Banach function spaces done right

TL;DR

This survey revisits the foundational definitions of Banach and quasi-Banach function spaces, emphasizing the original approach by Zaanen and Luxemburg and establishing basic properties within this framework.

Contribution

It advocates for the original definitions of Banach function spaces and extends fundamental properties to quasi-Banach spaces, clarifying theoretical foundations.

Findings

Original definitions avoid issues of later versions

Basic properties proven for quasi-Banach spaces

Clarifies theoretical framework for function spaces

Abstract

In this survey, we discuss the definition of a (quasi-)Banach function space. We advertise the original definition by Zaanen and Luxemburg, which does not have various issues introduced by other, subsequent definitions. Moreover, we prove versions of well-known basic properties of Banach function spaces in the setting of quasi-Banach function spaces.