Constructions of sequential spaces

TL;DR

This paper introduces variable exponent ll^p spaces with unique local geometric properties, providing new examples of Banach spaces that have a 1-unconditional basis, expanding the understanding of their structure.

Contribution

It presents a novel class of variable exponent ll^p spaces with specific local control of geometric properties, not rearrangement-invariant, and constructs examples with a 1-unconditional basis.

Findings

Spaces have good local geometric control

Examples of Banach spaces with 1-unconditional basis

Spaces are typically not rearrangement-invariant

Abstract

We introduce and study certain type of variable exponent \ell^p spaces. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. We obtain some interesting examples of Banach spaces with a 1-unconditional basis.