Teoremas clássicos da análise funcional

Abstract

This undergraduate thesis aims to present and detail the proofs of the classical theorems of Functional Analysis, serving as a comprehensive support material for readers and students in the field, in addition to establishing connections between the results so as to guide the reader throughout the development of the work. The research focuses on exploring essential results for the study of infinite-dimensional normed spaces and continuous linear operators. Initially, the text addresses the Hahn-Banach Theorem in its analytical and geometrics formulations. The analytical version is discussed in the context of extending linear functionals defined on subspaces, while the geometric versions are explored from the perspective of separating nonempty, convex and disjoint subsets using hyperplanes. Next, the study advances to Banach spaces, presenting three central results on linear operators that rely on the completeness of these spaces and Baire’s Theorem: the Uniform Boundedness Theorem, the Open Mapping Theorem, and the Closed Graph Theorem. It details how the first one relates pointwise bounds to global bounds for families of operators, and how the other two results provide equivalences and characterizations for isomorphisms and the continuity of operators. Finally, the work investigates the compactness of closed balls. Since these balls are not compact under the norm topology in infinite-dimensional spaces, the weak and weak star topologies associated with families of functions are introduced. In this context, the text demonstrates the Banach-Alaoglu-Bourbaki Theorem, which ensures the compactness of the closed unit ball in the dual space under the weak star topology, and Kakutani’s Theorem, which establishes the equivalence between the reflexivity of a normed space and the compactness of its closed unit ball under the weak topology.