Professor Hugo Tavares, gabinete 5.44, Pavilhão de Matemática, IST
Horário: Terça-feira, 9:00-10:30 (sala V1.12) e Quinta-feira 11:30-13:30 (sala VA.6).
Horário de atendimento: após cada aula, ou noutro horário (combinando com o professor diretamente)
Métodos de avaliação:
A nota final resulta dos seguintes elementos:
Exame final (50%). Os alunos poderão trazer duas folhas de papel (frente e verso) com o material que quiserem. Para serem aprovados à disciplina, é necessário ter pelo menos 8 valores no exame.
Quatro trabalhos de casa (40%) e outra avaliação contínua (10%).
Os enunciados dos TPCs serão entregues nas semanas 4, 7, 10 e 13 do semestre, com um prazo de entrega de 2 semanas.
Os 10% de avaliação contínua (0, 1 ou 2 valores) terão em conta a participação dos alunos nas aulas, resolução de exercícios das listas, bem como a interação professor-aluno após a entrega dos trabalhos de casa.
Trabalhadores-estudantes: as mesmas regras.
Época especial: O máximo entre a nota desse exame e a nota obtida com a fórmula descrita acima.
Estilo e filosofia de ensino: Haverá tempo para resolver exercícios quase todas as semanas (tipicamente às quintas-feiras); os exercícios a resolver serão indicados na semana anterior. A participação ativa dos alunos nas será fomentada em todas as aulas. Sendo Análise Funcional uma cadeira abstrata e teórica, tipicamente apresentarei vários exemplos dos conceitos.
Os trabalhos de casa são uma parte muito importante na compreensão ativa do material: vão conter tanto exercícios de rotina como outros exercícios mais difíceis e profundos (que não caberiam num exame). Irei sempre devolver as correções aos alunos, com sugestões de melhoria. Os exercícios dos TPCs serão diferentes dos exercícios das listas de exercícios. Se necessário, após a devolução da correção do trabalho de casa, pedirei que me clarifiquem ou expliquem partes dos trabalhos de casa.
Slides da primeira aula: aqui
Quadros das Aulas:
Aula 1 (quinta-feira, 19/02/2026)
Context and objectives:
Some history: If you want to know more about the history of Functional Analysis, check out this. For more specific and extensive material, you may want to check some parts of this book.
A nice chapter, in Portuguese, relating Functional Analysis (more specifically operator theory in Hilbert spaces) and Quantum Mechanics can be found in the introduction of this book (entitled "Nota histórica e considerações acerca da motivação física de alguns dos temas fundamentais da Análise Funcional"), or Chapter 11 of Kreyszig's book (entitled "Unbounded Linear Operators in Quantum Mechanics"). After taking Functional Analysis, you are ready to read the very nice book by Brian C. Hall, Quantum theory for Mathematicians.
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Program
Normed and Banach spaces. Linear operators. Generalities and review of classical material. Completion of a normed space. Compactness, density. Separable spaces. Hamel and Schauder basis. Linear and bounded operators. Duals. Examples, including Lebesgue and one-dimensional Sobolev spaces (~2 weeks).
Hilbert spaces. Inner product and Hilbert spaces. projections on closed convex sets and on closed sublinear spaces; orthogonal projections. The Riesz-Fréchet representation theorem. Hilbert sums and Hilbert basis; relation with Fourier series, full characterization of separable Hilbert spaces. The Hilbert adjoint operator. Self-adjoint, unitary and normal operators. (~3 weeks)
Compact operators and Spectral theory. Compact operators between Banach spaces. The approximation problem by finite rank operators. Ascoli-Arzela's theorem. Spectral theory: spectrum, point spectrum and resolvant of an operator; Fredholm's alternative, spectral decomposition of compact self-adjoint operators in Hilbert spaces. (2-3 weeks)
The four pilars of Functional Analysis. Hahn-Banach theorem: analytic and geometric forms. Applications: the adjoint of a bounded operator, the bidual of a normed space; reflexive spaces. Baire Category theorem. Uniform Boundedness Principle/Banach Steinhaus Theorem and applications to Fourier series and topological complements. Open and closed operators. The Open mapping theorem and the Closed graph Theorem. (4 weeks)
Weak topologies. The weak and weak-* topology: characterization and main properties. Banach-Alaoglu's theorem. (2 week)
Possible extra topics: Lax-Milgram theorem and examples of application. Unbounded operators.
Main bibliography
[1] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer 2011.
[2] J. B. Conway, A Course in Functional Analysis, Springer 2007.
[3] E. Kreyszig, Introductory Functional Analysis with Applications, Wiley 1978.
Secondary bibliography
[4] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, 1972.
[5] B. Rynne and M. Youngson, Linear Funcional Analysis, Springer 2008, 2nd edition.
[6] E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, Springer 1995.
[7] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Springer 1995.
Commented bibliography
In this course I will mostly follow [3], a masterpiece which provides an excellent and detailed introduction to Functional Analysis and where concepts are motivated and introduced in a very pedagogical way. However, it is a book that never uses measure theory nor topology. Therefore, it will be complemented with [1] (which is a bit oriented towards Partial Differential Equations) and [2] (the most mathematically advanced book of the main bibliography). I will (try to) write some lecture notes along the year.
All other references are complementary. Number [4] is a classic, the first volume of a collection oriented towards Mathematical Physics, and [6,7] treat Functional Analysis and its applications in an extensively way (treating, in particular, nonlinear problems). Reference [5] has a nice and simple chapter on the Lebesgue integral.
Recommended reading for those who don't know what is Lebesgue integration:
In this course we will see many examples. Some of them will use Lebesgue spaces, which I will recall briefly in a (possibly extra) class. It is not a requirement to know a priori Lebesgue integration, and I had students in the past in those conditions. In any case, I recommend taking a look at pages 20-30 of this book (pdf freely available), to complement the classes.
You may also check these handwritten notes for a selfcontained construction of the Riemann and Lebesgue integral in R^n (which do not pass through the definition of sigma-algebras, measures, etc).