A space with no unconditional basis that satisfies the Johnson-Lindenstrauss lemma

TL;DR

This paper constructs a unique twisted Hilbert space lacking an unconditional basis that still satisfies the Johnson-Lindenstrauss lemma, providing new insights into the structure of such spaces.

Contribution

It presents the first example of a nontrivial twisted Hilbert space satisfying the Johnson-Lindenstrauss lemma without an unconditional basis.

Findings

Existence of a twisted Hilbert space satisfying JL lemma

Space has no unconditional basis

Partial answer to Mascioni's question

Abstract

We give the first example of a nontrivial twisted Hilbert space that satisfies the Johnson-Lindenstrauss lemma. This space has no unconditional basis. We also show that such a space gives a partial answer to a question of Mascioni.