A topology on inequalities
Date
2006-08-02
Authors
D'Aristotile, Anna Maria
Fiorenza, Alberto
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
Abstract
We consider sets of inequalities in Real Analysis and construct a topology such that inequalities usually called "limit cases" of certain sequences of inequalities are in fact limits - in the precise topological sense - of such sequences. To show the generality of the results, several examples are given for the notions introduced, and three main examples are considered: Sequences of inequalities relating real numbers, sequences of classical Hardy's inequalities, and sequences of embedding inequalities for fractional Sobolev spaces. All examples are considered along with their limit cases, and it is shown how they can be considered as sequences of one "big" space of inequalities. As a byproduct, we show how an abstract process to derive inequalities among homogeneous operators can be a tool for proving inequalities. Finally, we give some tools to compute limits of sequences of inequalities in the topology introduced, and we exhibit new applications.
Description
Keywords
Real analysis, Topology, Inequalities, Homogeneous operators, Banach spaces, Orlicz spaces, Sobolev spaces, Norms, Density
Citation
D'Aristotile, A. M., & Fiorenza, A. (2006). A topology on inequalities. <i>Electronic Journal of Differential Equations, 2006</i>(85), pp. 1-22.
Rights
Attribution 4.0 International