\documentclass{article} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsmath} %\def\pbf{\par\bigpagebreak\flushpar} %\def\pmf{\par\medpagebreak\flushpar} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Sublemma}{Sublemma} \newtheorem{Definition}[Theorem]{Definition} \newtheorem{Problem}[Theorem]{Problem} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Corollary}[Theorem]{Corollary} \newtheorem{Example}[Theorem]{Example} \newtheorem{Examples}[Theorem]{Examples} % % \def\bbc{{\Bbb C}} \def\bbf{{\Bbb F}} \def\bbn{{\Bbb N}} \def\bbq{{\Bbb Q}} \def\bbr{{\Bbb R}} \def\bbt{{\Bbb T}} \def\bbz{{\Bbb Z}} \def\f{\flushpar} \def\un{\underbar} \def\elim{\{e_n\}^\infty_{n=1}} \def\suminf{\sum\limits^\infty_{n=1}} \def\suml{\sum\limits} \def\half{1/2} \def\intl{\int\limits} \def\a{\alpha} \def\b{\beta} \def\e{\varepsilon} \def\xinf{ \{X_n\}^\infty_{n=1}} \def\nnf{^\infty_{n=1}} \def\p{^{\prime}} \def\pp{^{\prime\prime}} \def\sumlr{\suml^\infty_{r=1}} \def\vp{\varphi} \def\gam{\gamma} \def\lag{\langle} \def\rag{\rangle} \begin{document} \setcounter{section}{-1} \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 $10$ 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 $$ (D_sf) (t) \, = \, \left\{\begin{array}{ll} f(t/s) &0\le t \le \min(1, s)\\ 0 &s1} \; {\log s\over \log\|D_s\|_X} $$ 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 $p 0$ 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. 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