An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James Space

TL;DR

This paper provides a constructive proof establishing a lower bound on the growth of the local unconditionality constants in the James space, improving understanding of its structural properties.

Contribution

It introduces a proof mining approach to derive a quantitative lower bound, replacing the non-constructive ultraproduct argument.

Findings

Established a lower bound related to the inverse Ackermann function

Provided a constructive method for analyzing local unconditionality constants

Enhanced understanding of the James space's geometric properties

Abstract

The proof that the James space is not locally unconditional appears to be non-constructive, since it makes use of an ultraproduct construction. Using proof mining, we extract a constructive proof and obtain a lower bound on the growth of the local unconditionality constants.