# A note on the symmetry of sequence spaces

## TL;DR

This paper provides a comprehensive and self-contained analysis of symmetric Banach sequence spaces, focusing on their symmetry properties and the existence of symmetric linear functionals, filling gaps in the literature.

## Contribution

It offers a detailed, self-contained treatment of symmetric Banach sequence spaces, highlighting properties often assumed but not thoroughly documented.

## Key findings

- Characterization of symmetry in Banach sequence spaces
- Existence conditions for symmetric linear functionals
- Compilation of known results with new insights

## Abstract

We give a self-contained treatment of symmetric Banach sequence spaces and some of their natural properties. We are particularly interested in the symmetry of the norm and the existence of symmetric linear functionals. Many of the presented results are known or commonly accepted but are not found in the literature.