Banach function spaces done right
Emiel Lorist, Zoe Nieraeth

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.
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Taxonomy
TopicsAdvanced Banach Space Theory · Advanced Harmonic Analysis Research · Holomorphic and Operator Theory
