On equivalence relations generated by Schauder bases

In this paper, a notion of Schauder equivalence relation $\mathbb R^\mathbb N/L$ is introduced, where $L$ is a linear subspace of $\mathbb R^\mathbb N$ and the unit vectors form a Schauder basis of $L$. The main theorem is to show that the following conditions are equivalent: (1) the unit vector basis is boundedly complete; (2) $L$ is $F_\sigma$ in $\mathbb R^\mathbb N$; (3) $\mathbb R^\mathbb N/L$ is Borel reducible to $\mathbb R^\mathbb N/\ell_\infty$. We show that any Schauder equivalence relation generalized by basis of $\ell_2$ is Borel bireducible to $\mathbb R^\mathbb N/\ell_2$ itself, but it is not true for bases of $c_0$ or $\ell_1$. Furthermore, among all Schauder equivalence relations generated by sequences in $c_0$, we find the minimum and the maximum elements with respect to Borel reducibility. We also show that $\mathbb R^\mathbb N/\ell_p$ is Borel reducible to $\mathbb R^\mathbb N/J$ iff $p\le 2$, where $J$ is James' space.