Functional Analysis

Recommended Prerequisites

Linear Algebra, Calculus, Mathematical Analysis, Complex Analysis, and Fourier Analysis.

Teaching Methods

Lectures have an expository character, being up to the professor the choice of the most appropriate way to do it and the degree of participation of the students. As an integral part of the learning process, it may be recommended or required the solution of exercises, course projects or oral presentations.

Learning Outcomes

The goal of this course is to provide a general knowledge of Functional Analysis and Measure Theory at an advanced level. It is intended that the students become familiar with the main basic techniques and results of this area of Mathematics, or at least attain enough familiarity with some of  them to be able to acquire others by themselves that may later prove to be useful.

Work Placement(s)

No

Syllabus

Complete metric spaces. Fixed point theorem. Theorem of  embedded balls. Baire theorem. Compactness: equivalent definitions. Arzela-Ascoli theorem. Normed linear spaces and applications: Banach spaces. Spaces of continuous linear applications. Dual space. Cauchy-Schwarz inequality. Parallelogram law. Orthogonalization. Hilbert spaces. Fourier series. Bessel's inequality. Parseval equality. Riesz-Fisher theorem. Hahn-Banach theorem. Minkowski function. Separation theorems. General form of linear functionals: functional in spaces of sequences; in Hilbert spaces. Riesz theorem. Weak convergence in a normed space and its dual. Banach-Steinhaus theorem. Compactness in the dual space. Topological vector spaces: Theorem of Kolmogorov. Weak topologies. Banach theorem of the inverse. Symmetric compact applications: Hilbert's Theorem. Fredholm theory. 

Head Lecturer(s)

Semyon Yakubovich

Assessment Methods

Assessment
Exam: 100.0%

Bibliography

A.N. Kolmogorov and S.V. Fomin, Elementos da Teoria das Funções e de Análise Funcional, Mir, Moscou, 1982.

J-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, New York, 1984.