On the pillars of Functional Analysis

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.

The Riemann-Cantor uniqueness theorem for unilateral trigonometric series via a special version of the Lusin-Privalov theorem

Using Baire's theorem, we give a very simple proof of a special version of the Lusin-Privalov theorem and deduce via Abel's theorem the Riemann-Cantor theorem on the uniqueness of the coefficients of pointwise convergent unilateral trigonometric series.

On Fuzzy Modular Spaces

The concept of fuzzy modular space is first proposed in this paper. Afterwards, a Hausdorff topology induced by aβ-homogeneous fuzzy modular is defined and some related topological properties are also examined. And then, several theorems onμ-completeness of the fuzzy modular space are given. Finally, the well-known Baire’s theorem and uniform limit theorem are extended to fuzzy modular spaces.

Some Properties of Fuzzy Quasimetric Spaces

Some properties of fuzzy quasimetric spaces are studied. We prove that the topology induced by any -complete fuzzy-quasi-space is a -complete quasimetric space. We also prove Baire's theorem, uniform limit theorem, and second countability result for fuzzy quasi-metric spaces.

A continuity principle, a version of Baire's theorem and a boundedness principle

We deal with a restricted form WC-N′ of the weak continuity principle, a version BT′ of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N′, BT′ + ¬LPO and BD-N + ¬LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience.

Some remarks on universal functions and Taylor series

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire's theorem. We also give a constructive proof, avoiding Baire's theorem, of the existence of universal Taylor series in any arbitrary simply connected domain.