Dichotomies pour les espaces de suites réelles

There is a general conjecture, the dichotomy (C) about Borel equivalence relations E: (i) E is Borel reducible to the equivalence relation ${\displaystyle E^X\_G}$ where X is a Polish space, and a Polish group acting continuously on X; or (ii) a canonical relation ${\displaystyle E_1}$ is Borel reducible to E. (C) is only proved for special cases as in [So].  In this paper we make a contribution to the study of (C): a stronger conjecture is true for hereditary subspaces of the Polish space ${\displaystyle {ℝ}^{\omega }}$ of real sequences, i.e., subspaces such that ${\displaystyle [y={\left(y_n\right)}_n\in X}$ and ∀n, ${\displaystyle \left|x_n\right|\le \left|y_n\right|]\Rightarrow x={\left(x_n\right)}_n\in X}$. If such an X is analytic as a subset of ${\displaystyle {ℝ}^{\omega }}$, then either X is Polishable as a vector subspace, or X admits a subspace strongly isomorphic to the space ${\displaystyle c_{00}}$ of finite sequences, or to the space ${\displaystyle {\ell }_{\infty }}$ of bounded sequences.  When X is Polishable, the metrics have a very simple form as in the case studied in [So], which allows us to study precisely the properties of those X's