# 1-Greedy renormings of Garling sequence spaces

## TL;DR

This paper demonstrates that Garling sequence spaces can be renormed to have a 1-greedy basis and explores their properties related to convexity and superreflexivity, using non-linear greedy approximation tools.

## Contribution

It introduces a new renorming of Garling sequence spaces making their basis 1-greedy and applies non-linear methods to analyze their linear structure.

## Key findings

- Garling sequence spaces admit a 1-greedy renorming
- The spaces exhibit properties related to uniform convexity and superreflexivity
- Non-linear tools from greedy approximation provide insights into their linear structure

## Abstract

Garling sequence spaces admit a renorming with respect to which their standard unit vector basis is 1-greedy. We also discuss some additional properties of these Banach spaces related to uniform convexity and superreflexivity. In particular, our approach to the study of the superreflexivity of Garling sequence space provides an example of how essentially non-linear tools from greedy approximation can be used to shed light into the linear structure of the spaces.