unconditional convergent I have to decide whether the following statement is true or false : every permutation of a basic sequence is equivalent to the entire sequence ! where a sequence $(x_n)$ in a Banach space $X$ is called basic if it's a basis of $[x_1,x_2,x_3,.........]$ (it's closed span). 
I think it's false, and I think that if it's true then  $\displaystyle\sum_{1}^{\infty}x_n$ converges unconditionaly . Any ideas. Thank you!
 A: A (Schauder) basis $(x_i)_{i=1}^\infty$ of a Banach space $X$ is unconditional if and only if for every $x\in X$, its expansion $\sum\limits_{i=1}^\infty \alpha_i x_i$ in terms of the basis converges unconditionally. This is equivalent to saying that any permutation of the basis is a basic sequence. 
There are, of course, bases that are not unconditional. For instance, it can be shown that  the sequence $\{e_1, e_1-e_2, e_1-e_3,\ldots\}$ is a basis of $\ell_1$ that is not unconditional.
The property that you require is stronger. A basis $(x_i)_{i=1}^\infty$ for which every permutation of $(x_i)_{i=1}^\infty$ is equivalent to $(x_i)_{i=1}^\infty$ is called symmetric (recall that two basic sequences $(x_i)$ and $(y_i)$ are equivalent if the convergence of $\sum\alpha_i x_i$ is equivalent to that of $\sum\alpha_i y_i$).
Of note:


*

*For $1<p<\infty$, the space $L_p(0,1)$ does not have a symmetric
basis (it in fact does not possess a subsymmetric basis)$\,^1$. So the
Haar system in these spaces gives an example of an unconditional
basis that is not symmetric.

*The sequence $(x_i)$ constructed in my answer to this
post
is an unconditional basic sequence in $\ell_1$.  It follows from the
results there that this sequence is not equivalent to the standard
unit vector basis of $\ell_1$. So from the last point below, it
follows that $(x_n)$ is not symmetric.

*If $X$ is one of $c_0$, $\ell_1$, or $\ell_2$, then every unconditional
basis of $X$ is symmetric. This follows from the fact that these
spaces have unique unconditional bases$\,^2$.

*If $X$ is one of $c_0$, $\ell_p$ $1\le p<\infty$, then every symmetric
basis of $X$ is equivalent to the unit vector basis of $X$.
In
fact,  any
symmetric basic sequence in $X$ is equivalent to the standard unit
vector basis of $X$$\,^3$.



$^1$ c.f., Ivan Singer, Bases in Banach Spaces, Chapter 22.
$^2$ This is a well known fact. See, e.g., Singer or Joeseph Diestel's Sequences and Series in Banach Spaces. 
$^3$ c.f. Lindenstrauss and Tzafriri Classical Banach Spaces I, proposition 3.b.5 and the remark following.
