On the equivalence between coarse and uniform embeddability of quasi-Banach spaces into a Hilbert space

TL;DR

This paper proves that for quasi-Banach spaces, coarse embeddability into a Hilbert space is equivalent to uniform embeddability, providing a direct proof of this fundamental relationship.

Contribution

It offers a direct proof establishing the equivalence between coarse and uniform embeddability of quasi-Banach spaces into Hilbert spaces.

Findings

Coarse and uniform embeddability are equivalent for quasi-Banach spaces.

Provides a direct proof of the equivalence.

Clarifies the relationship between different embedding notions.

Abstract

We give a direct proof of the fact that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it uniformly embeds into a Hilbert space.