On equivalence relations generated by Schauder bases

In this paper, a notion of Schauder equivalence relation $\mathbb R^\mathbb N/L$isintroduced,where$L$isalinearsubspaceof$\mathbb R^\mathbb N$ 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) ${\mathbb{R}}^{\mathbb{N}}\mathrm{/}L$ is Borel reducible to $\mathbb R^\mathbb N/\ell_\infty$. We show that any Schauder equivalence relation generalized by basis of ${\mathrm{\ell }}_2$ is Borel bireducible to ${\mathbb{R}}^{\mathbb{N}}\mathrm{/}{\mathrm{\ell }}_2$ itself, but it is not true for bases of $c_0$ or ${\mathrm{\ell }}_1$. 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 ${\mathbb{R}}^{\mathbb{N}}\mathrm{/}{\mathrm{\ell }}_p$ is Borel reducible to $\mathbb R^\mathbb N/J$iff$p\le 2$,where$J$ is James’ space.