We investigate Borel reducibility between equivalence relations $E(X;p)=X^{\mathbb{N}}/\ell_{p}(X)'s$ where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) 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 ∈ [1, +∞). We also answer a problem presented by Kanovei in the affirmative by showing that $\mathrm{C}\left({\mathrm{\mathbb{R}}}^{+}\right)/{\mathrm{C}}_{0}\left({\mathrm{\mathbb{R}}}^{+}\ri ght)$ (...) is Borel bireducible to $\mathbb{R}^{\mathbb{N}}/c_{0}$. (shrink)