Real and Functional Analysis

Recommended Prerequisites

Basic courses in Differential and Integral Calculus; Topology and Linear Analysis.

Teaching Methods

The classes are essentially of expository nature and should include examples and exercises that lead the students to understanding and applying the material being taught.  There must be room for the presentation of more elaborated exercises and more detailed constructions of concrete examples. The classes should be focused on the teaching of the reasoning processes, so that the students learn how to manipulate the objects presented along the course and more easily find out by themselves how to reach other results by independent reading or problem solving.

Learning Outcomes

The main goal of the course is to introduce the basic tools from Measure Theory and Functional Analysis, including the Lebesgue integral and the L^p spaces, as well as the main results on weak topologies and operator theory, namely in the framework of the theory of compact operators in Hilbert spaces. The relevance of the results presented in terms of the applications should be emphasized, especially in connection with Partial Differential Equations and Approximation Theory.

The course aims at developing the following skills: knowledge of mathematical results; ability to generalization and abstraction; logic reasoning;   rigorous oral and writing skills; computational ability. On the personal level it also allows to develop self-learning skills and independent thinking.

Work Placement(s)

No

Syllabus

Measures and measurable functions. Lesbesgue integral. Fundamental theorems: Fatou’s lemma, monotone convergence, dominated convergence and Fubini theorems. Differentiation of Radon measures. Radon-Nikodym theorem and Lebesgue decomposition theorem. Riesz representation theorem.

Hann-Banach, Banach-Steinhaus, open mapping and closed graph theorems (review and complements).

Weak topologies. Banach-Alaoglu theorem. Reflexive spaces. Separable spaces. Uniformly convex spaces.

L^p spaces: reflexivity, duality and separability. Convolution and regularization.

Hilbert spaces: projections, duality, Stampacchia and Lax-Milgram theorems.

Compact operators: Riesz-Fredholm theory; spectrum; spectral decompostion of compact self-adjoint operators.

Head Lecturer(s)

Júlio Severino das Neves

Assessment Methods

Final assessment
Exam: 100.0%

Continuous assessment
Evaluation during the semester includes one or more mid-term exams (75-100%) and presentations, either of proofs of more elaborate results or the solving of problems proposed in the form of homework sets (0-25%): 100.0%

Bibliography

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011(H. Brézis, Analyse Fonctionnelle, Masson, 1983).

L.C. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.

P. Fernandéz, Medida e Integração, Instituto de Matemática Pura e Aplicada, CNPq, 1976 (Projeto Euclides, nº2).

P.D. Lax, Functional Analysis, John Wiley and Sons, 2002.

M. Reed, B. Simon, Methods of Modern Mathematical Physics. Volume I: Functional Analysis, Academic Press, 1980.

L.C. Evans, Partial Differential Equations, American Mathematical Society, terceira edição, 1998.