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\begin{document}

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\title{Uniqueness of Structure in Banach Spaces} \author{Lior Tzafriri}
\maketitle

\section{Introduction}


In this chapter, we consider Banach spaces which can be represented
as spaces of sequences or functions with some specific properties, and
study the natural question
 whether such a representation is unique, up to a notion of equivalence
which can vary from case to case.  The typical space of sequences is
derived from a Banach space $X$ with a normalized
 Schauder basis $\elim$, where to each element $x =\suminf a_n e_n \in X$,
there corresponds the sequence of coefficients $(a_1, a_2, \cdots, a_n,
\cdots )$.  Of particular interest in this context will be the spaces
with a normalized unconditional or symmetric
 basis.

The notion of uniqueness mentioned  above means a different thing in
each of the cases under consideration.  A Banach space $X$ will be said
to have a {\bf unique}  general or unconditional or symmetric
 basis, up to equivalence, if, for any two normalized bases of the {\bf
 same} type, there exists an automorphism $T$ of $X$ which
takes one basis into the other.

The existence of a normalized Schauder basis
 $\{e_n\}^\infty_{n=1}$ in a Banach space $X$ allows the representation
 of $X$ as a space of {\bf ordered} sequences.
In the special case where $\{ e_n\}^\infty_{n=1}$  is an unconditional
basis any permutation $\{ e_{\pi(n)}\}^\infty_{n=1}$ of $\elim$ is again
a basis which need not be equivalent to $\elim$, for every permutation
$\pi$, unless $\elim$ is a symmetric basis.  Therefore, in  the case of
spaces with a normalized unconditional basis (which is not symmetric)
we have the option of considering a different type of uniqueness of
the normalized
 unconditional basis, namely that of uniqueness up to a permutation
 and equivalence.
More precisely, a space $X$ has a {\bf unique} normalized unconditional
basis, up to a {\bf permutation} (and equivalence) whenever, for each
pair of normalized unconditional
 bases of $X$, there exists an automorphism $T$ of $X$ which takes
 the first
basis into a {\bf permutation} of the second.  Of course, both notions
of uniqueness coincide in the case of symmetric bases.

In the continuous case, we typically consider Banach lattices which can
be represented
 as spaces of measurable functions on a suitable measure space.
Most interesting is the special case of the so-called rearrangement
invariant (r.i.)  function spaces  over a measure space $(\Omega,
\Sigma, \mu)$.  The main property of such a space $X$  of functions
 is that automorphisms of the underlying measure space transform
elements of $X$ into elements of $X$ with the {\bf same} norm. In the
Basic Concepts article such spaces are called {\bf symmetric lattices }.

The notion of uniqueness can be also studied in the context of finite
dimensional spaces, but, in this case, we have to be more careful
since for
 spaces of the same dimension all the structures are obviously unique.
However, for families of Banach spaces $\xinf$, with $\dim X_n=n$, for
all $n$
 it makes perfect sense to inquire whether, for any two normalized bases
 of $X_n$, of  the {\bf same} type, there exists an automorphism $T_n$
 of $X_n$ which
takes the first basis into the second and, most importantly, the
quantity $\|
 T_n \| \cdot \| T_n^{-1} \|$ is bounded by a constant independent
 of $n$ but possibly dependent on the structure constant (by structure
 constant we mean either the  basis constant or unconditional or symmetric
 constant, according to the case under consideration).

\section{Uniqueness of general and unconditional bases}

It is quite easy to prove that a separable Hilbert space has a unique
normalized
 unconditional basis, up to equivalence.
Indeed, if $\{ u_n \}^\infty_{n=1}$ is a normalized $K$-unconditional
basis in $\ell_2$ then, by the parallelogram identity, $$
\mathop{Ave}\limits_{\sigma_n=\pm 1} \| \suml^{N}_{n=1} a_n \sigma_n u_n
\|^2 = \suml^{N}_{n=1} |a_n|^2 $$ for every choice of $N$ and scalars $\{
a_n\}^{2^N}_{n=1}$.  This of course implies that $$ K^{-1} \Big (\suminf
|a_n|^2 \Big )^{1/2} \le \| \suminf a_n u_n \| \le K\left(\suminf |a_n
|^2 \right)^{\half}, $$ for any    choice of $\{ a_n\}^\infty_{n=1}$,
i.e. $\{ u_n\}^\infty_{n=1}$ is $K$-equivalent to the unit vector basis
of $\ell_2$.

It turns out that a separable Hilbert space does not have unique basis,
up to equivalence.  This fact, which is less trivial, was first proved
by Babenko [ B ].  His construction is based on the simple
observation that the characters $\{e^{int}\}_{n\in \bbz}$, in the order
$\{ 1, e^{it}, e^{-it}, e^{2it}, e^{-2it}, \cdots \}$, form a basis
in the space $L_p (\bbt, W(t))$, where $W(t)$ is an integrable weight
function on the circle $\bbt$, if and only if the Riesz projection,
defined
 by
$$ P_+ \left(\suml^{+\infty}_{n=-\infty} a_n e^{int}\right) =
\suml^\infty_{n=0} a_n e^{int} $$ is bounded on this space.  This latter
question  has been extensively studied in harmonic analysis and a well
known necessary and sufficient condition for the boundedness of the
Riesz projection (or, as a matter of fact, of the Hilbert transform)
in the space $L_p (\bbt, W(t))$ is the so-called $A_p$-condition ( see
e.g. [ G p.254 ] ) 
$$ \mathop{\sup}\limits_I \Bigl({1\over \mu(I)} \intl_I
W(t) dt\Bigr) \; \Bigl({1\over \mu (I)} \intl_I \Big({1\over W(t)}
\Big)^{1\over p-1} dt\Bigr)^{p-1}
 < \infty,
$$ where $\mu$ is the Lebesgue measure and the supremum is taken 
over all intervals $I\subset \bbt$ of
positive measure.

The weight function considered by Babenko is $$ W_\a (t) = |t|^{2\a}, $$
where $\a$ is a suitable number.  In order to ensure the integrability
of $W_\a(t)$, one has to require that $2\a + 1
 > 0$ i.e. that $\a > - \half$.
Furthermore, it is easily verified that the $A_2$-condition is satisfied
by $W_\a$ iff $1-2\a >0$, i.e.  when $\a < \half$.  It follows that, for
$-\half < \a < \half$, the characters $\{e^{int}\}_{n\in \bbz}$, in the
order described above, form a basis of the Hilbert space $L_2
 (\bbt, W_\a (t))$.
Equivalently, the sequence $$ \Big\{ \Big({2\a+1\over
2\pi^{2\a+1}}\Big)^{\half} |t|^\a e^{int} \Big\}_{ n \in \bbz}$$ forms
a normalized Schauder basis in $L_2 (\bbt)$,
 for any value of $-\half < \a < \half$.
For $\a = 0$, we recover the (unique) unconditional basis of the separable
 Hilbert space while, for the remaining
values of $\a$, we obtain mutually non-equivalent conditional bases  of
the Hilbert space.  Indeed, for $\a\neq \b$ in the interval $(-\half,
\half)$ and $0 < \lambda < \pi$,
 we can find scalars $\{ a_n \}_{n \in \bbz} $ so that
$$ f(t) = \suml^{+\infty}_{n=-\infty} a_n e^{int} = \chi_{[0, \lambda)}
(t).  $$ in $L_2 (\bbt)$ and a.e. on $\bbt$.  Then the fact that $$ {\|
f |t|^\a\|\over \| f |t|^\b\|} = \lambda^{\a-\b}; \; \; 0 < \lambda <
\pi, $$ shows that the bases $\{ |t|^\a e^{int}\}_{n\in \bbz}$ and
 $\{ |t|^\b e^{int} \}_{n\in \bbz}$ are not equivalent.

Another construction of conditional bases in a separable Hilbert space,
due to C.A. McCarthy
 and J. Schwartz  [ Mc - S ], is presented in detail in [ L - T I p. 74 ].

The discussion above can be summarized in the following proposition.
\begin{Proposition} A separable Hilbert space has, up to equivalence,
a unique
 normalized unconditional basis and uncountably many mutually
 non-equivalent conditional
basis.  \end{Proposition}

In fact, the following more general result was proved by Pelczynski and
Singer [ P - S ].

\begin{Theorem} Any Banach space with an infinite Schauder basis has
uncountably many mutually non-equivalent bases.  \end{Theorem}

It turns out that also the spaces $\ell_1$ and $ c_0$ have a unique
unconditional basis, up to equivalence, but this fact, which was proved
by J. Lindenstrauss and A. Pelczynski [ L - P ], is more difficult
and requires
 some consequences of the famous
Grothendieck inequality.  We refer to well-known corollaries of this
inequality that every bounded linear operator $T\colon c_o \to \ell_1$
is 2-absolutely summing and its 2-summing norm  $\pi_2 (T)$ satisfies
the inequality $\pi_2(T) \le K_G \| T \|$, and that every
 bounded linear operator $U \colon \ell_1 \to \ell_2$ is
absolutely summing and its 1-summing norm $\pi_1 (U) $ satisfies $\pi_1
(U) \le K_G \| U \|$.  In both  inequalities above, $K_G$ denotes as
usual the Grothendieck constant. The precise statement of Grothendieck's
inequality together with a simple proof can be found in the Basic
Concepts article.

In order to prove e.g. that $\ell_1$ has a unique unconditional basis,
up to equivalence,
 let $\{ x_n \}^\infty_{n=1}$ be a normalized $K$-unconditional basis
 of $\ell_1$, for some $K \ge 1 $, and fix a sequence of scalars $\{
 a_n\}^\infty_{n=1}$ such
that the series $\suminf a_n x_n$ converges.  Then notice that the
operator $T\colon c_o \to \ell_1$, defined by $Tt= \suml^{\infty}_{n=1}
a_nt_nx_n$, for $t = \{ t_n\}^\infty_{n=1} \in c_0$, is of norm $\|
T\| \le 2 K \| \suml^\infty_{n=1} a_n x_n \| $ (in the real case, $2K$
can be replaced by $K$).  Hence, by the aforementioned estimate of the
2-summing norm of $T$,
 we conclude that $\pi_2 (T) \le 2 K_G K \| \suml^\infty_{n=1} a_n
 x_n \|$.
Then the definition of $\pi_2 (T)$, applied for the unit vectors in
$c_0$, together with
 the fact that
$\{ x_n\}^\infty_{n=1}$ is assumed to be normalized imply  that $$ \left(
\suminf |a_n|^2\right)^{\half} = \left(\suminf \| T e_n\|^2\right)^{\half}
 \le 2K_G K \| \suminf a_n x_n \|.
$$

This inequality shows that the operator $U\colon \ell_1 \to \ell_2$,
 defined by $U (\suminf a_nx_n) = \{ a_n \}^\infty_{n=1}$, for any
 convergent series $\suminf a_n x_n \in \ell_1$, is bounded by $2K_G K$.
Then, by using the estimate,  discussed above, of the 1-summing norm
$\pi_1(U)$
 of $U$, it follows
that $\pi_1 (U) \le 2 K^2_G K$.  Hence, for any $x = \suminf
a_nx_n\in\ell_1$, $$ \suminf |a_n| = \suminf \| U (a_n x_n) \| \le \pi_1
(U) \sup\limits_{\e_n=\pm 1} \| \suminf
 \e_na_nx_n\| \le
4K^2_G K^2 \| \suml_{n=1} a_n x_n \|, $$ which completes the proof of
the fact that $\{ x_n\}^\infty_{n=1}$
 is equivalent to the unit vector basis of $\ell_1$.

The case of $c_0$ is proved by using a simple duality argument.

\bigskip

The summary of this discussion is

\begin{Proposition} Both spaces $\ell_1$ and $c_0$ have, up to
equivalence,
 a unique normalized unconditional basis.
\end{Proposition}

An alternative proof of Proposition 1.3, which does not use
Grothendieck's inequality, is presented in Section 5 of the Basic Concepts 
article.

It turns out that $\ell_2, \ell_1$ and $c_o$ are the only Banach spaces
with a unique
 unconditional basis, up to equivalence.
This result was proved by Lindenstrauss and Zippin [ L - Z ].

\begin{Theorem} A Banach space has, up to equivalence, a unique
unconditional basis if and only if it is isomorphic to one of the spaces
$\ell_2, \ell_1$ or $c_0$.  \end{Theorem}

Instead of the original proof from [ L - Z ], we present a shorter proof
based on an argument due to W.B. Johnson. Suppose that a space $X$ has,
up to equivalence, a unique normalized unconditional
 basis $\{ x_n\}^\infty_{n=1}$.
Since, for any choice of $\e_n =\pm 1, n = 1, 2, \cdots, \{
\e_nx_n\}^\infty_{n=1}$ is an unconditional basis of $X$ it must be
equivalent to $\{x_n\}^\infty_{n=1}$ and, thus, $\{x_n\}^\infty_{n=1}$
 is symmetric.
Let now $\{u_m\}^\infty_{m=1}$ be an arbitrary normalized block basis
with constant
 coefficients of $\{x_n\}^\infty_{n=1}$ and denote by $U $ its span.

If we prove that $X$ is isomorphic to $X\oplus U$, then 
$$ \{ x_1, u_1, x_2, u_2, \cdots, x_n, u_n, \cdots \}
$$ forms a normalized unconditional basis of a space isomorphic 
to $X$. 
Hence, by  the uniqueness of $\{x_n\}^\infty_{n=1}$,
$\{u_n\}^\infty_{n=1}$ is equivalent
 to $\{ x_n\}^\infty_{n=1}$ i.e. $\{x_n\}^\infty_{n=1}$ 
is a perfectly homogeneous basis in the terminology
of [ Z ]. Thus, by the well 
known
result of Zippin [ Z ], $\{x_n\}^\infty_{n=1}$ is 
equivalent to the unit vector
basis of $\ell_p$; $ 1 \le p < \infty$ or of
 $c_0$.  In order
to complete the proof, recall e.g. that the Rademacher functions
$\{r_n\}_{n=1}^\infty$ span a Hilbert space in $L_p(0,1)$, for any
value of $p\ge1$, and their span $[r_n]_{n=1}^\infty$ in complemented
in $L_p(0,1)$, whenever $p>1$.  In particular, $\ell_p^{2^n}$
contains a uniformly complemented copy of $\ell_2^n$, for any $n$
and $p>1$.  Hence, $\ell_p$ contains a complemented copy of the
direct sum $\left(\suml_{n=1}^\infty\oplus\ell_2^n\right)_p$
and, by the decomposition method of Pelczynski,
$\ell_p\approx\left(\suml_{n=1}^\infty\oplus\ell_2^n\right)_p$, for
$p>1$, which shows that the unit vector basis of $\ell_p$ is not the
unique unconditional basis of this space, up to equivalence, for
$p>1, p\neq 2$.

In order to prove that $X \approx X \oplus U$, Johnson suggested
to use the decomposition method, as follows:  Let $V$ be 
the span of normalized blocks with constant coefficients of
$\{x_n\}^\infty_{n=1}$
which has the universality property that every possible normalized
block with constant
coefficients of $\{x_n\}^\infty_{n=1}$ appears infinitely many times 
in the sequences
generating $V$. Then, since any block basis with constant coefficients
spans a complemented subspace in a space with a symmetric basis ( see
e.g. [ L ] or [ L - T I p. 123 ] $X \oplus V$ is isomorphic to a
complemented  in $X$, i.e. that $$ X \approx X \oplus V \oplus W, $$
for some space $W$.  Since $V \oplus U \approx V$ because of the above
universality property, it follows that $$ X\approx X \oplus V \oplus
U\oplus W \approx X \oplus U, $$ which completes the argument.



\noindent {\bf Remark}: It can be shown that any space with an
unconditional basis which is not
 unique, up to equivalence, actually has uncountably many  mutually non
 equivalent unconditional bases.


Suppose now that $X$ is a space with a unique, up to equivalence,
normalized Schauder
 basis $\{ x_n\}^\infty_{n=1}$.
Then $\{x_n\}^\infty_{n=1}$ is equivalent to $\{\e_nx_n\}^\infty_{n=1}$,
for every choice of $\e_n = \pm 1$, $n = 1, 2, \cdots$,
 i.e. $\{ x_n\}^\infty_{n=1}$, is unconditional and thus equivalent to
 the unit vector basis of $\ell_2, \ell_1$ or $c_o$.
However, in each of these three cases, 
$X$ also has a conditional basis.  Simple
examples  of conditional bases are $\{ e_n - e_{n-1}\}^\infty_{n=1}$ in
$\ell_1$ (where $e_0 = 0$ and $\{ e_n\}^\infty_{n=1}$ denotes the unit
vector basis in $\ell_1$) and the summing basis $\{ \suml^\infty_{i=n}
e_n\}^\infty_{n=1}$
 in $c$, which of course is isomorphic to $c_0$.
These facts together with Proposition 1.1 provide a partial proof of
Theorem 1.2.


\section{Uniqueness of symmetric bases}

An immediate consequence of Theorem 1.3 is the fact that $\ell_2, \ell_1$
and $c_0$
 have also a unique symmetric basis, up to equivalence.
It turns out that there are considerably more Banach spaces with a
unique, up to equivalence, symmetric basis.  

\begin{Theorem} The spaces
$c_0$ and $\ell_p ; 1 \le p < \infty$, have, up to equivalence, a unique
symmetric basis.  \end{Theorem} \noindent {\bf Proof}: Fix $p > 1$,
let $\{ e_n \}^\infty_{n=1}$, denote the unit vector basis of $\ell_p,
\{e^*_n\}^\infty_{n=1}$ the biorthogonal
 sequence associated to $\{ e_n\}^\infty_{n=1}$ and let
 $\{x_m\}^\infty_{m=1}$
be another normalized symmetric basis of $\ell_p$.  By a simple
diagonalization argument, one can find a subsequence $\{ x_{m_i}\}^\infty_{
i=1}$ of $\{x_m\}^\infty_{m=1}$ such that $\lim\limits_{i\to\infty} e^*_n
x_{m_i}$ exists for any choice of $n$.  If all these limits are equal to
zero then one can find a further subsequence of $\{x_m\}^\infty_{m=1}$,
which is equivalent to a block basis of $\{e_n\}^\infty_{n=1}$. 
Since any normalized block basis of $\{e_n\}^\infty_{n=1}$ is equivalent
to  $\{e_n\}^\infty_{n=1}$ itself in $\ell_p$ it follows 
that $\{ x_m\}^\infty_{m=1}$ is
equivalent to $\{e_n\}^\infty_{n=1}$.

In case there exists a value of $n$ for which $\lim\limits_{i\to\infty}
e^*_nx_{m_i} = \a \neq 0$ then one can easily find an infinite subsequence
of $\{ x_m\}^\infty_{m=1}$ which is equivalent to the unit vector basis
of $\ell_1$, thus completing the proof.  \hfill $\square $

\bigskip

The class of spaces with a unique symmetric basis is considerably larger
than that of $\ell_p$-spaces and it contains e.g. all the Orlicz sequence
spaces $\ell_M$ for which the limit $\lim\limits_{t\to 0}tM\p(t)/M(t)$
exists.  A concrete such function is  $M(t) = t^p/(1+|\log  t |); p
\ge 1$.  Another class of spaces with a unique symmetric basis is that
of Lorentz sequence
 spaces $d(w, p)$ where $p\ge 1$ and $w=\{w_n\}^\infty_{n=1}$ is a
 non-increasing
sequence of  positive weights satisfying $w_1 =1$,
$\lim\limits_{n\to\infty} w_n=0$ and $\suminf w_n=\infty$.  Recall that
$d(w,p)$  in the space of all sequences $\a = \{\a_n\}^\infty_{n=1}$
of scalars so that $$ \| \a \| = \sup\limits_{\pi} \left(\suminf
|\a_{\pi(n)}^p w_n | \right)^{1/p}
 < \infty,
$$ where the supremum is taken over all permutation $\pi$ of the integers.

The proofs that these two classes of spaces do have a unique symmetric
basis are not very hard but we omit them. They can be 
found e.g. in
[ L - T I  Section 4 ].

An example of a space with ``many" mutually non-equivalent symmetric bases
is the so-called space $U_1$ of Pelczynski [ P ], which is universal for
all unconditional bases in the sense that it has an unconditional basis
$\{u_n\}^\infty_{i=1}$, and, quite remarkably, every unconditional basic
sequence $\{ v_j\}^\infty_{j=1}$ in any Banach space $V$ is equivalent
to a subsequence
of $\{ u_n\}^\infty_{n=1}$. A simple application of the decomposition
method shows that the property defining $U_1$ characterizes it up to 
isomorphism.

A simple way of constructing this space was suggested by Schechtman [ S ].
To this end, let $\{ f_n\}^\infty_{n=1}$ be a sequence of continuous
functions which is dense in the space $C(0, 1)$ of all continuous
functions on $[0, 1]$ and, for any sequence $a=\{a_n\}^\infty_{n=1}$ which
is eventually zero, i. e. $a\in c_{oo}$,  define 
$$ ||| a |||=\sup\{\|\suml^\infty_{n=1} \e_n
a_n f_n \|_{C(0, 1)}; \; \; \e_n = \pm 1, n = 1, 2 \cdots \}.  $$ The
unit vectors $\{u_n\}^\infty_{n=1}$ clearly form an unconditional basis
in the completion $U_1$ of $c_{oo}$ relative to the norm $||| \cdot |||$.
If $\{ v_k \}^\infty_{k=1}$ is an unconditional basic sequence in an
arbitrary Banach space $V$ one can assume without loss of generality
that $V$ is a subspace of $C(0, 1)$.  Hence, one can find a subsequence
$\{f_{n_k}\}^\infty_{k=1}$ of $\{ f_n\}^\infty_{n=1}$ so that $\| v_k
- f_{n_k} \|_{C(0, 1)} \to 0$,
 as $k \to \infty$, ``fast enough"  as to  imply that $\{ v_k
 \}^\infty_{k=1}$ is
equivalent to $\{ f_{n_k}\}^\infty_{k = 1}$.  Therefore, $\{
f_{n_k}\}^\infty_{k=1}$ is unconditional and thus equivalent to $\{
u_{n_k} \}^\infty_{k=1}$, which completes the proof.

In order to show that $U_1$ has a symmetric basis, a fact which is not
a priori
 obvious, one needs an interpolation argument of Davis [ D ]: let
 $\{x_n\}^\infty_{n=1}$
be a normalized 1-unconditional basis in a Banach space $X$ and, for
$m>1$ and $p\ge 1$, define
 the Orlicz function $M_p(t) = t^p/(1+|\log t |)$; $t >0$, and the norm
$$ \| \a \|_{m, p} = \inf \{ ( \|\beta\|^2_{\ell_{M_p}}+ \| \gamma
\|^2_{\ell_p})^{\half}
 ; \a = \beta m^{-1} + \gamma m,
\; \; \hbox{\rm with} \; \; \beta \in \ell_{M_p} \; \; \hbox{\rm and} \;
\; \gamma \in \ell_p \}, $$ for all $\a \in \ell_{M_p}$.  Then, whenever
$\{ m_n\}^\infty_{n=1}$ is a sequence of numbers $>1$ satisfying
 the condition $\suminf m^{-1}_n < \infty$,
the expression $$ \| \a \|^{(p)} = \| \suminf \| \a \|_{m_{n, p}} x_n
\|_X, $$ defines a norm on a space $\widetilde Y_p$ whose unit vectors
form a 1-symmetric basis in the completion $Y_p$ of $\widetilde Y_p$
so that $$ K^{-1}_p \| \a \|_{\ell_{M_p}} \le \| \a \|^{(p)} \le
K_p \| \a \|_{\ell_p}, $$ for all $\a \in \ell_p$.  It turns out
that $\{ m_n\}^\infty_{n=1}$ can be selected as to ensure that $\{
x_n\}^\infty_{n=1}$  is equivalent to a block basis with constant
coefficients
 of the unit vector basis of $Y_p$.
This proves that $X$ is isomorphic to a complemented  subspace of
the space
 $Y_p$, which of course has a symmetric basis.
This fact can be also proved by using the fact proved in [ L ] that
every space with an unconditional basis is isomorphic to a complemented
subspsce of a space with a symmetric basis.  The advantage of the present
proof is that it shows that $X$ can be complementably
 embedded in uncountably many spaces with mutually non-equivalent
 symmetric bases.

By applying this argument to $U_1$, one constructs, for each value of
$p \ge 1$,
 a space $Z_p$ with a symmetric basis $\{ z_n^{(p)} \}^\infty_{n=1}$ so
 that $U_1$ is isomorphic to a complemented subspace of $Z_p$.
Since both $U_1$ and $Z_p$ are isomorphic to their own square one can
apply the decomposition
 method and conclude that $U_1 \approx Z_p$, for any $p \ge 1$.
This further implies that $U_1$ has, quite surprisingly, a symmetric basis
which is equivalent to $\{ z_n^{(p)}\}^\infty_{n=1}$.  As we have noticed
before, for $p\neq q$, $\{ z_n^{(p)} \}^\infty_{n=1}$ is not equivalent
 to $\{z_n^{(q)}\}^\infty_{n=1}$, i.e. $U_1$ has even a continuum of
 mutually non-equivalent
symmetric bases.

With some additional effort, one can construct more ``natural" spaces,
namely Orlicz sequence
 spaces, which also have a continuum of mutually non-equivalent symmetric
 bases (cf. [ L - T O III ] or [ L - T I p. 153 ]).
The fact that all the examples of spaces with a  symmetric basis
considered so far
 have either a unique symmetric basis, up to equivalence, or uncountably
 many
mutually non-equivalent symmetric bases lead inevitably to the question
whether this
 is always the case.
It turns out that the answer is negative: C. Read [ R ] has constructed,
for every $n=1, 2, \cdots $ or $n = \aleph_0$, a space $X_n$ with a
symmetric basis which has precisely $n$ different symmetric bases.

It should be added that we are very far form being able to characterize
the class of spaces with a unique symmetric basis, up to equivalence.
In fact, we do not have even a reasonable conjecture.

\section{Uniqueness of unconditional bases, up to a permutation}


The notion of uniqueness of the unconditional basis of a Banach space
$X$ can be interpreted as asserting  that $X$ can be represented in a
{\bf unique} manner as
 a space of sequences $a = \{a_n\}^\infty_{n=1} \in X$ with the property
 that
$$ \{ a_n\}^\infty_{n=1} \in X \Rightarrow \{\e_na_n\}\nnf \in X, $$ for
any choice of $\e_n = \pm 1;\;  n=1, 2, \cdots $ .  Obviously, in this
representation $X$ is considered as a set of {\bf ordered } sequences.

Now, if instead we consider $X$ as a space of {\bf unordered} sequences
with the property
 described above then uniqueness of the representation has a different
meaning: uniqueness up to permutation and equivalence.  Recall that, as
it was mentioned in the introduction, a Banach space $X$
 with a normalized unconditional basis $\{x_n\}^\infty_{n=1}$
is said to have {\bf
 unique}  unconditional basis, {\bf up to equivalence and permutation},
 if, whenever $\{y_n\}\nnf$ is another normalized unconditional basis
 of $X$ then $\{y_n\}\nnf$ is  equivalent to $\{x_{\pi(n)}\}\nnf$,
for some permutation $\pi$ of the integers.

It turns out that there are considerably more spaces with a unique
unconditional basis,
 up to permutation, than the three space $\ell_2, \ell_1$ and $c_o$,
 which are known to have, up to equivalence, a unique unconditional basis.

One such class of spaces was found by I.S. Edelstein and
 P. Wojtaszczyk [ E - W ], who showed that finite direct sums of $\ell_2,
 \ell_1$ and $c_0$ have the uniqueness property mentioned before.

\begin{Theorem}
 Each of the Banach spaces $\ell_1 \oplus \ell_2, \, \ell_1 \oplus c_0,
 \, \ell_2 \oplus c_0$ and $\ell_1 \oplus \ell_2 \oplus c_0$ has, up to
 permutation and equivalence, a unique unconditional basis.
\end{Theorem}

In order to describe the ideas used in the proof, consider e.g. the
case of the space $\ell_1 \oplus \ell_2$ and assume that $\{ z_n = x_n
+ y_n\}\nnf$, where $x_n \in \ell_1$
 and $y_n \in \ell_2$ for all $n$, is a normalized $K$-unconditional
 basis of this space.
Put $$ N_1 = \{ n; \| x_n \| \le 1/2K\} \; \; \hbox{\rm and } \; \; N_2
= \{ n; \| x_n \| > 1/2k \} $$ and notice that both $[z_n]_{n\in N_1}$
and $[z_n]_{n\in N_2}$ are complemented subspaces
 in $\ell_1 \oplus \ell_2$ whose direct sum is the whole space.
In order to study these two complemented subspaces of $\ell_1 \oplus
\ell_2$, we need a result which ``rotates" any complemented subspace of a
``nice" direct sum into a ``correct"
 position.
To this end, recall that an operator $T$ from a Banach space  $X$ into
a space $Y$
 is {\bf strictly singular} if the restriction of $T$ to any infinite
 dimensional
subspace of $X$ is not an isomorphism.  Compact operators are of course
strictly singular but these two notions
 are very different.
Clearly, every bounded linear operator from $\ell_1$ into $\ell_2$ or,
vice-versa, from $\ell_2$ into $\ell_1$ is strictly singular.

Now, we can state the theorem of Edelstein and Wojtaszczyk [ E - W ]
which provide
 the ``rotation" into a ``correct" position.
\begin{Theorem} Suppose that $X$ and $Y$ are two Banach spaces so that
every operator from $X$ into $Y$ is strictly singular, and let $P$ be
a bounded projection
 form $X \oplus Y$ onto a subspace $Z$.
Then one can find an automorphism $S$ of $X \oplus Y$ and complemented
subspaces $X_0 $
 of $X$ and $Y_0$ of $Y$ such that
$$ SZ = X_0 \oplus Y_0.  $$ \end{Theorem} We  omit the proof of the
theorem which is based on a good understanding of several facts on
Fredholm operators.

We return now to the proof of Theorem 3.1 in the case of the direct
sum $\ell_1 \oplus \ell_2$.  The proof will be  completed once we show
e.g. that $[z_n]_{n\in N_1} \approx \ell_2$
 and $[z_n]_{n\in N_2} \approx \ell_1$ since both $\ell_1$ and $\ell_2$
 have a
unique unconditional basis, up to equivalence.  \ Notice that if $$
x_n = \suml^\infty_{j=1} a_{n,j} z_j \; \; \hbox{\rm and} \; \; y_n=\
\suml^\infty_{j=1} b_{n,  j} z_j $$ then $a_{n, n} + b_{n,n} = 1$, for
all $n$.  However, for $n\in N_1$, $$ |a_{n,n} | \le K \| x_n \| \le
1/2, $$ and thus $|b_{n,n} | > 1/2$.

Now consider the linear operator $U$ from the subspace $[\e_ny_n]_{n
\in N_1}$ of
 $L_\infty (\ell_2)$, where $\e_n(t);\;  n = 1, 2, \cdots \; , $ denote
 the Rademacher
functions, into $[z_n]_{n\in N_1}$ which maps $\e_ny_n$ to $z_n$, for
all $n \in N_1$.  Since both $\{ \e_ny_n\}_{n\in N_1}$
 and $\{ z_n\}_{n\in N_1}$ are unconditional bases, a well known diagonal
 argument shows
that the corresponding diagonal operator is bounded by $K \| U \|$.
This means that 

$$ \| \suml_{n\in N_1} c_n b_{n, n} z_n \| \le K \| U \| 
\sup\limits_{\scriptstyle \e_n=\pm 1 \atop \scriptstyle n=1, 2, \cdots}
\| \suml_{n\in N_1} c_n \e_n y_n \|, $$ for any choice of $\{ c_n \}_{n
\in N_1}$.


By Theorem 3.2, if $[z_n]_{n \in N_1}$ is not isomorphic to $\ell_2$ then
it contains a complemented  subspace isomorphic to $\ell_1$.  Hence, one
can find a normalized block basis $w_j = \suml_{\ell \in \sigma_j}
 d_\ell z_\ell$; $ j = 1, 2, \cdots$, of $\{ z_n\}_{n\in N_1}$,
which is equivalent to the unit vector basis in $\ell_1$.  By passing to a
subsequence if necessary, one can assume that the subspaces $[y_\ell]_{
\ell \in \sigma_j} ; j = 1, 2, \cdots $ are ``almost"  supported on 
mutually orthogonal subspaces
 of $\ell_2$ in the sense that the unit vector basis $\{ e_n \}\nnf $
 of $\ell_2 $ can be split into disjoint subsets $\{ e_n\}_{n \in n_j}
 \colon j = 1, 2, \cdots $ so that, essentially speaking, $[y_\ell]_{\ell
 \in \sigma_j} \subset [e_n]_{n\in \eta_{j}}$, for all $j$.
Therefore, for any sequence $\{c_j\}^\infty_{j=1}$ with\break
$\suml^\infty_{j=1} |c_j |^2 <\infty $ and any choice of $\e_n = \pm 1$,
for all $n$, 

$$ \begin{array}{ll} &\|\suml^\infty_{j=1} c_j \suml_{\ell
\in \sigma_j} d_\ell \e_\ell y_\ell \| \sim \left(\suml^\infty_{j=1}
|c_j|^2 \| \suml_{\ell \in \sigma_j} d_\ell \e_\ell y_\ell
\|^2\right)^{1/2} \le K\left(\suml^\infty_{j=1}
 |c_j |^2\right)^{1/2}.\\
\end{array} $$ Hence, the series $\suml^\infty_{j=1} c_j \suml_{\ell\in
\sigma_j} d_\ell \e_\ell
 y_\ell$ converges unconditionally and  so does  the series
$\suml^\infty_{j=1} c_j \suml_{\ell\in \sigma_j} b_{ \ell, \ell} d_\ell
z_\ell$.  This
 implies the convergence of the series
$\suml^\infty_{j=1} c_j w_j$, whenever $\suml^\infty_{j=1} |c_j|^2 <
\infty$, which of course contradicts the fact that $\{ w_j\}^\infty_{j=1}$
 is equivalent to the unit vector basis  of $\ell_1$.
Consequently $\{ z_n\}_{n\in N_1}$ is equivalent to the unit vector
basis of $\ell_2$.

In a similar manner but using also a duality argument, one proves that
$[z_n]_{n\in N_2} \approx
 \ell_1$, and thus that $\{ z_n\}_{n\in N_2}$ is  equivalent to the
 unit vector
basis of $\ell_1$.


While the finite direct sums constructed out of the three spaces $\ell_1,
\ell_2$
 and $c_0$ have, up to equivalence and permutation, a unique unconditional
 basis, this is not always the case for infinite direct sums constructed
 out of the same set of
spaces.  On the positive side we have e.g. the following result from [ B
- C - L - T ].  
\begin{Theorem} Every normalized unconditional basis of
an infinite dimensional complemented subspace of the direct sum $(\ell_2
\oplus \ell_2 \oplus \cdots \oplus \ell_2 \oplus \cdots)_0$,
 is equivalent to a permutation of the unit vector basis of one of
 the following
  six  spaces:
$$ \ell_2, c_0, \ell_2 \oplus c_0, \left(\suml^\infty_{n=1} \oplus
\ell^n_2\right)_0, \ell_2 \oplus \left(\suml^\infty_{n=1} \oplus
\ell^n_2\right)_0, (\ell_2 \oplus \ell_2 \oplus \cdots \oplus \ell_2
\oplus \cdots)_0.  $$ A similar statement holds for complemented subspaces
of the dual space \newline $(\ell_2 \oplus \ell_2 \oplus \cdots \oplus \ell_2
\oplus \cdots)_1$.

Consequently, the six spaces appearing above and their duals have, up to
permutation and equivalence, a unique unconditional basis.  \end{Theorem}

\bigskip

The proof of Theorem 3.3 is not extremely difficult but still beyond the
scope of this article and therefore we omit it here.  Another result
from [ B - C - L - T ] is: 

\begin{Theorem}  Every normalized unconditional basis of an infinite
dimensional complemented subspace of the direct sum $(\ell_1 \oplus
\ell_1 \oplus \cdots \oplus \ell_1 \oplus \cdots)_0$ is equivalent to a
permutation of the unit vector basis of one of the following six spaces:
$$ c_0, \ell_1, c_0 \oplus \ell_1, \left(\suminf \oplus \ell^n_1\right)_0,
\ell_1 \oplus \left(\suminf \oplus \ell^n_1\right)_0, (\ell_1 \oplus
\ell_1 \oplus \cdots \oplus \ell_1 \oplus)_0 $$

Consequently, each of these six spaces has, up to permutation and
equivalence, a unique unconditional basis.  \end{Theorem}

Though Theorems 3.3 and 3.4 have a similar formulation, the proof of
3.4 in [ B - C - L - T ] is considerably harder than that of 3.3. This
is reflected also by the fact that 
the function $M=M(K)$ (so that if $\{x_n\}^\infty_{n=1} $ is a normalized
$K$-unconditional basis of one
 of the spaces $X$ appearing above, then $\{x_n\}^\infty_{n-1} $ is
 $M(K)$-equivalent to a permutation of the unit vector basis of $X$)
 behaves differently in the two
cases discussed above.  While the proof of 3.3 gives $M(K)$ as a power
of $K$, the proof of 3.4 yields $M=M(K)$ as an exponential function of
$K$ and examples show that exponential function is really needed.

Very recently, Casazza  and Kalton [ C - K 2 ] provided a simpler proof
of the uniqueness,
 up to a permutation, of the unit vector basis in the space
$(\ell_1 \oplus \ell_1\oplus \cdots \oplus \ell_1 \oplus \cdots)_0$ and
its dual.

Quite surprisingly, the other infinite direct sums which can 
be constructed out the the
three spaces $\ell_1, \ell_2$ and $c_o$ do not have the the uniqueness
property exhibited in Theorems 3.3 and 3.4.  
\begin{Theorem}  {\rm
( cf. [ B - C - L - T ] )} The direct sums $$ \left (\suminf \oplus
\ell^n_\infty \right )_2, \; \,  c_0  \oplus \left (\suminf \oplus
\ell^n_\infty \right )_2, \; \; (c_0 \oplus c_0 \oplus \cdots \oplus c_0
\oplus \cdots )_2 $$ and their duals fail to have a unique unconditional
basis, up to equivalence
 and permutation.
\end{Theorem}

\noindent {\bf Proof}:  We shall treat here only the case of the dual
direct sum $X=(\suminf \oplus \ell^n_1)_2$;
 the other cases can be easily derived from it without too much
 difficulty.

In order to exhibit a normalized unconditional basis of $X$ which is
not permutatively equivalent to the unit vector basis of $X$, we first
 fix $n$ and let ${\cal F}_i = \{ A_{i, j} \}^n_{j=1}$; $ 1 \le i \le
 n$, the independent partitions of the interval $[0, 1]$ into sets of
 measure equal to $1/n$ (i.e., for $A \in {\cal F}_i$, $B \in {\cal
 F}_j$ and $i \neq j, \mu (A\cap B) = \mu (A) \mu (B) = 1/n^2)$.
With $\{ e_i\}^n_{i=1}$ denoting the unit vector basis of the space
$\ell^n_1$, consider the vector valued functions 
$$ f_{i,j} (t) = n^{1/2}
\chi_{A_{i, j}}(t) e_i; \; \; 1 \le i, j \le n, $$ which form a normalized
1-unconditional basis in a subspace $Y_n$ of the space $L_2
 ([0, 1], \ell^n_1)$ since
$$ \| \suml^n_{i, j=1} a_{i, j} f_{i, j} \| = n^{1/2} \left(\intl^1_0
\left (\suml^n_{i = 1}|\sum^n_{j = 1 }a_{i,j} \chi_{A_{i, j}} (t)| \right)^2 dt\right)^{1/2},$$ 
for any choice of $\{ a_{i, j}\}^n_{i,j=1}$. Indeed, for each 
$1 \le i \le n$,
the functions 
$$ | \sum^n_{j = 1 }a_{i,j} \chi_{A_{i, j}} (t) |$$ and
$$| \sum^n_{j = 1 } |a_{i,j} |\chi_{A_{i, j}} (t) |$$ are both  ${\cal F}_i$
measurable and have the same distribution. Hence, the independence of   
${\cal F}_i, 1 \le i \le n$ ensures that that the functions 
$ | \sum^n_{j = 1 }a_{i,j} \chi_{A_{i, j}} (t) |$ and
$ \sum^n_{j = 1 } |a_{i,j} |\chi_{A_{i, j}} (t) |$ have the same norm in
the space  $L_2 ([0, 1], \ell^n_1)$. Now notice that if ${\Bbb E}_i$ 
denotes the conditional expectation operator associated to the
partition ${\cal F}_i$, i.e. the operator defined by $${\Bbb E}_i \varphi =
\suml^n_{j=1} \left (\intl_{A_{i,j}} \varphi(t)dt \right )\chi_{A_{i,j}},
$$ for $\varphi \in L_1 (0,1)$, then one can define a projection $P_n$
 from $L_2 ([0,1], \ell^n_1) $ onto $Y_n$ by setting
$$ P_n =\suml^n_{i=1} ({\Bbb E}_i\varphi_i) e_i, $$ whenever $f(t) =
\suml^n_{i=1} \varphi_i (t) e_i $ is an element of $ L_2 ([0,
1],\ell^n_1)$.  Then, by using the independence of the partitions $\{
{\cal F}_i\}^n_{i=1}$,
 one gets that
$$ \begin{array}{ll} &\| P_n f \|^2 = \intl^1_0 (\suml^n_{i=1} | {\Bbb E}_i
\varphi_i|)^2  dt = \suml^n_{i=1} \intl^1_0 | {\Bbb E}_i \varphi_i |^2 dt +\\
&\suml^n_{\scriptstyle i,j = 1\atop i \scriptstyle \neq j} \intl^1_0 |E_i
\varphi_i | |E_j \vp_j|dt = \suml^n_{i=1} \intl^1_0 |E_i \vp_i |^2 dt
+\\ &\suml^n_{\scriptstyle i,j=1\atop \scriptstyle i\neq j} (\intl^1_0
|E_i\vp_i| dt) (\intl^1_0 |E_j \vp_j | dt),\\ \end{array} $$ 
for any $f$
as above.  Hence, $$ \| P_n f\|^2 \le 2 \suml^n_{i=1} \intl^1_0  | \vp_i
|^2 dt = 2 \| f \|^2, $$ i.e. $\| P_n \| \le \sqrt{2}$, which shows that
$Y_n$ is $\sqrt{2}$-complemented  in $L_2 ([0, 1], \ell^n_1)$, or even
in the subspace
 $(\ell^n_1 \oplus \ell^n_1 \oplus \cdots \oplus \ell^n_1)_2$  ($n^n$
 factors) of $L_2(
[0, 1], \ell^n_1)$.  It follows that the space $Y = (\suml^\infty_{n=1}
\oplus Y_n)_2$ is isomorphic to a complemented subspace of $X$.

On the other hand, the block basis $y_i = n^{-1/2} \suml^n_{j=1} f_{i,j}$;
\ $1 \le i \le n$, of $\{ f_{i,j} \}^n_{i, j =1}$ is $1$-equivalent to
the unit vector basis of $\ell^n_1$, and thus its span is complemented in
$Y_n$, since 
$\ell_1$ is block injective, i.e. 1-complemented, whenever
it is embedded as a block
 basis of a 1-unconditional basis.
Consequently, $Y$ contains a 1-complemented copy of $X$ and thus, by
the decomposition
 method, $X\approx Y$.

The proof will  be completed once we show that the natural basis of $Y$
is not equivalent
 to a permutation of the unit vector basis in $X$.
To this end, notice that the  unit vector basis of $X$ has the property
that any of its subsets of finite codimension
 contains, for each $n$,  a subsequence 1-equivalent to the unit vector
 basis of $\ell^n_1$.
The natural basis of $Y$ has,  however, a diametrically opposite behavior.
Indeed, for each $\e>0$ and each integer $k$, there exists
 an integer $n=n(\e,k)$ such that any subset of $\{ f_{i,j} \}^n_{i,j=1}$
of cardinality
 $k$ is $1+\e$-equivalent to the unit vector basis of $\ell^k_2$ and,
 moreover, this property remains valid in $Y$ provided that, for any such
 $k$, we eliminate a suitable finite set of vector from the basis of $Y$.

\bigskip

Another quite well-known space for which the question of uniqueness,
 up to a\break permutation, could be settled is the so-called Tsirelson
 space, introduced in [ T ] (see also [ F - J ]).
This space is the completion of the space of sequences  $x \in c_{0, 0}$
under the minimal norm $\| \cdot \|_T$ satisfying the conditions: $$
(i) \quad  \| x\|_T \ge \| x \|_0, $$ for all $x \in c_{00}$, and $$
(ii) \quad  \| \suml^\ell_{i=1} x_i \|_T = {1\over 2} \suml^\ell_{i
= 1} \| x_i \|_T,$$ whenever $\ell \le $ support $x_1 < $ support $x_2
< \cdots <$ support $x_\ell$; $ \ell = 1, 2, \cdots, $ .  The fact that
such a norm exists is proved in [ F - J ].

\begin{Theorem} Every complemented subspace of $T$ with an unconditional
basis has, up to permutation
 and equivalence, a unique unconditional basis.
\end{Theorem}


Actually, this result was preceded by a theorem proved in [ B - C -
L - T ]  which asserts
 that the 2-convexification $T^{(2)} $ of $T$, as well as its complemented
 subspaces with an unconditional basis,  have a unique unconditional
 basis, up to equivalence and permutation (recall that $T^{(2)}$ is
 obtained from $T$ by defining the norm in it
as follows: $\| x \|_{T^{(2)}} = \|\; \;  |x|^2 \; \|^{1/2}_T$, for
vectors $x$ having the property that the square of their absolute value
belongs to $ T$).  The proof is quite difficult.  The proof for $T$ ( i.e.
of Theorem 3.6 ), due to 
Casazza and Kalton [ C - K 1 ], is the byproduct of a more general
study of the uniqueness property in spaces which do not contain uniformly
complemented copies of $\ell^n_2$, for all $n$.


For $1<p\not=2$, the $p$-convexification $T^{(p)}$ of $T$ does not
have a unique unconditional basis, up to permutation and equivalence,
since, as was pointed out by Kalton, $T^{(p)}$ can be represented a
Tsirelson sum of $\ell_p^n$'s and, in this sum, the factor $\ell_p^n$
has an unconditional basis containing among its vectors an $\ell_2^k$
with $k\sim\log n$.

\bigskip

The Banach spaces with a unique unconditional basis, up to permutation
and equivalence, which were considered so far in this section have
the additional property that also their complemented subspaces with an
unconditional basis share the uniqueness feature.  For such spaces one
can introduce the notion of genus (cf. [ B - C - L - T] ).

A Banach space with an unconditional basis is said to be of {\bf genus}
$n$ if in all its complemented subspaces with an unconditional basis,
the normalized unconditional basis is unique, up to equivalence and
permutation, and there are exactly $n$ different isomorphic types of
complemented subspaces with an unconditional basis.

The spaces $\ell_1,\ell_2$ and $c_0$ are clearly of genus $1$.  It turns
out  ( cf. [ B - C - L - T ]  ) that these three spaces are the only
ones of genus $1$.

\begin{Theorem} $\ell_1,\ell_2$ and $c_0$ are the only spaces with a
(unique) unconditional basis of genus $1$.  \end{Theorem}

The idea of the proof from [ B - C - L - T ] is the following: if $X$
is a space with a normalized unconditional basis $\{x_n\}_{n=1}^\infty$
of genus $1$ then every infinite subsequence of $\{x_n\}_{n=1}^\infty$
is equivalent to a permutation of $\{x_n\}_{n=1}^\infty$.  Then, by
using a sort of compactification argument involving spreading models
and ultraproducts, one can show that $\{x_n\}_{n=1}^\infty$ is, up
to permutation, a subsymmetric basis.  Therefore, one can assume
w.l.o.g.. that $\{x_n\}_{n=1}^\infty$ itself is subsymmetric.

Since $\{x_n\}_{n=1}^\infty$ is subsymmetric any block basis with constant
coefficients of $\{x_n\}_{n=1}^\infty$ spans a complemented subspace $U$
of $X$.  Moreover, by a slightly more complicated argument than that used
in the case when $\{x_n\}_{n=1}^\infty$ is symmetric, it can be easily
shown that $X\approx X\oplus U$.  Hence, any normalized block basis
with constant coefficients of $\{x_n\}_{n=1}^\infty$ is equivalent to
$\{x_{\pi(n)}\}_{n=1}^\infty$, for some permutation $\pi$ of the integers.
Then, by  a simple modification of Zippin's characterization of perfectly
homogeneous bases from [ Z ], one concludes that $\{x_n\}_{n=1}^\infty$
is equivalent to the unit vector basis in $c_0$ or $\ell_p$, for some
$p\ge1$.  The cases when $1<p\not=2$ can be easily dismissed, as shown
before.

\bigskip

Theorems 3.3 and 3.4 above show that the sums
$\left(\suml_{n=1}^\infty\oplus \ell_2^n\right)_0$,
$\left(\suml_{n=1}^\infty\oplus \ell_1^n\right)_0$ and their duals are
spaces of genus $2$.  However, it is not known if there exist other
spaces of genus $2$.

The spaces $\ell_1\oplus\ell_2$, $\ell_1\oplus c_0$ and $\ell_2\oplus c_0$
are of genus $3$ but, again, we do not have a complete characterization
of this class.

It is quite possible that the class of spaces of finite genus coincides
with that of spaces which are obtained from Hilbert space by taking
repeated finite or infinite direct sums in the sense of $c_0$ or $\ell_1$.  
Casazza and Lammers [ C -
L ] obtained many results on 
spaces of finite genus, e.g. that the unconditional basis contains a subsequence
equivalent to the unit vector basis in either $\ell_1$, or $\ell_2$
or $c_0$.  There is a feeling that this class will be well understood
once spaces of genus $2$ are characterized.

The Tsirelson space $T$ and its $2$-convexification are of infinite genus.

\bigskip

The question whether the uniqueness of the unconditional basis, up
to permutation and equivalence, is a hereditary property has
a negative answer.  More precisely, Casazza and Kalton [ C - K 1 ]
constructed the first example of a Banach space with a unique
unconditional basis, up to equivalence and permutation, which has
complemented subspaces with an unconditional basis lacking the uniqueness
property.  Their starting point in the Orlicz sequence space $\ell_M$,
where $M(t)=t/(1+|\log t|)$, for $t\ge0$.  In this, as in any other Orlicz
space, a normalized block basis with constant coefficients of the from
$u_j=\left(\suml_{n=q_{j-1}+1}^{q_{j+1}}e_n\right)/\|\suml_{n=q_{j-1}+1}^{q_{j+1}}e_n\|$;
$j=1,2,\ldots$ generates a so called modular space $\ell_M[s_j]$,
where $s_j=1/\|\suml_{n=q_{j-1}+1}^{q_{j+1}}e_n\|$, for all $j$.  This
subspace, which is clearly complemented in $\ell_M$, can be described
as the space of all sequences $a=(a_1,a_2,\ldots,a_j,\ldots)$ so that
$\suml_{j=1}^\infty M_{s_j}(|a_j|)<\infty$, where $M_s(t)=M(st)/M(s)$.

It can be easily verified that if $s_j\to0$, very fast as $j\to\infty$,
then $\ell_M[s_j]$ is isomorphic to $\ell_1$, and, on the other hand, if
$\inf\limits_js_j>0$ then $\ell_M[s_j]\approx \ell_M$.  By manipulating
between these diametrically opposite situations, one can select a
normalized block basis $\{u_j\}_{j=1}^\infty$ of the unit vector basis in
$\ell_M$ which is permutatively equivalent to its square but $\ell_M[s_j]$
is not isomorphic to either $\ell_1$ or $\ell_M$ itself.  Casazza and
Kalton have shown in [ C - K 1 ] that in this case $\{u_j\}_{j=1}^\infty$,
is, up to permutation and equivalence, the unique unconditional basis
of $\ell_M[s_j]$ and, moreover, $\ell_M[s_j]$ contains complemented
subspaces with a non-unique unconditional basis.

The fact that direct sums in the sense of $c_o$ of spaces such as $\ell_2$
 or $\ell_1$, which do have a unique unconditional basis, have also the
uniqueness property, led to the question (stated explicitly in [ B - C -
L - T ]) whether, whenever a space $X$ has unique unconditional basis, up
to permutation, then $(X \oplus X \oplus \cdots \oplus X \oplus\cdots)_0$
also has a unique unconditional basis.  It turns out that the answer to
this question is negative: Casazza and Kalton [ C - K 1 ] have proved
that direct sums of $T$ or $T^{(2)}$ in the $c_0$-sense do not have a 
unique unconditional
basis, up to permutation, in spite of the fact
 mentioned above that  both $T$ and $T^{(2)}$ have this property.
%\input banach2.tex %\end{document}


\section{Uniqueness in finite dimensional spaces}

Since any two bases of a finite dimensional space are always equivalent
the question of uniqueness of the basis makes no sense in the framework
 of a single finite dimensional Banach space but rather in that of
 families of such spaces.  As in the case of infinite dimensional spaces,
 the most interesting results are obtained for families of spaces with
 an unconditional or symmetric basis.
\begin{Definition} Let ${\cal F}$ be a family of finite dimensional spaces
 each of which has a normalized 1-unconditional basis.
We say that the members of ${\cal F}$  have a unique unconditional basis,
up to equivalence
 (and permutation), if there exists a function $\psi\colon
 [1,\infty)\to[1, \infty)$
such that, whenever a space $E \in {\cal F}$ has another normalized
unconditional basis
 $\{ e_j\}^n_{n=1}$ whose unconditional constant is $\le K$, then $\{
 e_j\}^n_{j=1}$ is $\psi
(K)$-equivalent to (a permutation of) the given 1-unconditional basis
of $E$.

By replacing in the above definition the word ``unconditional" with
``symmetric", one defines the notion of uniqueness of the symmetric
basis for the elements
 of ${\cal F}$.
\end{Definition}

Typical families studied in this section are $$ {\cal F}_p=\{ \ell^n_p;
n = 1, 2, \ldots \} ; \; \; 1\le p \le \infty.  $$ The same argument,
involving the parallelogram identity, which was used in the case
of $\ell_2$, shows that the members of ${\cal F}_2$ have a unique
unconditional basis, up to equivalence.  Furthermore, by using the
same argument involving 1-summing and 2-summing operators, as in the
case of $\ell_1$ and $c_0$, one can show that also ${\cal F}_1$ and
${\cal F_\infty}$ have the same uniqueness property as ${\cal F}_2$.
\begin{Proposition} The members of the families ${\cal F}_1, {\cal
F}_2$ and
 ${\cal F}_\infty$ have a unique unconditional basis, up to equivalence.
\end{Proposition}

The analogy with the infinite dimensional case might lead one to believe
that these three
 families above are the only ones having, up to equivalence, a unique
 unconditional basis.

It turns out that this fact is false.  In order to produce examples of
other families whose member have a unique unconditional
 basis, up to equivalence, one needs a result of Dor and Starbird [ D
 - S ] asserting that,
for $p>2$, a normalized unconditional basis of $\ell_p$ is either
equivalent
 to the unit vector basis of $\ell_p$ or it admits, for each $n$, a block
basis which is 2-equivalent to the unit vector basis of $\ell^n_2$.
By using this fact together with the known assertion that Hilbert space
is uniformly complemented in $L_p$-spaces, for $p>2 $ (cf. [ P - R ] or
[ M ]), it is deduced in [ J - M - S - T p. 48 ]
 that:
\begin{Proposition} There exist constants $C <\infty$ and
 $M < \infty
$ such that every normalized $K$-unconditional basis of $\ell_p$;
 $p > 1$, with
$$ K \le C \max \{ \sqrt{p}, \; \sqrt{p/(p-1)} \}, $$ for some constant
$C<\infty$, is $M$-equivalent to the unit vector basis of $\ell_p$.
\end{Proposition} We omit the details of the proof but just mention
that the bound for $K$, appearing in the right hand side of the above
inequality, is actually equal to $\gamma_p (\ell_2)/ \sqrt{2}$, where
$\gamma_p (\ell_2)$ denotes as usual the factorization constant of
$\ell_2$ through $L_p$-spaces.  \vskip.2cm

An immediate consequence of Proposition 4.3 is the fact that, for each
sequence $p_n\to\infty$, the members of the family $$ {\cal F} = \{
\ell^n_{p_n}; \; \; n = 1, 2, \ldots \} $$ have a unique unconditional
basis, up to equivalence.

We pass now to some questions concerning the uniqueness of the symmetric
basis.  We begin with subspaces of $L_p$.  Among the finite dimensional
subspaces of $L_p$ with a symmetric basis one can find the families
${\cal F}_p$ and ${\cal F}_2$.  Other interesting subspaces of $L_p$
are the ``diagonals" of $\ell^n_p \oplus
 \ell^n_2$ generated by vectors of the form $\{ e_j^{(p)} + w_je_j^{(2)}
 \}^n_{
j=1}$, where  $\{e_j^{(p)} \}^n_{j=1}$ and $\{ e_j^{(2)}\}^n_{j=1}$
denote the unit vectors
 in $\ell^n_p$, respectively $\ell^n_2$, and $\{ w_j\}^n_{j=1}$
is an arbitrary sequence of scalars.  Spaces of this type were studies
by H.P. Rosenthal [ Ro ] who coined  for them
 the name $X_p$-space.
If $w_j =w; \; j = 1, 2, \ldots n$ then obviously we deal with  symmetric
$X_p$-space.

It turns out that every symmetric basic sequence in $L_p$; $ p > 2$, is
equivalent to
 a symmetric $X_p$-space.
This characterization has been proved in the Memoir [ J - M - S -
T p.34 ].  
\begin{Theorem} For every $p>2$ and $K\ge 1$, one can find a
constant $D=D(p, K) <\infty$ so that any normalized basic sequence $\{
x_j\}^n_{j=1}$ in $L_p (0,1)$, whose symmetry constant is $\le K$, is
$D$-equivalent to the symmetric $X_p$-space generated  by $\{e_j^{(p)}
+ we_j^{(2)} \}^n_{j=1}$,
 where
$$ w=\|\sum^n_{j=1} x_j\|/\sqrt{n}.  $$ \end{Theorem}

\noindent{\bf Proof}:  In the first step of the proof, the symmetric basic
sequence in $L_p$  is replaced by a sequence of so-called ``symmetrically
exchangeable" random
 variables, i.e. a sequence of functions in $L_p$ whose joint distribution
 in
$\bbr^n$ remains invariant under permutation and change of sign.
In order to describe this construction, let $\{ x_j\}^n_{j=1}$ be a
normalized $K$-symmetric
 basic sequence in  $L_p (0, 1)$; $ p >2$, and $H_n$ the family of all
 distinct $n!2^n$-pairs $\{ \pi, (\e_j)^n_{j=1}\}$, where $\pi $ is a
 permutation  of $\{ 1, 2, \ldots, n\}$ and $\e_j = \pm 1$, for all $1
 \le j \le n$.

The elements of $H_n$ can be put in a one-to-one correspondence with
unit subintervals
 of $[0, n! 2^n]$ of the form $[k, k+1]$, with $k$ being an integer.
If a  pair $\{ \pi, (\e_j)^n_{j=1}\}$ is in correspondence with the
subinterval $I$
 of $[0, n! 2^n]$ then we define
$$ f_j (t) = \e_j x_{\pi(j)} (t); \; t \in I; \; 1 \le j \le n.  $$ In
order to make the functions $\{ f_j\}^n_{j=1}$, which are
 defined on the interval $[0, n! 2^n]$,  into an exchangeable sequence
 we compress the interval $[0, n! 2^n]$ into $[0, 1]$ in the obvious
 linear way thus obtaining a new sequence
$\{g_j\}^n_{j=1}$ which is symmetrically exchangeable and $K$-equivalent
to the original sequence $\{x_j\}^n_{j=1}$.

It is easily seen that there is no loss of generality in assuming that
each point of
 $[0, 1]$ belongs to the support of at least one of the functions $\{
 x_j\}^n_{j=1}$.
Then, by a corresponding change of density, one can make the expression
$(\Sigma^n_{j=1} |x_j|^2)^{1/2}$ into a constant $C$.  This however
implies that also the square function $(\Sigma^n_{j=1}|g_j|^2)^{1/2}$
 is a constant equal to $C=\| (\Sigma^n_{j=1} |x_j|^2)^{1/2}\|$.

Since $\{ g_j\}^n_{j=1} $ is a 1-unconditional basic sequence in
$L_p(0,1)$ and $L_p(0, 1)$ is of cotype $p$, for $p>2$, it follows that
$$ \| \suml^n_{j=1} a_j g_j\| \ge \left (\suml^n_{j=1} |a_j |^p \right
)^{1/p}, $$ for any choice  if $\{a_j\}^n_{j=1}$.  On the other hand,
since $\{g_j\}^n_{j=1} $ is an orthogonal sequence in $L_2(0, 1)$
with $\| g_j\|_2 = C/\sqrt{n}$, for all $1 \le j \le n$, we get that
$$\| \suml^n_{j=1} a_j g_j\|\ge \|\suml^n_{j=1} a_j g_j\|_2 = \left
(\suml^n_{j=1} | a_j|^2 \| g_j \|^2_2 \right )^{1/2} = {C\over {\sqrt
n}} \left (\suml^n_{j=1}
 | a_j|^2 \right )^{1\over 2},
$$ for all $1\le j \le n$.  This proves one part of 4.3 since $$ \|
\suml^n_{j=1} a_j x_j\| \ge K^{-1} \| \suml^n_{j=1} a_jg_j\| \ge K^{-1}
\max \left \{ \left (\suml^n_{j=1} |a_j|^p \right )^{1/p}, \; \; {C\over
\sqrt{n}} \left (\suml^n_{j=1}
 |a_j|^2 \right )^{1/2} \right \},
$$ for all $\{a_j\}^n_{j=1}$, and, by  Khinchine's inequality, $$ \|
\suml^n_{j=1} x_j \| \le K \left (\int \| \suml^n_{j=1} \e_j x_j\|^p d
\e \right )^{1/p} =K\| \left (\int|\sum \e_j x_j|^pd\e \right )^{1/p}
\| \le KB_p C.  $$

The opposite inequality is more difficult and requires some ideas of
H.P. Rosenthal
 [ Ro ] and a certain averaging procedure.
We omit this argument which is described in detail in [ J - M - S - T ].

The significance of Theorem 4.4 becomes clear only after we fully
understand the meaning
 of the expression $w = \| \Sigma^n_{j=1} x_j \| /\sqrt{n}$.
\begin{Proposition} For every $K \ge 1$, there exists a constant $M=M(K)$
 such that, whenever $\{x_j\}^n_{j=1}$ is a normalized $K$-symmetric
 basis in a finite dimensional subspace $X$ of $L_p (0, 1)$, then
 the Banach-Mazur distance $d(X, \ell^n_2)$  is $M$-equivalent to the
 expression $\| \Sigma^n_{j=1} x_j\| /\sqrt{n}$ if $1 \le p \le 2 $ and
 to $\sqrt{n} /\| \Sigma^n_{j=1} x_j\|$ if $p>2$.
\end{Proposition}

\noindent{\bf Proof}:  Suppose that $p>2$.  It follows from 4.4 that $$
\| \suml^n_{j=1} a_jx_j\| \ge D( K, p)^{-1} {\| \suml^n_{j=1} x_j\|\over
{\sqrt{n}}} \left (\suml^n_{j=1} |a_j|^2 \right )^{1/2}, $$ for any choice
of $\{ a_j\}^n_{j=1}$.  Furthermore, by using Khinchine's inequality in
$L_p(0,1)$ and the 2-convexity
 of $L_p (0, 1)$, for $p>2$, we also get that
$$ \begin{array}{ll} \| \suml^n_{j=1} a_j x_j\| &\le K(\int\|
\suml^n_{j=1} a_j \e_j x_j\|^p d\e)^{1/p} = K\| (\int|\suml^n_{j=1} a_j
\e_j x_j|^p d\e )^{1/p} \| \le\\ &\le K  B_p \| (\suml^n_{j=1} |a_j|^2|x_j
|^2)^{1/2} \| \le KB_p (\suml^n_{j=1}
 \a_j|^2)^{1/2},\\
\end{array} $$ again for any choice of $\{ a_j\}^n_{j=1}$.  Combining
these two inequalities, we conclude that $$ d(X, \ell^n_2) \le KB_p D(K)
\; {\sqrt{n}\over \| \suml^n_{j=1} x_j\|}.  $$

Next notice that, for every sequence $\{ a_j\}^n_{j=1}$ of scalars,
we also get $$ {1\over 2K} \max\limits_{1\le j \le n} |a_j| \le \|
\suml^n_{j=1} a_j x_j\| \le
 K \| \suml^n_{j=1} x_j \| \max\limits_{1\le j \le n} |a_j|
$$ i.e.  $$ d(X, \ell^n_\infty) \le 2 K^2 \| \suml^n_{j=1} x_j\|.
$$ Hence, $$ \sqrt{n} \le d (X, \ell^n_2)d(X, \ell^n_\infty) \le 2K^2
\| \suml^n_{j=1} x_j \| d(X, \ell^n_2) $$ i.e.  $$ d(X, \ell^n_2) \ge
(2K^2)^{-1} \;{\sqrt{n}\over \| \suml^n_{j=1} x_j \|}, $$ which completes
the proof for $p>2$.  The case $1\le p \le 2 $ is treated in a similar
manner.  \hfill $\square $

Proposition 4.5 shows that, for a given finite dimensional subspace
$X$ of $L_p (0, 1); 1 \le p \le \infty$, with a symmetric basis, the
expression $\| \suml^n_{j=1} x_j\|$ is, up to a constant depending only
on the symmetry constant of $\{ x_j\}^n_{j=1}$, an invariant
 of the space $X$ rather than that of the particular symmetric basis
$\{ x_j\}^n_{j=1}$. 
This fact together with Theorem 4.3 imply the following consequence from
[ J - M - S - T  p.39 ].  
\begin{Corollary} For $p>2$, let $G_p$ denote the
family of all finite dimensional subspaces of $L_p(0, 1)$ which have a
1-symmetric basis.  Then the members of $G_p$ have for fixed $p$
a unique symmetric basis, up to equivalence.  
\end{Corollary} 
A trivial duality argument,
together with 4.2, shows that, for $1 \le p \le \infty$,
 each member of the family ${\cal F}_p$ has, up to equivalence, a unique
 symmetric basis.
This means that, for every $1 \le p \le \infty$, there is a function
$\psi_p(K)$\
 which corresponds to the family ${\cal F}_p$ by Definition 4.1.
It turns out that these functions can be selected as not to depend on
$p$ and the
following result from [ J - M - S - T p.39 ] is true.
\begin{Theorem} The members of the family 
$\cup_{1 \le p \le \infty} {\cal F}_p$ 
have, up to equivalence, a unique symmetric basis.
\end{Theorem} 
We omit the proof which is given in detail  in the Memoir
[ J - M - S - T ].

We already see that the class of families whose members have, up to
equivalence,
 a unique symmetric basis is quite large.
In an attempt to discover the largest family of space with a unique
 symmetric basis, Sch\"utt studied in [ Schu ] the family ${\cal D}_\a$
 of all finite dimensional Banach spaces $X$ with a normalized 1-symmetric
 basis $\{x_j\}^n_{j=1}$ which satisfy the condition
$$ d(X, \ell^n_2) \ge n^\a, $$ and proved the following beautiful result
which we quote here without giving
its proof.
\begin{Theorem}  For any $\a >0$, each member of the family ${\cal D}_\a$
 has, up to equivalence, a unique symmetric basis.
\end{Theorem}

The abundance of families of finite dimensional spaces with a unique
symmetric basis leads naturally to the question, which was raised in the
Memoir [ J - M - S - T ], whether this is not always the case.  Theorem
4.8, mentioned above, shows that the construction of a counterexample is
not  an easy matter.  Eventually, Gowers produced in [ Go ]  ingenious
examples of  finite dimensional
 normed spaces with two (asymptotically) non-equivalent symmetric bases.
The precise  statement is as follows.  \begin{Theorem}  For each integer
$k$, there exists a Banach space
 of dimension $n=2^k$ with two 2-symmetric normalized symmetric bases $\{
 e_i\}^n_{i=i}$ and $\{ e\p_2\}^n_{i=1}$ whose constant of equivalence
 is at least $\exp (\log \log n/8 \log \log\log n)$.
\end{Theorem}

We shall describe here only the main ideas of Gowers' construction; the
details can be found in the paper [ G ].

Fix $n$ and let $A\colon \bbr^n \to \bbr^n$ be a linear map defined by
an orthogonal matrix which will be specified later.
Let $\{ e_i\}^n_{i=i} $ denote the unit vector basis of $\bbr^n$ and put
$e\p_i = Ae_i$, for all $1 \le i \le n$.  These will be the two bases
under consideration.

The next step is to define a norm $\| \cdot \|$ on $\bbr^n$ so that both
$\{ e_i\}^n_{i=i}$ and $\{e\p_i\}^n_{i=1}$ become 2-symmetric bases.
To this end, let $\Omega$ be the group of symmetries associated to the
first basis
 $\{ e_i\}^n_{i=1}$, i.e. of the linear maps $w$ of the form
$$w\left(\suml^n_{i=1} a_i e_i\right) \; = \; \suml^n_{i=1} \e_i a_i
e_{\pi(i)}, $$ where $\e_i = \pm 1$, for all $1\le i \le n$, and $\pi$
is a permutation of the integers
 $\{ e_i\}^n_{i=1}$.
To the second basis $\{ e_i\p\}^n_{i=1}$, we associate the group $\Omega\p
= \{ A w A^{-1}; w \in \Omega\}$.  Then put $$ X_0 = \{ \e_i e_i ; \e_i
= \pm 1, \; 1 \le i \le n \} $$ and define sets $\{ X_j\}^\infty_{j=1}$
by induction, as follows: if $j \ge 1$ is odd then $$ X_j = \{ w\p x ;
x \in X_{j-1}, w\p \in \Omega\p\}, $$ and if $j \ge 1$ is even, then $$
X_j = \{ wx; x \in X_{j-1}, w \in \Omega \}.  $$

Once the sets $\{ X_j\}^\infty_{j=0}$ have been introduced, we define
the norm of a vector $x \in \bbr^n$ by setting $$ \| x \| = \max \{
2^{-j} | \lag x, x_j\rag |; x_j \in X_j, \,  \;
 j = 0, 1, 2, \dots \}.
$$ The fact that $\| \cdot \|$ is indeed a norm on $\bbr^n$ is trivial
since the above maximum is always attained on a finite set of indices
depending of course on the vector $x$ under consideration.

In order to prove that $\{ e_i\}^n_{i=1}$ is a 2-symmetric  basis, fix
a vector
 $x = \suml^n_{i=1} a_i e_i \in \bbr^n$ and assume that $\| x \| = 2^{-j}
 | \lag x, x_j\rag |$, for some integer $j$ and some $x_j \in X_j$.
If $j$ is odd then, for any $w \in \Omega, w x_j \in X_{j+1}$ so that
$$ \| wx\| \ge 2^{-(j+1)} | \lag wx,  wx_j\rag | = 2^{-(j+1)} |\lag x,
x_j\rag |=2^{-1} \| x\|.  $$ On the other hand, if $j$ is even then $x_j
= \tilde w x_{j-1}$,
 for some $\tilde w \in \Omega$ and $x_{j-1} \in X_{j-1}$.
Hence, for any $w \in \Omega, w x_j = (w \tilde w) x_{j-1} \in X_j$ so
that $$ \| w x \| \ge 2^{-j} | \lag wx, wx_j\rag | = 2^{-j} | \lag x,
x_j\rag | = \| x\|, $$ which proves that $\{ e_i\}^n_{i=1} $ is indeed
2-symmetric.  The proof of the fact that $\{ e_i\p\}^n_{2=1}$ is also
2-symmetric is done in exactly the same way.

The above construction is independent of the particular  choice of the
orthogonal matrix $A$.  The idea now is to select such a matrix $A$ so
that $\| e\p_1\|= \| Ae_1 \|$ is
 as small as possible and  it turns out that one can construct an
 orthogonal matrix $A$ for which the corresponding vector $e_1^{\prime}$
 satisfies
$$ \| e\p_1 \| \le \exp (-\log \log n /8 \log \log\log n).  $$ This
will suffice since it is not too difficult to show that $\{ e_i /\| e_i
\|\}^n_{i=1}$ and $\{ e\p_i /\| e\p_i \|\}^n_{u=1}$ are at best ${1\over
\|e\p_1\|}$-equivalent; this fact follows by comparing the expectations
$\frac{1}{\| e_1\|}{\Bbb E} \ \|\suml^n_{i=1} g_i e_i\| $ and $\frac{1}{\|
e_1\p\|}{\Bbb E} \ \|\suml^n_{i=1} g_i e\p_i \|$, where $\{ g_i\|^n_{i=1}$
is  a sequence of independent identically distributed random variables.

The construction of the orthogonal matrix $A$ so that $\| e\p_1 \| = \|
Ae_1 \| $ is ``small" is done by induction on its size.  For $k = 0$  and
therefore $n =2^0 =1$, we let $A\p_0 = (1) $ while, for $k>0$, $A\p_k$
is defined by $$ A\p_k = \left(\begin{array}{cc} A\p_{k-1}  &I_{k-1}\\
I_{k-1} &-A\p_{k-1}\\ \end{array}\right), $$ where $I_{k-1}$ denotes
the $2^{k-1} \times 2^{k-1}$-identity matrix.  Once the $2^k\times
2^k$-matrix $A\p_k$ is defined, we put $A_k = k^{-\half}
 A\p_k$.
It is easily seen that the matrices  $\{ A_k\}^\infty_{k=0}$ are not
only  orthogonal
 but also symmetric.

The most difficult part of the proof is to show that, for $n =2^k$, the
matrix $A=A_k$
 has the property that $\| e\p_1\| \le \exp (-\log \log n/8 \log
 \log\log n)$.
This argument is quite technical and it relies on a lemma of Harper
[ H ], Bernstein [ Be ], Hart [ Ha ] and Lindsey [ Li ] which gives
an estimate from below for the number of edges joining vertices of the
$k$-cube of a given cardinality $r$.  We omit these details which, as
we mentioned before, can be found in [ Go ].  \vskip.2cm

We conclude this section with some remarks on two notions of uniqueness
which are in the spirit of the so-called ``proportional'' theory
of finite dimensional spaces.  The main definition introduced in
[ C - K - T ] is the following.  \begin{Definition}\label{4.10}  Let
${\cal F}$ be a family of finite dimensional spaces each of which has
a normalized 1-unconditional basis.  We say that the members of ${\cal
F}$ have an almost (somewhat) unique unconditional basis provided there
exists a function $\varphi (K, \lambda)$, defined for all $K \ge 1$
and $0 < \lambda < 1$, such that, whenever $X \in {\cal F}$ with the
given normalized 1-unconditional basis $\{ x_i \}^n_{i = 1}$ has also
another normalized $K$-unconditional basis $\{ y_i \}^n_{i = 1}$ then,
for any (some) $0 < \lambda < 1$, there exist a subset $\sigma \subset
\{ 1, 2, \cdot, n \}$ and a one to one function $\pi : \sigma \to \{
1, 2, \cdot, n \}$ so that $\{ x_i \}_{i \in \sigma}$ is $\varphi
(K, \lambda)$-equivalent to $\{ y_{\pi (i)} \}_{i \in \sigma}$.
\end{Definition}

These notions are obviously a generalization of the notion of unique
unconditional basis, up to equivalence and permutation, introduced above.
A thorough study of almost and somewhat uniqueness of unconditional
basis is made in the paper [ C - K - T ].

We quote here, without proof, the following result from
[ C - K - T ], which is clearly a generalization of Theorem 4.8.
\begin{Theorem}\label{4.11}  For any $\alpha > 0$, each member of the
family $D_\alpha$ has an almost unique unconditional basis.  \end{Theorem}

\section{Uniqueness of rearrangement invariant structures}

While in the preceding sections we focused on the uniqueness question
only in the
 setting of sequence spaces, in the present one we pass to a study of
 similar problems
in the framework of rearrangement invariant (r.i.)  function spaces on
a non-atomic measure space.

The main requirement imposed on an r.i. function space $X$ on a finite
or $\sigma$-finite
 non-atomic measure space $(\Omega, \Sigma,\mu)$ is that, for any
 automorphism $\tau$  of $\Omega$ and every measurable function $f \in
 X$, the function $f(\tau^{-1})$ also belongs to $X$ and has the same
 norm as $f$.
If the measure space $(\Omega, \Sigma, \mu)$ is assumed to be non-atomic and
separable (i.e.  $\Sigma$ endowed with the usual metric $\rho(\tau,
\eta) = \mu (\sigma \Delta \eta);
 \sigma, \eta \in \Sigma$,  is a separable metric space) then it is well
 known that $(
\Omega, \Sigma, \mu)$ is isomorphic to a finite or infinite interval
endowed with the usual
 Lebesgue measure.
Hence, in principle, we can restrict our attention to the canonical
cases $\Omega
 = [0, 1]$ and $\Omega = [0, \infty)$, both endowed with the Lebesgue
 measure.
In the Basic Concepts article such spaces are called symmetric lattices.

In the case when a function space $X$ is invariant with respect to the
automorphisms of $\Omega$
 then the same is true for $X\p$, the subspace of the dual
$X^*$ of $X$ which consists of ``integrals", i.e. of functionals of the
form $$ x^*_g (t) = \intl_\Omega f g  d \mu, \; \; f \in X.  $$

In most of the interesting cases that appear in analysis, $X\p$ is a
norming subspace
 of $X^*$.
This occurs if and only if $0 \le f_n (w) \uparrow f (w)$ a.e. on $\Omega$
 implies that $\lim\limits_{n\to\infty} \| f_n\| = \| f \|$.
The proof of this simple assertion can be found in [ L - T II  1.b.18 ].
For convenience, we shall assume here that this is always the case.
The formal definition of the notion of r.i. function space, which will be
used in the sequel, is the following.  
\begin{Definition} A r.i. function
space $X$ on the interval $\Omega = [0, 1]$ or on the interval
 $\Omega = [0, \infty)$ is a Banach space of classes of equivalence of
 measurable 
functions  on $\Omega$ such that:

(i)  For any automorphism $\tau$ of $\Omega$, a function $f \in X$ if
and only if $f(\tau^{-1}) \in X$, and if this is the case then $\| f
(\tau^{-1}) \| = \|f\|$.

(ii) \ $X\p$ is a norming subspace of the dual $X^*$ of $X$ and thus $X$
is order isometric to a subspace of $X^{\prime\prime}$.  As a subspace
of $X^{\prime\prime}$, the space $X$ is either minimal (i.e. $X$ is
 the closure of the simple integrable functions on $\Omega$) or it is
 maximal (i.e.  $X = X^{\prime\prime}$).

(iii) \ If $\Omega = [0, 1]$ then $$ L_\infty(0, 1) \subset X \subset L_1
(0, 1), $$ with the inclusion maps being of norm one, i.e.  $$\| f\|_1
\le \| f \|_X \le \| f \|_\infty, $$ for all $f \in L_\infty(0, 1)$.

(iii$^{\prime}$) \ If $\Omega = [0, \infty)$ then $$ L_1  (0, \infty)
\cap L_\infty(0, \infty) \subset X \subset L_1 (0, \infty) + L_\infty
(0, \infty), $$ again with the inclusion maps being of norm one.
\end{Definition}

Recall that the norm of a function $f$ in the space $L_1 (0, \infty)
\cap L_\infty (0, \infty)$
 is defined as
$$ \| f \| = \max ( \| f \|_1    \ , \| f \|_\infty).  $$ 
The space $L_1
(0, \infty) + L_\infty(0, \infty)$ is often used in interpolation
theory and the norm of a function $f$ in this space is usually defined
by the formula $$ \|f\| = \inf \{ \| g \|_1 + \| h \|_\infty; \; \; f
= g+h\}, $$ the infimum being taken over all decompositions $f = g+h$,
with $g \in L_1 (0, \infty)$ and $h \in L_\infty (0, \infty)$.
 It is easily verified that if \  $Y = L_1 (0, \infty) + L_\infty
 (0,\infty)$ then $Y\p = L_1 (0, \infty) \cap L_\infty(0, \infty)$.
The norm in the space $Y$ can be alternatively defined with the aid of
the notion
 of {\bf decreasing rearrangement} of a function $f$ on either $\Omega  =
 [0, 1]$\ or on $\Omega = [0, \infty)$.
The decreasing rearrangement $f^*$ of a function $f \ge 0$ is defined
as the right
 continuous inverse of the distribution function  $d_f $ of $f$, which
 is defined by
$$ d_f (t) = \mu\{ w \in \Omega; \; \; f(w) > t\}.  $$ In other words,
$$ f^* (x) = \inf \{t>0; d_f (t) \le x \}; \; \; 0 \le x < \mu(\Omega).
$$ If $f$ is not $\ge 0$ then $f^*$ is defined as the decreasing
rearrangement of the absolute
 value $|f|$ of $f$.

It turns out that the norm of a function $f \in Y = L_1 (0, \infty) +
L_\infty (0, \infty)$ is equal to $$ \intl^1_0 f^* (x) dx.  $$ Indeed,
for any decomposition $f = g+h$, as above, and any subset $\sigma \subset
[0, \infty)$, we have that $$ \intl_\sigma |f(t) |dt \le \| g \|_1 + \|
h \|_\infty \mu(\sigma).  $$ Hence, $$ \intl^1_0f^* (x) dx = \sup \{
\intl_\sigma |f(t)|dt; \mu (\sigma) = 1\} \le \| f\|.  $$ Conversely,
fix $f \in Y$ and put $\lambda = \| f^* - f^* \chi_{[0, 1]} \|_\infty$.
Then $$ \begin{array}{ll} \| f \| \, &=\,\| f^* \| \le \| f^* -\min
(\lambda, f^*) \|_1 + \| \min (\lambda, f)
 \|_\infty =\\
&=\| \, (f^* - \lambda) \chi_{[0, 1]} \|_1 + \lambda = \intl^1_0 f^*
(x) dx.\\ \end{array} $$

As in the case of spaces  with a symmetric basis, the  question of
uniqueness ( this time of the
 r.i. structure ) is studied most extensively in $L_p$-spaces.
A result on $r.i.$ function spaces on [0, 1], which is quite useful
in the sdudy of uniqueness in $L_p$-spaces, was proved
in the Memoir [ J - M - S - T p.41 ].  
\begin{Theorem} A r.i. function space
$X$  on $[0, 1]$, which is isomorphic to a subspace of $L_p(0, 1); p >
2$, coincides with either $L_p(0, 1)$ or $L_2(0,1)$,
 up to an equivalent renorming.
\end{Theorem}

\noindent {\bf Proof}: Let $X$ be an r.i. function space on $[0, 1]$
and let $T$ be an isomorphism from $X$
 into $L_p(0, 1)$; $p>2$.
For every $n$ and $1 \le i \le 2^n$, denote by $\vp_{n, i}$ the
characteristic function
 of the interval $[(i-1)/2^n, i/2^n$).
Since, for every $n$, the sequence $\{ T\vp_{n, i}\}^{2^n}_{i=1}$ is
a $K$-symmetric
 basic sequence in $L_p(0, 1)$, with $K\le \| T\|\cdot \| T^{-1} \|$,
 one can use
Theorem 4.4 from the previous section and conclude the existence of a
constant $D <\infty$,
 depending only on $p$ and on $T$, so that, for every choice of scalars
 $\{ a_{i}\}^{2^n}_{i=1}$, we have
$$ \| \suml^{2^n}_{i=1} a_i {\vp_{n, i}\over \| \vp_{n, i}\|_X} \|_X
\overset{D}{\sim} \max \left \{ \left (\suml^{2^n}_{i=1} | a_i |^p
\right )^{1/p}, \; \; w \left ( \suml^{2^n}_{i=1} |a_i|^2 \right )^{1/2}
\right \}, $$ where $$ w={\| \suml^{2^n}_{i=1} {\vp_{n, i}\over \|
\vp_{n, i} \|_X} \|_X\over \sqrt{2^n}} \, = \, {1\over \| \vp_{n, 1} \|_X
\sqrt{2^n}}.  $$ Hence, for any simple function $f$ on the field generated
by the intervals\break $[(i-1)/2^n, i/2^n)$; $1 \le i \le 2^n$, we have
that $$ \| f\|_X \overset{D}{\sim} \max \{ \| \vp_{n, 1} \|_X 2^{n/p} \|
f \|_p, \; \; \| f \|_2\}.$$ 
Taking $f \equiv 1$ we get that the
sequence $\{ \| \vp_{n, 1} \|_X 2^{n/p} \}^\infty_{n=1}$ is bounded by
$D$ and thus, with $$ \a \, = \, \mathop{\lim \inf}\limits_{n\to\infty}
\| \vp_{n, 1} \|_X 2^{n/p}, $$ one concludes that, $$ \| f \|_X
\overset{D}{\sim} \max \{ \a \| f \|_p, \; \; \| f \|_2\}, $$ for any
simple function $f$ over the dyadic intervals in $[0, 1]$.  If $\a =
0$ then obviously $X=L_2 (0, 1)$ and if $\a > 0$ then, since
 $p>2$, $X = L_p (0, 1)$, both equalities up to an equivalent renorming.
\vskip.2cm

Theorem 5.2 has been generalized in [ H - K ] to a quite large class of
pairs $X$ and $Y$, where $Y$ is an r.i. function space on $[0, \infty)$
and $X$ is a non-atomic Banach lattice isomorphic to a subspace of $Y$.
Under different conditions on $Y$, stated mostly in terms of $p$-convexity
and $q$-concavity-notions which are described in the Basic Concepts 
aticle-it is
shown in [ H - K ] that $X$ is isomorphic to a {\it sublattice} of $Y$.
For instance, this is the case when $Y$ is $p$-convex and $q$-concave, for
some $p >2$ and $q < \infty$, and $X$ is $r$-convex, for some $r > 2$.
The same type of assumptions imply that if $Y$ is an r.i. function space
on $[0, 1]$ then $X$ contains a non-trivial band lattice isomorphic to
a sublattice of $Y$.  The paper [ H - K ] contains also some results on
complemented spaces.  For instance, if $Y$ is a separable r.i. function
space on either $[0, 1]$ or $[0, \infty)$, which contains no $\ell_2$ as a
complemented sublattice, and $X$ is a $p$-convex Banach lattice, for some
$p > 2$, which is isomorphic to a complemented subspace of $Y$ then $X$ is
even lattice-isomorphic to a complemented sublattice of $Y$.  \vskip.2cm

In exactly the same manner as in the proof of Theorem 5.2, one can prove 
the following version for $[0,
\infty)$ ( cf. [ J - M - S - T p.43 ] ).

\begin{Theorem} An r.i. function space $X$ on $[0, \infty)$, which is
isomorphic to a subspace of $L_p(0, \infty)$; $p>2$, coincides with one
of the spaces $L_p(0, \infty)$, $L_2 (0, \infty)$
 or $L_2 (0, \infty) \cap L_p (0, \infty)$, up to an equivalent renorming.
\end{Theorem}

The norm of a function $f \in L_2(0, \infty) \cap L_p (0, \infty)$ is of
 course defined by $\| f \| = \max (\| f \|_2, \| f \|_p)$.

Theorem 5..2 above implies the uniqueness of the r.i. structure on $[0,
1]$ of $L_p(0, 1)$,
 for $p >2$, and thus, by duality, also of $1 < p <2$.
The uniqueness of the r.i. structure of $L_2 (0, 1)$ is quite trivial,
in view of 1.1 above.  The uniqueness of the r.i. structure on $[0, 1]$
of $L_\infty (0, 1)$ can be easily
 reduced to that of $L_1 (0, 1)$.

In  order to prove the uniqueness of the r.i. structure on $[0, 1]$ of
the space $L_1 (0, 1)$, we need the fact that if $X$ is an r.i. function
space on $[0, 1]$, which is isomorphic
 to $L_1 (0, 1)$, then, for every $n$, the sequence of characteristic
 functions $\vp_{n, i} = \chi_{[(i-1)/2^n, i/2^n)}$; $ 1 \le i \le
 2^n$, forms a 1-unconditional basic sequence in $X$ whose span is
 1-complemented in $X$, by the conditional expectation operator relative
 to the field generated by the intervals $[(i-1) /2^n, i/2^n)$;
  $ 1 \le i \le 2^n$ (i.e. by the operator defined by ${\Bbb E}_nf = 2^n
  \suml^{2^n}_{i=1} (\intl^{
i/2^n}_{(i-1)/2^n} f dx) \vp_{n, i})$.  Then, the same argument, as the
one used in section 1 to prove that $\ell_1$ has a unique unconditional
basis,
 shows that $\{ \vp_{n, i} / \| \vp_{n, i} \|_X \}^{2^n}_{i=1}$ is
 equivalent to the unit vector basis of $\ell^{2^n}_1$, which constant
 of equivalence
independent of $n$.

We summarize the above observations in the following theorem which, again,
has been proved in [ J - M - S - T ].  \begin{Theorem} The space $L_p
(0, 1)$ has a unique structure as an r.i.
 function space on $[0, 1]$, for any value of $1\le p \le \infty$.
\end{Theorem} \noindent {\bf Remark}: In the case $p  =1$, Kalton [ K
] proved a much stronger result: an r.i.
 function space on $[0, 1]$, which contains a copy of $L_1 (0, 1)$,
 already\
coincides with $L_1 (0, 1)$, up to an equivalent norm, provided it does
not contain
 uniformly isomorphic copies of $\ell^n_\infty$.

\bigskip

Contrary to an initial belief, the space $L_p(0, \infty)$ does not have
a unique structure
 as an r.i. function space on $[0, \infty)$, unless $p = 1, 2$ or
 $\infty$.
In fact, the following result can be easily deduced from Theorem 5.3.
\begin{Theorem}  The space $L_p (0, \infty); 1 < p \neq 2 < \infty$,
 has exactly two distinct representations as an r.i. function space on
 $[0, \infty)$:
$$ \begin{array}{ll} &L_p(0, \infty) \; \;\hbox{\rm and} \; \; L_2(0,
\infty) \cap L_p(0, \infty), \; \; \hbox{\rm if} \; \; p > 2, \;
\; \hbox{\rm and}\\ &L_p(0, \infty) \; \;  \hbox{\rm and} \; \; L_2
(0,\infty) + L_p (0, \infty), \; \; \hbox{\rm if} \; \; 1 < p <2.\\
\end{array} $$ \end{Theorem} \noindent{\bf Proof}: The proof of 5.5 is
completed once we show that, for $p>2$, the spaces $L_p(0, \infty)$ and
$Z=L_2(0, \infty) \cap L_p (0, \infty)$ are isomorphic.  To this end,
notice that the restriction $Z_{|[0, 1]}$ is isomorphic to $L_p(0, 1)$ and
thus $Z$ contains a complemented subspace isomorphic to $L_p(0, \infty)$.
Conversely, for any $n $ and $m$, the sequence $\{ \chi_{[(i-1)/2^n,
i/2^n)}\}^{2^n}_{i=1}$ spans
 in $Z$ a symmetric $X_p$-space, and thus its span embeds complementably
 in $L_p
(0, \infty)$, in a uniform manner.  Hence, by a compactness argument
using, for instance, ultraproducts, one can show that $Z$ is isomorphic to
a complemented subspace of $L_p (0, \infty)$.  Then the decomposition
method shows that $Z \approx L_p (0, \infty)$.

Many  other classes of spaces which admit a unique representation as an
r.i. function
 space on $[0, 1]$ were exhibited in the Memoir [ J - M - S - T ] and
 [ K ].
We shall state some of these results without proof.

\begin{Theorem}  A r.i. function space $X$ on $[0, 1]$, which is
$q$-concave
 for some $q<2$, has unique structure as an r.i. function space on
 $[0, 1]$.
\end{Theorem}

The other result that we want to state below involves the notion
of the Haar basis.  Recall that the Haar system $\{\chi_n\}^\infty_{n=1}$
on $[0, 1]$ is defined by $\chi_1 (t) \equiv 1$ and, for $\ell = 1, 2,
\cdots, 2^k$ and $k = 0, 1, \cdots, $ by $$ \chi_{2^k+\ell}(t) \, = \,
\left\{\begin{array}{ll} 1 &\hbox{\rm if } \; \; t \in [(2\ell-2)/2^{k+1},
(2\ell -1)/2^{k+1})\\ -1 &\hbox{\rm if} \; \; t \in [(2\ell-1) (2^{k+1},
2\ell/2^{k+1})\\ 0 &\hbox{\rm otherwise}.\end{array}\right.  $$ It is an
immediate consequence of basic interpolation theorems that the Haar system
 forms a (monotone) basis in any separable  r.i. function space on
 $[0, 1]$.

We now present a result on uniqueness which was originally proved in [
J - M - S - T ]  under some additional assumptions and whose definitive
form, stated below, is due to Kalton [ K ].

\begin{Theorem} Let $X$ be a separable r.i. function space on $ [0, 1]$
such that
 the Haar basis of $X$ is not equivalent to a sequence of mutually
 disjoint functions in $X$ whose linear span is complemented in $X$.
Then $X$ has unique structure as a r.i. function space on $[0, 1]$.
\end{Theorem}

For using interpolation between two $L_p$-spaces other than $L_1 $ and
$L_\infty$ it is useful to consider 
 the so-called Boyd indices.
In order to define these indices for an r.i. function space $X $ on $
[0, 1]$, we need the
 dilation operator $D_s$ restricted to $[0, 1]$, which is defined for
 $0 < s <\infty
$ and $f$ on $[0, 1]$, by $$ (D_sf) (t) \, = \, \left\{\begin{array}{ll}
f(t/s) &0\le t \le \min(1, s)\\ 0 &s<t\le 1 \; \; (\hbox{\rm in the
case} \; \; s < 1).\end{array}\right.  $$

In fact, the operator $D_s$ is a dilation by the ratio $s\colon 1$ in the
positive direction of the $t$-axis followed by restriction to $[0, 1]$.
It is easily verified that, for every choice of $r$ and $s$, and every
r.i. function
 space $X$ on $[0, 1]$,
$$ \| D_{rs}\|_X \le \| D_r \|_X \| D_s \|_X, $$ which eventually ensures
that the following limits exist: $$ p_X = \lim\limits_{s\to\infty} \;
{\log s\over \log \|D_s\|_X} \, = \, \sup\limits_{s>1} \; {\log s\over
\log\|D_s\|_X} $$ and $$ q_X \, = \, \lim\limits_{s\to 0^+} \; {\log
s\over\log\| D_s\|_X} = \sup\limits_{0 < s < 1}
 \; {\log s \over \log \| D_s \|_X}.
$$

The importance of these indices stems from the fact, proved by Boyd [ Bo
], that, whenever $1 \le p < p_X$ and $q_X < q \le \infty$ for some
r.i. function space $X$
 on $[0,1]$, then every linear operator $T$, which is bounded on both $L_p
 (0, 1)$ and $L_q (0, 1)$, is also bounded on $X$.

A result due to Paley [ Pa ], proved in detail
e.g. in [ L - T II p.155 ], ( cf. also the article of Burkholder and 
the article of Figiel and Wojtaszczyk  
in this handbook ) asserts that the
 Haar system forms an unconditional basis in every $L_p (0, 1)$-space,
 if $1 < p < \infty$.
It follows immediately that the Haar system is also unconditional  in
every separable r.i. function space $X$ on $[0, 1]$,  whose Boyd indices
are non-trivial i.e. $1 < p_X$ and $q_X < \infty$.

In the case that one of the Boyd indices is trivial, Kalton [ K ] proved
the following uniqueness result.  
\begin{Theorem} Let $X$ be a separable
r.i. function space on $[0, 1]$ such that
 either $p_X = 1$ or $q_X = \infty$.
Then $X$ has unique structure as an r.i. function space on $[0, 1]$.
\end{Theorem} Finally, we mention the following result which was proved in
[  J - M - S - T p.169 ] , in the reflexive case, and follows from 5.8 in the
non reflexive one. 
\begin{Theorem} A separable Orlicz function space
on $[0, 1]$ has unique structure as an r.i. function space on $[0, 1]$.
\end{Theorem}

It turns out that there are also many r.i. function spaces on $[0, 1]$
which fail
 to have a unique structure as a r.i. function space on $[0, 1]$.
The construction of such examples has something in  common with that of
a space with ``many" mutually non-equivalent symmetric bases, presented
in section 2.

We start, as in section 2, with the space $U_1$ defined there, whose
normalized unconditional basis $\{ u_n\}^\infty_{n=1}$ is universal for
all normalized unconditional
 basic sequences in the sense that each such sequence is equivalent to
 a subsequence of $\{ u_n\}^\infty_{n=1}$.

We could proceed as in section 2 and interpolate $U_1$ between two
distinct $L_p$-spaces but the r.i. space constructed in this way will
not be isomorphic
 to $U_1$.
In order to avoid this problem, we shall first  $p$-convexify and
$2$-concavify
 the space $U_1$, for some $1 < p <2$, thus obtaining a new space whose
 unconditional basis is universal for all $p$-convex and 2-concave
 normalized unconditional basic sequences.
To this end, with $1 < p < 2 $ and $q=p^* = p/(p-1)$, we introduce a
new norm $\|| \cdot \|| $ on $U_1$ by setting for $u \in U_1$, $$ |\| u
|\| \, = \, \sup \left \{  \left ( \suml^n_{k=1} \| v^{(k)} \|^q \right
)^{1/2};\; \;
 u \, = \, \left (\suml^n_{k=1} | v^{(k)}|^q \right )^{1/2} \right \},
$$ where the supremum is taken over all decompositions of $u$, as above.
It is easily checked that  $\{u_n\}^\infty_{n=1}$ is in $(U_1, |\|
\cdot |\|$) $q$-concave and universal for all $q$-concave normalized
unconditional basic sequences.  Hence, the biorthogonal functionals form
in the dual $V$ of $(U_1, |\| \cdot |\|)$
 a $p$-convex unconditional basis which is universal for all $p$-convex
 normalized unconditional basic
sequences.  Repeating this procedure, this time with 2 instead of $q$,
we eventually get a space $W_p$, with a normalized unconditional basis
$\{w^{(p)}_n \}^\infty_{n=1}$, which is $p$-convex and $2$-concave
and, moreover, each $p$-convex and $2$-concave normalized unconditional
sequence is equivalent to a subsequence of $\{ w_n^{(p)} \}^\infty_{n=1}$.

Next, with $p\le r < 2$, we interpolate $W_p$ between $L_r(0, 1)$ and
$L_2(0, 1)$ thus obtaining a $p$-convex and $2$-concave r.i. function
space $Y_{p, r}$ on $[0, 1]$.  The Boyd indices of this space are
non-trivial and therefore the Haar system in $Y_{p, r}$ is unconditional
and this unconditional structure is $p$-convex and $2$-concave.
Hence, $Y_{p, r}$ is isomorphic to  a complemented subspace of $W_p$.

The interpolation procedure discussed above is done by choosing a
sequence $\{n_k\}^\infty_{k=1}$
 and by defining
$$ \| f\|_k = \inf \{ n_k \| g\|_r + n^{-1}_k \| h \|_2; \; \;
 f = g +h\}
$$ and $$ \| f\|_{Y_{p, r}} \, = \,\| \suml^\infty_{\ell = 1} \| f \|_k
w^{(p)}_k \|_{W_p}, $$ for any $f \in L_r (0, 1)$.  One can easily show
that if $\{ n_k\}^\infty_{k=1}$ tends ``fast" to $\infty$
 then the corresponding space $Y_{p, r}$ contains a complemented subspace
 isomorphic to $W_p$.
Then, by using the decomposition method, we conclude  that $Y_{p,
r}\approx
 W_p$, for any choice of $p<r<2$.
Finally,  it is not very hard to verify that, for different values of
$p<r<2$, we obtain mutually non-equivalent r.i. function spaces $Y_{p,r}$
on $[0, 1]$, which are all isomorphic to $W_p$.

\medskip

While the class of r..i. function spaces on a non-atomic measure space
can be considered
 as a continuous variant  of the class of spaces with a symmetric basis,
 non-atomic Banach lattices are the continuous analogue of spaces with
 an unconditional basis.
The first result on uniqueness of structure for non-atomic Banach lattices
was proved by Abramovich and Wojtaszczyk [ A - W ] who showed that $L_1$
and $L_2$ have unique structure as non-atomic Banach lattices.



It turns out that, contrary to the case of spaces with unique
unconditional basis, there exist other spaces with unique non-atomic
structure.  For example, the Orlicz space $L_M (0, 1)$, with $M(t) \sim
t(\log t)^\a$, for large
 $t$ and $0<\a<{1\over2}$, is such an example (cf. [ K ]).
In particular, this is an example of an r.i. function space on $[0, 1]$
which is not
 isomorphic to an r.i. function space on $[0, \infty]$.
\section{Uniqueness of bases in non-Banach spaces}

As we have seen in the previous sections, it is quite rare for an
unconditional basis in a Banach space to be unique, even up to a
permutation.  It turns out that in spaces other than Banach spaces
one can find relatively many unconditional bases having the uniqueness
property and , moreover, some of these uniqueness results have quite
interesting applications.

One class of spaces with a rich structure is that of so-called {\bf
quasi-Banach} spaces.  A quasi-Banach space is a vector space $X$ endowed
with a quasi-norm $\| \cdot \|$ which satisfies the usual axioms of the
norm except that the triangle inequality is replaced by the inequality
$$ \| x + y \| \le C (\| x \| + \| y \| ), $$ for all $x, y \in X$ and
some fixed constant $C \ge 1$.  A survey of the theory of these
spaces is [ K - P - R ] or the article of Kalton in this handbook.
In the class of quasi-Banach spaces the uniqueness of unconditional bases,
up to equivalence, is quite a common occurrence.  For instance, it was
shown by Kalton [K 1] that the spaces $\ell_p$ have, up to equivalence, a
unique unconditional basis, for every value of $0 < p < 1$.  As we have
seen in the previous sections, this result is false for $1 < p \neq 2$,
even up to a permutation.  In the same paper [ K 1 ] , the author also
exhibits a larger class of non-locally convex Orlicz sequence spaces
with a unique unconditional basis, up to equivalence.  One can consider
the problem of uniqueness for the so-called non-locally convex Lorentz
sequence spaces $d (w, p)$, where $p > 0$ and $w = (w_n)^\infty_{n =
1}$ is a non-summable monotone decreasing sequence of positive numbers
and the quasi-norm $\| x \|_{w, p}$ of an element $x \in d (w, p)$ is
defined by $$ \| x \|_{w, p} =\sup\limits_\pi (\sum\limits^\infty_{n =
1} | x_{\pi (n)} |^p w_n)^{1/p} < \infty.$$

In this definition, the supremum is taken over all permutation $\pi$ of
the integers.

The uniqueness of the basis in $d (w, p)$ was studied in
[ N - O ] and further in [ K - L - W ].  One result in this direction,
which is proved in [ K - L - W ] and provides the solution to an open
question raised in [ N - O ], asserts that the space $d(w, p)$ has,
up to equivalence, a unique unconditional basis, whenever $0<p<1$ and
the sequence of weights $w$ satisfies the condition.  $$ \lim\limits_{n
\to \infty} \frac{1}{n} (w_1 + \dots + w_n)^{1/p} = \infty. $$ Another
remarkable result proved in [ K - L - W ] states that also the spaces
$\ell_p (\ell_q)$ have a unique unconditional basis, up to equivalence
and permutation, as long as $0<p, q < 1$.  The same is true for the
spaces $c_0 (\ell_p)$, as long as $0 < p < 1$ (cf.~[Le]).

Perhaps, the most interesting class of quasi-Banach spaces is that of the
Hardy spaces $H_p ({\Bbb T}), 0 < p < 1$, where ${\Bbb T}$ denotes the
circle, and their $m$-dimensional version $H_p ({\Bbb T}^m)$, where ${\Bbb
T}^m$ denotes the $m$-dimensional torus.  The fact that these quasi-Banach
spaces have an unconditional basis is not obvious.  For $m =
1$, this follows from the fact, proved in [W], that $H_p ({\Bbb T})$ is
isomorphic to the {\bf dyadic} Hardy space $H_p (0, 1)$, for all $0 < p <
1$. The usual Haar basis $\{ h_n \}^\infty_{n=1}$ forms an unconditional
basis in $H_p (0, 1), 0 < p < \infty$, since the dyadic $H_p (0, 1)$
consists of all distribution of the form $f = \sum\limits^\infty_{n = 1}
a_n h_n$ for which the expression $$ \| f \|_p = \Big ( \int\limits^1_0 (
\sum\limits^\infty_{n = 1} | a_n h_n |^2)^{p/2} \Big )^{1/p} < \infty. $$
For higher dimensions, one verifies that the $m$-times tensor product
of the Haar basis is an unconditional basis in the dyadic Hardy space
$H_p (0, 1)^m$, for every value of $0 < p < 1$, and that this space is
isomorphic to the usual space $H_p ({\Bbb T}^n)$ (cf. [W]). A remarkable
result from [W1] asserts that every normalized unconditional basis in $H_p
({\Bbb T}); 0 < p < 1$, is equivalent to the Haar basis in some order
 i.e. the space $H_p ({\Bbb T})$ has, up to
equivalence and permutation, a unique unconditional basis.

A nice application of the uniqueness result in non-locally convex Hardy
spaces is the fact ( proved first by Kalton, Leranoz and
Wojtaszczyk [ K - L - W ] ) that, for $0 < p < 1$, the spaces $H_p ({\Bbb
T}^m), m = 1, 2, \cdots$,  are mutually non-isomorphic.  If, for example,
$H_p ({\Bbb T}^m)$ were isomorphic to $H_p ({\Bbb T})$, for some $0< p <
1$ and some integer $m$, then the corresponding normalized unconditional
bases of these two spaces would be equivalent, up to a permutation.
That this is false is checked directly.

One should point out that the result asserting that the spaces $H_1
({\Bbb T}^m),  m = 1, 2, \cdots$, are mutually non-isomorphic was proved
before by Bourgain ([ Bo 1 ], [ Bo 2 ] ).

Uniqueness of bases can be also studied in the more general context
of locally convex spaces ( l.c.s. ), i.e. vector spaces which admit a
fundamental system of convex neighborhoods of $0$.  The topology of
such a space can be defined with the aid of a family of semi-norms.
Of particular interest are the Fr\'echet spaces.  (recall that these
are l.c.s.  whose topology is both metrizable and complete).

The main difficulty faced in studying uniqueness in l.c.s. is the fact the
notion of ``normalized basis'' does not make any sense in these spaces.

The only Fr\'echet spaces which have a unique unconditional basis are
$\omega$, the space of all scalar sequences, and its dual $\omega^\ast$, the
space of all scalar sequences which are eventually zero.
The uniqueness assertion for $\omega$ and $\omega^\ast$ is due to
K\"othe and T\"oplitz. This is the result that in some sense initiated
the research on uniqueness of bases. That $\omega$ and $\omega^\ast$
are the only spaces with this uniqueness property is due to
Dragilev ( cf. [ Dr 3] ).

Less restrictive is the notion of quasi-equivalence.  Two unconditional
bases $\{x_n \}^\infty_{n =1}$ and $\{ y_n \}^\infty_{n = 1}$ in
l.c.s. $X$, respectively $Y$, are said to be {\bf quasi-equivalent} if
there exists a permutation $\pi$ of the integers and a sequence $\lambda
= (\lambda_n)^\infty_{n = 1}$ of scalars such that $\{ \lambda_n x_{\pi
(n)} \}^\infty_{n = 1}$ and $\{ y_n \}^\infty_{n = 1}$ are equivalent,
i. e. there exists an isomorphism $T$ from $X$  onto $Y$ so that $y_n = T
(\lambda_n x_{\pi (n)})$, for all $n$.

A remarkable result, also due to Dragilev [ Dr 1 ],  asserts that the
Fr\'echet space $A (D)$ of all the functions which are analytic on the
open disk $D$ (endowed with the topology of uniform convergence on each
compact subset of the disk) has the property that all its unconditional
bases are quasi-equivalent.  This fact is also true in the class of all
nuclear power series space (cf. [ Mi 1 ] ).

A systematic study of quasi-equivalence in l.c.s. has been carried out
by Dragilev [ Dr 2 ], Dubinsky [Du], Mityagin [ Mi 2 ] and recently by
Zahariuta [Za].

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