# On Garling sequence spaces

## TL;DR

This paper introduces a new class of Banach spaces inspired by Garling's example, demonstrating unique subsymmetric bases and properties like $ ext{l}_p$-saturation and complementable homogeneity, thus providing novel examples in Banach space theory.

## Contribution

It constructs and analyzes a new class of Banach spaces with unique subsymmetric bases and specific structural properties, filling a gap in existing examples.

## Key findings

- Existence of Banach spaces with unique subsymmetric bases
- Spaces are $ ext{l}_p$-saturated and complementably homogeneous
- First known examples with both properties coexist

## Abstract

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leqslant p<\infty$ and each nonincreasing weight $\textbf{w}\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous, and uniformly subprojective Banach space $g(\textbf{w},p)$. We also show that $g(\textbf{w},p)$ admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist.