On the Properties of Quasi-Banach Function Spaces
Aleš Nekvinda, Dalimil Peša
Abstract
Abstract In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they possess 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.
Topics & Concepts
MathematicsApproximation propertyPure mathematicsBanach spaceBanach manifoldFunction spaceInterpolation spaceFinite-rank operatorEberlein–Šmulian theoremLp spaceCompleteness (order theory)Unbounded operatorOperator theoryFunctional analysisDiscrete mathematicsMathematical analysisBiochemistryChemistryGeneAdvanced Harmonic Analysis ResearchAdvanced Banach Space TheoryApproximation Theory and Sequence Spaces