On the pillars of Functional Analysis

Abstract

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.