A topology on inequalities

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. In order 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 of them 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.