Non-Baire Proof of Uniform Boundedness Theorem and Its Applications in the Proof of Some Grand Theorems of Functional Analysis

Abstract

The proving of three big theorems, known as the uniform boundedness theorem, the open mapping theorem, and the closed graph theorem, is the pinnacle of any first functional analysis course. All three rely on the completeness of some or all of the spaces involved, and their proofs are based on Baire's theorem (or, the Baire category theorem), a topological conclusion. The aim of this paper is to prove the Uniform Boundedness Theorem without using Baire’s Theorem and to show the logical dependence of these above three results on each other.