  We investigate Borel reducibility between equivalence relations
$E(X,p)=X^{\Bbb N}/\ell_p(X)$'s where $X$ is a separable Banach space. We show
that this reducibility is related to the so called H\"older$(\alpha)$
embeddability between Banach spaces. By using the notions of type and cotype of
Banach spaces, we present many results on reducibility and unreducibility
between $E(L_r,p)$'s and $E(c_0,p)$'s for $r,p\in[1,+\infty)$.
  We also answer a problem presented by Kanovei in the affirmative by showing
that $C({\Bbb R}^+)/C_0({\Bbb R}^+)$ is Borel bireducible to ${\Bbb R}^{\Bbb
N}/c_0$.
