Application of Baire's Theorem

Abstract

This work deals with a literature review, in which are presented results obtaine from a project of Scientific Initiation of Functional Analysis. Firstly, it was discussed about the concepts of Banach Space, Dual Space, Never Dense Set, First Category Set, Second Category Set. With these definitions it was possible to enunciate and to demonstrate Baire’s Theorem together with its corollaries, which constituted the basis for the two Banach-Steinhauss’ Theorem, the second being the reciprocal of the first one, with addition of the hypotesis of X being a Banach Space. These two theorems, in their turn, are fundamental to demonstrate the Principle
of Uniform Limitation presented here. We use this principle in the following result: in a Banach Space X, where a function f belongs to Dual Space, that is, X* , if the direct image of a set, f(B), is a limited set, then B will also be limited.