No Borel upper bound for the F_σ equivalence relations

We prove that the family of coset relations of F_σ real linear subspaces of Banach space ₁ has no common upper bound among Borel equivalence relations in the sense of Borel reducibility. These relations are indexed by a well-founded tree. Consequently, on a Polish space, a universal Σ⁰_α equivalence relation exists exactly when $α=1$, and a universal Π⁰_α equivalence relation exists exactly when $α=1$ or $2$. This strengthens the earlier nonexistence theorem for a universal F_σ equivalence relation. This also answers a question of Kanovei and Lyubetsky about universal equivalence relations in Borel classes.