^{-1})_{j\in\mathbb{N}}]$.

In order to define the extended norming set as in Subsection 5.1 we consider
the sets

$K^{2j}=\\{m_{2j}^{-1}\sum_{i\in F}e^{*}_{i}\mathrel{\mathop{\mathchar
58\relax}}F\,\,\text{is maximal $\mathcal{S}_{n_{2j}}$-set}\\}$

and a partition $(K_{i}^{2j})_{i}$ of them such that for every $i$,
$\sup\\{\min(f)\mathrel{\mathop{\mathchar 58\relax}}f\in
K_{i}^{2j}\\}=+\infty$. The sets $W_{mT}^{2j}$ and the coding functions
$\varrho_{i}^{2j}\mathrel{\mathop{\mathchar 58\relax}}W_{mT}^{2j}\to
K_{i}^{2j}$ are also defined as in Subsection 5.1.

The definition of the paired special, semi paired special and special
sequences are the same using
$\mathcal{S}_{n_{2j-1}}^{f}\odot\mathcal{A}_{2}$-admissibility instead of the
$\mathcal{A}_{n_{2j}}$-admissibility. Let us give the definition of the
$(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$\- paired special sequence.

######  Definition 9.5.

Let $j,n\in\mathbb{N}$ and $V\subset W_{mT}$. We call
$(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$\- paired special sequence
any block sequence $(f_{1},\widehat{g}_{1},\dots,f_{p},\widehat{g}_{p})$
satisfying the following:

  1. (1)

it is $\mathcal{S}_{n_{2j-1}}^{f}\odot\mathcal{A}_{2}$-admissible and for
every $k$ there exists $j_{k}\in\mathbb{N}$ such that $f_{k}\in V\cap
W_{mT}^{2j_{k}}$ and
$\widehat{g}_{k}=\widehat{\varrho^{{2j_{k}}}_{i_{k}}(f_{k})}$ for some
$i_{k}\leq n$.

  2. (2)

$(g_{1},g_{2},\dots,g_{p})$ is a $(2j-1)$-special sequence of $W_{hi}$.

######  Remark 9.6.

Let us point out that the $\mathcal{S}_{n_{2j-1}}$-admissibility of
$(g_{1},g_{2},\dots,g_{p})$ implies the
$\mathcal{S}_{n_{2j-1}}^{f}\odot\mathcal{A}_{2}$-admissibility of the sequence
$(f_{1},\widehat{g}_{1},\dots,f_{p},\widehat{g}_{p})$ (see Remark 9.4).

The definition of the $(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$\-
paired special sequences and
$(2j-1,V,(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$\- special sequence are as
in Subsection 5.1.

Next we prove that $W_{ex|\mathbb{N}_{ev}}=\widehat{W}_{hi}$ proving the
analogues of Lemmas 5.11,5.13.

######  Lemma 9.7.

For every $n\in\mathbb{N}$ and every $f\in W_{n}$ we have that
$f_{|\mathbb{N}_{ev}}\in\widehat{W}_{hi}$.

###### Proof.

As in Lemma 5.11 we use induction. The case $n=0$ is identical. Assume that
the result holds for all $k\leq n$ and let $f\in W_{n+1}$.

If $f$ is the result of an $(\mathcal{S}^{f}_{n_{2j}},m_{2j}^{-1})$ operation
of elements of $W_{n}$, i.e., $f=m_{2j}^{-1}\sum_{i=1}^{d}f_{i}$, from the
inductive hypothesis we have that for every $i$ there exists $\phi_{i}\in
W_{hi}$ such that $f_{i|\mathbb{N}_{ev}}=\widehat{\phi_{i}}$. Lemma 9.2 yields
that $\\{\operatorname{minsupp}(\phi_{i})\mathrel{\mathop{\mathchar
58\relax}}i\leq d\\}$ is $\mathcal{S}_{n_{2j}}$-admissible. Since $W_{hi}$ is
closed in $(\mathcal{S}_{n_{2j}},m_{2j}^{-1})$-operations we get that
$f_{|\mathbb{N}_{ev}}=\widehat{m_{2j}^{-1}\sum_{i=1}^{d}\phi_{i}}\in\widehat{W}_{hi}$.

The result in the case where $f$ is the result of an
$(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$-special sequence is
analogue to the previous one using Remark 9.6. The case is $f$ is a rational
convex combination follows readily. ∎

######  Lemma 9.8.

For every $f\in W_{hi}$ there exists $\phi\in W_{ex}$ such that
$\phi_{|\mathbb{N}_{ev}}=\widehat{f}$.

###### Proof.

Similar to the proof of Lemma 5.13 we proceed by induction.

For $n=1$ and $f=m_{2j}^{-1}\sum_{i\in G}\pm e_{i}^{*}$,
$G\in\mathcal{S}_{n_{2j}}$ we observe from Lemma 9.2 that
$\widehat{G}\in\mathcal{S}_{n}^{f}$. Using this observation, as in Lemma 5.13
we conclude that $\widehat{f}\in W_{ex}$

We pass now to the case where $f=m_{2j-1}^{-1}\sum_{i=1}^{p}g_{i}$ and
$(g_{1},\dots,g_{p})$ is a $(2j-1)$-special sequence of $W_{hi}^{1}$. As in
Lemma 5.13 the proof is based on the following claim.

######  Claim.

Let $g_{i}\in K_{n_{i}}^{2j_{i}}$, $i\leq m$ be such that
$(g_{1},\dots,g_{m})$ is a $(2j-1)$-special sequence of $W_{hi}^{1}$. Then
there exist $0\leq l\leq m$ and $f_{1},\dots,f_{l}\in W_{ex}$ such that
$(f_{1},\widehat{g}_{1},\dots,f_{l},\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m})$
is a $(2j-1,V_{n},(\varrho_{k}^{n})_{k=1}^{n},i=0,\dots)$ semi-paired special
sequence where $n=\max\\{n_{i}\mathrel{\mathop{\mathchar 58\relax}}i\leq
m\\}$.

Sketch of the proof. Let $l\leq m$ be maximal with the properties

  1. (1)

$g_{i}\in\varrho_{n_{i}}^{2j_{i}}(V_{n}\cap W_{mT}^{2j_{i}})$ for every $i\leq
l$.

  2. (2)

$F=((\varrho^{2j_{i}}_{n_{i}})^{-1}(g_{i}),\widehat{g}_{i})_{i=1}^{l}$ is a
$(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1\dots)$ -paired special
sequence, where $n=\max\\{n_{i}\mathrel{\mathop{\mathchar 58\relax}}i\leq
m\\}$.

For every $i$, as in the first case we have $\widehat{g}_{i}\in W_{ex}$, and
from Remark 9.4 we get
$(\varrho^{2j_{1}}_{n_{i}})^{-1}(g_{1}),\widehat{g}_{1},\dots,(\varrho^{2j_{l}}_{n_{i}})^{-1}(g_{i}),\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m})$
is $\mathcal{S}_{n_{2j-1}}^{f}\odot\mathcal{A}_{2}$-admissible in $W_{ex}$.
Considering the cases $l=0,l=m$ and $0<l<m$, as in the proof of Lemma 5.13 we
get that
$(\varrho^{2j_{1}}_{n_{i}})^{-1}(g_{1}),\widehat{g}_{1},\dots,(\varrho^{2j_{l}}_{n_{i}})^{-1}(g_{i}),\widehat{g}_{l},\widehat{g}_{l+1},\dots,\widehat{g}_{m})$
is a $(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$-semi paired
special sequence. $\square$

Returning back to the proof, let $f=m_{2j-1}^{-1}\sum_{i=1}^{p}g_{i}$ where
$(g_{1},\dots,g_{p})$ is a $(2j-1)$-special sequence of $W_{hi}^{1}$, and
$m\leq p$ be maximal with the property $g_{i}\in K_{2n_{i}}^{2j_{i}}$ for
every $i\leq m$. It follows from the Claim and the definition of the special
sequences that
$((\varrho^{2j_{1}}_{n_{1}})^{-1}(g_{1}),\widehat{g}_{1},\dots,(\varrho^{2j_{l}}_{n_{m}})^{-1}(g_{m}),\widehat{g}_{m},\widehat{g}_{m+1},\dots,\widehat{g}_{p})$
is a $(2j-1,V_{n},(\varrho_{k}^{2i})_{k=1}^{n},i=0,1,\dots)$-special sequence.
Hence
$\phi=m_{2j-1}^{-1}(\sum_{i=1}^{m}((\varrho^{2j_{l}}_{n_{i}})^{-1}(g_{i})+\widehat{g}_{i})+\sum_{m+1}^{p}\widehat{g}_{i})\in
W_{ex}$ and $\widehat{f}=\phi_{|\mathbb{N}_{ev}}$.

Assume that we have proved the result for every $k\leq n$ and let $f\in
W_{hi}^{n+1}$.

In the case $f$ is the result of an
$(\mathcal{S}_{n_{2j}},m_{2j}^{-1})$-operation of elements of $W_{hi}^{n}$,
using Lemma 9.2 and the property $W_{ex}$ is closed in the
$(\mathcal{S}^{f}_{n_{2j}},m_{2j}^{-1})$-operations, we show, as in Lemma
5.13, that $\widehat{f}=\phi_{|\mathbb{N}_{ev}}$ for some $\phi\in W_{ex}$.

In the case $f=m_{2j-1}^{-1}\sum_{i=1}^{p}g_{i}$ where $(g_{1},\dots,g_{p})$
is a $(2j-1)$-special sequence of $W_{hi}^{n+1}$, is proved as in Lemma 5.13
using the above Claim. ∎

As a corollary we get the following.

######  Proposition 9.9.

For the norming sets $W_{hi}$ and $W_{ex}$ of the spaces $X_{hi}$ and $X_{ex}$
we have that

$W_{ex|\mathbb{N}_{ev}}=\widehat{W}_{hi}.$

As a consequence the space $X_{hi}$ is isometric to the subspace of $X_{ex}$
generated by the sequence $(e_{2n})_{n\in\mathbb{N}}$.

The results of the Sections 4,5,7 with the obvious modifications imposed by
the usage of the Schreier families in place of the $\mathcal{A}_{n_{j}}$
families, hold also for the new spaces with similar proofs (see [10] for a
detailed exposition of the related results).

The above yield the following.

######  Theorem 9.10.

A standard asymptotic $\ell_{1}$ HI Banach space is isometric to an infinite
codimensional subspace of a reflexive asymptotic $\ell_{1}$ HI Banach space.

######  Remark 9.11.

Applying the analogue modifications we get that any standard asymptotic
$\ell_{p}$ HI space, $1<p<+\infty$, (see [12]), admits an HI extension.

##  10\. Final discussion

We discuss some open problems that are related to this article. We begin with
the one that motivated this paper and remains open in its full generality.

######  Problem 1 (Pełczyński).

Does every HI space $X$ embed into an HI space $Y$ such that $Y/X$ is infinite
dimensional?

Problem 1 was posed by Pełczynski at a conference where the first author was
present, however we were unable to find it in the literature. Yet, he posed a
strongly related one in [17, Problem 3.3, page 348].

######  Problem 2 (Pełczynski).

Does every separable HI space $X$ embed into an HI space $Y$ with a Schauder
basis?

It is remarkable that in the same paper [17] Pełczyński posed the problem of
the existence of a $\mathcal{L}_{\infty}$-space that is HI, a fact that the
first author was unaware of. This is why this paper was not cited in [2],
where the scalar-plus-compact problem was solved using such a space. Let us
point out that the methods developed in the current paper show promise in
constructing HI extensions of $\mathcal{L}_{\infty}$-HI spaces, such as the
Argyros-Haydon space.

After sharing a preliminary version of this article to W. B. Johnson, he
pointed out to us a more general form of Pełczyński’s HI extension Problem 1,
based on the following definition (see also [11, page 4701] for a formulation
in the setting of twisted sums).

######  Definition 10.1.

Let $X$ be a Banach space $Y$ be a superspace of $X$. We call $Y$ a singular
extension of $X$ if $Y/X$ is infinite dimensional and the quotient map
$Q\mathrel{\mathop{\mathchar 58\relax}}Y\to Y/X$ is strictly singular.

######  Problem 3 (W. B. Johnson).

Does every infinite dimensional separable Banach space $X$ that is not
isomorphic to $c_{0}$ admit a singular extension $Y$?

For example, as proved by Kalton and Peck, the space $X=\ell_{p}$,
$1<p<\infty$, admits a singular extension, namely the Banach space $Z_{p}$
(see [16, Theorem 6.4, page 27]). It is not hard to see that if $X$ is HI and
it admits a singular extension $Y$ then $Y$ must be HI as well. Therefore, a
positive answer to Problem 3 would imply a positive answer of Problem 1, at
least in the separable setting (whether every separable HI space admits a
singular extension was also asked by Castillo, Ferenczi, and Gonzalez in [11,
page 4701]).

In [11] it is proved that the uniformly convex HI space $\mathcal{F}$
constructed by V. Ferenczi in [13] admits an HI short exact sequence
$0\to\mathcal{F}\to\mathcal{F}_{2}\to\mathcal{F}\to 0$, such that
$\mathcal{F}_{2}$ is HI, i.e., $\mathcal{F}_{2}$ is an HI twisted sum of
$\mathcal{F}$. This is achieved by using the fact that $\mathcal{F}$ is a
space resulting from complex interpolation. Note that in particular
$\mathcal{F}_{2}$ is an HI extension of $\mathcal{F}$.

It has long been an open question whether it is possible to construct a
uniformly convex HI space by directly defining a norming set instead of using
complex interpolation. It is therefore relevant to ask whether it is possible
to find an HI space $X$ and an HI twisted sum $Y$ of $X$ by defining a norming
set $W$. The natural approach would be to try to construct a rationally convex
norming set $W$ that induces an HI space $Y$ such that

$\\{f|_{\mathbb{N}_{ev}}\mathrel{\mathop{\mathchar 58\relax}}f\in
W\\}=\\{f\circ\pi\mathrel{\mathop{\mathchar 58\relax}}f\in W\text{ with
}\mathrm{supp}(f)\subset\mathbb{N}_{od}\\},$

where $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{N}_{ev}\to\mathbb{N}_{od}$ is the order preserving
bijection. Then, by taking
$X=\overline{\langle\\{e_{2n}\mathrel{\mathop{\mathchar
58\relax}}n\in\mathbb{N}\\}\rangle}$, the Hahn-Banach theorem would yield that
$Y/X$ is isometrically isomorphic to $X$.

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