Analyse Fonctionnelle
Course Info
The lectures will be every Monday from 11:30-13:00 in Maison du Nombre room 1.050. The exercise sessions will be held by Nicolas on Fridays from 8:00-9:30 at the same place.
You can email me at nathaniel(dot)sagman(at)uni(dot)lu or nathaniels1729(at)gmail(dot)com. My office is MNO E06 0615-170. If you'd like to meet to discuss anything,
I will have designated office hours/meeting time on Mondays from 3-4, or
you can send me an email and we can arrange a meeting, or
come to my office at any time and if I'm not busy we can chat.
Resources on mental health at the university: see here and here.
Course catalogue description
The course will cover the following topics: Banach spaces and bounded linear functionals. Hanh-Banach theorem and Baire Category theorem. Uniform boundedness principle. Open mapping theorem. Closed graph theorem. Unbounded linear operators. Issues of classical compactness. Weak topology. Weak* topology. Reflexive and separable spaces. Lp spaces (reflexivity, separability, duality, strong compactness). Hilbert spaces and their duals. Theorems of Stampacchia and Lax-Milgram. Hilbert sums. Orthonormal bases. Compact operators. Riesz-Fredholm theory. Spectrum.
The more accurate course description
The main topics we plan to cover, roughly in order, are:
(Possibly review of) metric spaces, completeness, and compactness theorems in metric spaces (ex. Arzelà-Ascoli).
Normed spaces (ex. linear operators, the Weak* topology, Hanh Banach theorem)
Hilbert spaces (ex. orthonormal bases and projections, Riesz representation theorem).
Banach spaces (ex. Baire Category theorem, Uniform boundedness principle, open mappings and closed graphs).
We will put an emphasis on examples. Time permitting, we will discuss more advanced topics
For assessments, there will be a) small homework sets every 2 weeks and b) a midterm test on April 15, and c) a final exam, some time in June. The grading scheme will be:
25% homework sets (each weighted equally).
25% midterm test.
50% final exam.
Course Materials
Homeworks
Homework set 1. Due date: Friday, March 8. Problems 1-3 are doable after Lecture 2 (February 26), while Problem 4 is doable after Lecture 3 (March 4). Solutions.
Homework set 2. Due date: Friday, March 22. Solutions.
Homework set 3. Due date: Friday, April 19. Solutions.
Homework set 4. Due date: Monday, May 13 (there are no classes on May 10). Solutions.
Homework set 5. Due date: Friday, May 30. Solutions.
Lecture notes
Course notes. Beware of typos. For future instructors: if you want access to the tex file, just contact me.
Laurent Loosveldt's previous course notes (in French).
Literature
I do not plan to follow one text, and will mainly draw from the sources of course notes above, together with some sections from the course notes for similar material taught at McGill University, found here and here.
Standard textbooks that might be helpful:
Brezis, Functional Analysis (very nice, and connects to applications).
Rudin, Functional Analysis (terse, but quite good).
Lax, Functional Analysis (I've never looked at it, but it seems nice).
A basic reference for background material could be Rudin's Principles of Mathematical Analysis.
For a historical perspective, from which you would certainly learn a lot (but is not necessary for our course), I recommend Dieudonné's History of Functional Analysis.
Exercise sheets (for Friday sessions)
Exercise sheet 5. Parts 1 and 2 were covered in the session, and part 3 will be covered next time.
Exercise sheet 8. This includes the solutions to most of the midterm problems.