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 may bring to the exam 2 sheets of paper (front and back) with whatever material they wish. To pass the course, it is necessary to have at least 8 out of 20 in the exam. 

• Four “Homework assignments” and other continuous evalutaion (50%). Homeworks will be assigned at the end of 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.

Teaching and assesment  style and philosophy: I will use my tablet to write the lessons, putting the resulting files in this webpage. Every week there will be time to solve exercises (tipically on thursdays). Students are asked to participate activelly in all classes. Tipically, theoretical results will be illustrated with many examples. The homeworks are an important part in understanding the material: they will contain not only routine but also hard and deep exercises; I will correct and grade the homeworks, returning the correction. The exercises of the homeworks are different from the ones in the exercise lists. If necessary, I will ask during office hours to present, clarify or explain part of your homework.

Context and objectives:

Check out this file 

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").

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Program

1. 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)

2. 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)

3. Spectral theory: general notions, compact operators, Fredholm's alternative, spectral decomposition of compact self-adjoint operators in Hilbert spaces. (~2-3 weeks weeks)

4. 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. (4 weeks)

5. Weak topologies and Banach-Alaoglu's theorem. (2 week)

6. (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.

[4] H. Tavares, lecture notes (under construction).

Secondary bibliography

[5] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, 1972

[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], 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). I will (try to) write some lecture notes along the year.

All other references are complementary. Number [5] 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).

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).