Professor Hugo Tavares, office 5.44, Pavilhão de Matemática, IST
Schedule: Tuesdays, 9:00-10:30 (room V1.12) and Thursdays 11:30-13:30 (room VA.6).
Office hours: immediately after each lesson, or in a different schedule (contact directly the professor)
Assesment methods:
The final grade is a combination of the following elements:
Final exam (50%). The students should have at least 8 out of 20 in the exam. *The students may bring to the exam 2 sheets of paper (front and back) with whatever material they wish.*
Four “Homework assignments” (50%). They will be assigned on weeks 4, 7, 10 and 13, with a deadline of two weeks.
Trabalhadores-estudantes: the same rules apply.
Época especial: The maximum between the grade of that exam and the grade obtained with the previous formula.
Slides of the first lecture: here
Exams:
Pautas do exame e nota da 1ª época
Exercise sheets:
Written assignments
Homework 1. Deadline: March 27, 2025, at 11:30. Grades!
Homework 2. Deadline: April 16, 2025, at 20:00. Grades.
Homework 3. Deadline: May 29, 2025, at 20:00 Grades
Homework 4. Deadline: June 20, 2025, at 20:00 Grades
To read:
Rationale of the homeworks: The exercises were thought to help you follow and consolidate the material of the course and to see some applications; some exercises would not fit in a regular exam. I will correct your solutions with my tablet and return them to you with corrections/remarks.
Do's and Don'ts: It is ok to look for information, but you have to write your own solutions. You can (and should) use the office hours to discuss the homework: I will give you hints and try to help you in case you are stuck in some part. On the other hand, it is not acceptable to copy nor to discuss detailed solutions with one another; this will not help you to make progress.
Recommended reading for those who don't know what is Lebesgue integration:
Read pages 20-30 of this book (pdf freely available)
You may also check these handwritten notes for a selfcontained construction of the Riemann and Lebesgue integral in R^n (without passing through the definition of sigma-algebras, measures, etc).
Handwritten Notes from the classes:
** NEW: all lessons in just one file **
Lesson #2 - 2nd half (February 19, 2025)
Lesson #3 (February 25, 2025)
Lesson #4 (February 28, 2025)
Lesson #5 (March 6, 2025)
Lesson #6 (March 7, 2025). Video of the lesson.
Lesson #7 (March 11, 2025)
Lesson #8 (March 13, 2025)
Lesson #9 (March 18, 2025)
Lesson #10 (March 20, 2025)
Lesson #11 (March 25, 2025)
Lesson #12 (March 27, 2025)
Lesson #13 (April 1, 2025)
Lesson #14 (April 3, 2025)
Lesson #15 (April 8, 2025): theory (with video) and exercises (with video).
Lesson #16 (April 29, 2025)
Lesson #17 (May 6, 2025)
Lesson #18 (May 8, 2025)
Lesson #19 (May 13, 2025)
Lesson #20 (May 15, 2025). Extra: See also this link.
Lesson #21 (May 23, 2025)
Lesson #22 (May 27, 2025)
Lesson #23 (May 29, 2025)
Lesson #24 (June 2, 2025), with video here
Lesson #25 (June 3, 2025): blackboard only - exercises.
Lesson #26 (June 5, 2025)
Lesson #27 (June 12, 2025)
** NEW: all lessons in just one file **
Context and objectives:
Check out this file (updated Feb 18)
Some history: If you want to know more about the history of Functional Analysis, check out this or 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").
.
Program and tentative planification
Normed and Banach spaces: generalities and review of classical material. Examples, among which Lebesgue and first order Sobolev spaces. Linear and bounded operators. (~2 weeks)
Inner product and Hilbert spaces, projections on closed convex sets and orthogonal projections, duals - Riesz-Fréchet representation theorem, Lax-Milgram and examples of application. Hilbert sums and Hilbert bases; relation with Fourier series. The Hilbert adjoint. (~3 weeks)
Spectral theory: general notions, compact operators, Fredholm's alternative, spectral decomposition of compact self-adjoint operators in Hilbert spaces. (~2 weeks)
The four pilars of Functional Analysis: Hahn-Banach theorem, Uniform Closedness/Banach Steinhaus theorem, Open mapping theorem and the Closed graph Theorem. The adjoint of a bounded operator. (6 weeks)
Weak topologies and Banach-Alaoglu's theorem. (1 week)
(If time allows/opcional) 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] E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, Springer 1995.
[6] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Springer 1995.
Commented bibliography
In this course I will mostly follow [3], which is a masterpiece, as it is an excellent and detailed introduction to Functional Analysis, in which the concepts are very well motivated and introduced in a 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 three main ones).
All other references are complementary. Number [4] is a classic, the first volume of a collection oriented towards Mathematical Physics, and [5,6] treat extensively Functional Analysis (in particular, nonlinear problems) and its application.