On the properties of quasi-Banach function spaces

Abstract

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.

References in corpus (4)

• Banach function spaces done right
• Extrapolation in general quasi-Banach function spaces
• Sobolev embeddings, rearrangement-invariant spaces and Frostman measures
• Basic functional properties of certain scale of rearrangement-invariant spaces

Cited by in corpus (11)

• Lorentz--Karamata spaces
• Reduction principle for Gaussian $K$-inequality
• Basic functional properties of certain scale of rearrangement-invariant spaces
• On Bilinear Hardy Inequality and Corresponding Geometric Mean Inequality
• On the properties of rearrangement-invariant quasi-Banach function spaces
• Maximal noncompactness of embeddings into Marcinkiewicz spaces
• Embeddings between Lorenz sequence spaces are strictly singular
• Non-improvability of sharp endpoint estimates
• Absolute continuity of the (quasi)norm in rearrangement-invariant spaces
• Wiener-Luxemburg amalgam spaces
• Quasi Grand Lebesgue Spaces