## Borel reducibility and Hölder$$(\alpha )$$ embeddability between Banach spaces.(English)Zbl 1250.03082

Summary: 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$$(\alpha )$$ embeddability between Banach spaces. 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 $$C(\mathbb R^{+})/C_{0}(\mathbb R^{+})$$ is Borel bireducible to $$\mathbb R^{\mathbb N}/c_{0}$$.

### MSC:

 03E15 Descriptive set theory 46B20 Geometry and structure of normed linear spaces
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### References:

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