On the properties of quasi-Banach function spaces

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they posses a generalised version of Riesz–Fischer property, that embeddings between them are always continuous and that the dilation operator is bounded on them. We also provide a characterisation of separability for quasi-Banach function spaces over the Euclidean space. Furthermore, we extend the classical Riesz–Fischer theorem to the context of quasinormed spaces and, as a consequence, obtain an alternative proof of completeness of quasi-Banach function spaces.