Uniform boundedness of norms of convex and nonconvex processes

The lower limit of a sequence of closed convex processes is again a closed convex process. In this note we prove the following uniform boundedness principle: if the lower limit is nonempty-valued everywhere, then, starting from a certain index, the given sequence is uniformly norm-bounded. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions

A uniform boundedness principle in pluripotential theory

For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above

On the pillars of Functional Analysis

Funding for open access charge: Universidad de Granada/CBUA. This work is supported in part by Grupo de Investigacion FQ199 of the Junta de Andalucia (Spain) and by the IMAG-Maria de Maeztu Grant CEX2020-001105-M/AEI/10.13039/501100011033..Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. The first one is derived from Zorn’s Lemma, while the latter two usually are obtained from Baire’s Category Theorem. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform Boundedness Principle, the Open Mapping Principle and another five theorems are equivalent, as we show in a very elemental way. Since one can give an almost trivial proof of the Uniform Boundedness Principle that does not require the Baire’s theorem, we conclude that this is also the case for the other equivalent theorems that, in this way, are simultaneously proved in a simple, brief and concise way that sheds light on their nature.Universidad de Granada/CBUAJunta de Andalucia FQ199IMAG-Maria de Maeztu Grant CEX2020-001105-M/AEI/10.13039/50110001103

A remark on condensation of singularities

Recently Alan D. Sokal gave a very short and completely elementary proof of the uniform boundedness principle. The aim of this note is to point out that by using a similiar technique one can give a considerably short and simple proof of a stronger statement, namely a principle of condensation of singularities for certain double-sequences of non-linear operators on quasi-Banach spaces, which is a bit more general than a result of I.\,S. G\'al.Comment: 7 page