Tópicos de análise funcional (vias clássica e não-standard)

This thesis concerns aspects of the functional analysis from both the classical and the nonstandard viewpoints. Nowadays functional analysis plays an increasing role in the applied sciences as well as in mathematics itself. Besides deepening the study of the subject itself, from a classical point of view, we took the opportunity to analyse some questions and try to make some proofs by the methods of nonstandard analysis, by applying some notions and techniques learned in the Master of Science in Applied Mathematics course on Nonstandard Analysis. We begin by presenting the Nonstandard Analysis axiomatic foundations (Nelson’s axiomatics), make some immediate applications to the real line and give nonstandard (or external) characterizations of several notions, namely the shadow of a real number and the limit of a sequence of real numbers. In order to study the spectral theory we also present some topological notions needed in a nonstandard framework