Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 85, pp. 1-22.
Title: A topology on inequalities
Authors: Anna Maria D'Aristotile (Univ. di Napoli, Italy)
Alberto Fiorenza (Univ. di Napoli, Italy)
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.
Submitted July 12, 2006. Published August 2, 2006.
Math Subject Classifications: 46E99, 54C30, 54A20, 54B15.
Key Words: Real analysis; topology; inequalities;
homogeneous operators; Banach spaces;
Orlicz spaces; Sobolev spaces; norms; density.