Geometry of Banach spaces and biorthogonal systems

TL;DR

This paper characterizes the geometric properties of separable Banach spaces using biorthogonal systems, linking the presence of $ ext{ell}_1$ and Schur property failures to specific biorthogonal system structures.

Contribution

It establishes new equivalences between geometric properties of Banach spaces and the existence of bounded wc_0^*-stable biorthogonal systems, advancing the understanding of Banach space geometry.

Findings

Separable Banach space contains $ ext{ell}_1$ iff it has a bounded wc_0^*-stable biorthogonal system.

Dual space fails Schur property iff the space has a bounded wc_0^*-biorthogonal system.

Abstract

A separable Banach space X contains $\ell_1$ isomorphically if and only if X has a bounded wc_0^*-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded wc_0^*-biorthogonal system.