The Non-Definability of the Spaces of Tsirelson and Schlumprecht

TL;DR

This paper demonstrates that the complex Banach spaces of Tsirelson and Schlumprecht cannot be explicitly characterized using finitary definitions within continuous first-order logic.

Contribution

It establishes the non-definability of Tsirelson and Schlumprecht spaces in a formal logical framework, highlighting their inherent complexity.

Findings

No explicit finitary definitions exist for these spaces.

The spaces cannot be characterized in continuous first-order logic.

This reveals limitations of logical definability for certain Banach spaces.

Abstract

We prove the impossibility of finding explicit finitary definitions of the spaces of Tsirelson and Schlumprecht in continuous first-order logic.