Part II, Functional Analysis
Lectures, F1-F7 (Tuesdays): Annemarie Luger, Stockholm University (luger-at-math-dot-su-dot-se)
Exercise sessions (Fridays, 3-5pm): Luca Sodomaco.
Topics:
Metric spaces; normed spaces; Banach spaces, including L^p spaces.
The Baire category theorem.
Bounded linear operators.
The Banach-Steinhaus theorem, the open mapping theorem, the closed graph theorem, the Hahn-Banach theorem. Hilbert spaces.
In this course we will in principle follow the coursebook [Friedman, chapters 3-4, 6.1-4], but I will also use other sources, mainly
- "Topics in Real Analysis", by Gerald Teschl, which is available online here:
https://www.mat.univie.ac.at/~gerald/ftp/book-ra/index.html - "Topics in Linear and Nonlinear Functional Analysis" by Gerald Teschl, which is available online here:
https://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html
More precise information will be given below in the file "homework".
Preliminary planning of the lectures:
10/10 Lecture F1: L^p spaces, metric spaces, completion. homework1 Download homework1
7/11 Lecture F2: The Baire category theorem. Compactness and its consequences. homework2 Download homework2
14/11 Lecture F3: Normed spaces. Banach spaces. Subspaces. homework3 Download homework3
21/11 Lecture F4: Bounded linear operators. The Banach-Steinhaus theorem (principle of uniform boundedness). homework4 Download homework4
28/11 Lecture F5: The open mapping and closed graph theorems. homework5 Download homework5
5/12 Lecture F6: The Hahn-Banach theorem and a bit about Hilbert spaces. homework6 Download homework6
12/12 Lecture F7: A bit more about Hilbert spaces and Applications. (we will look at some of the applications there) Notes on Applications
Download Notes on Applications
Homework assignments
As in the first part of the course there will be two sets of homework assignments, each passed one giving 1 p for the final exam (to part A). The assignments will be posted under Assignments in the left column (one week before the due date).
Assignment 3: Due 21 November
Assignment 4: Due 12 December
The assessment of the homework assignments is as in the first part. On each (of the two) homework assignment ONE of the problems is randomly chosen. This (and only this) problem is graded (the same problem for all students). If this problem is correctly solved, with a complete solution, one gets 1p.
Solutions to the problems will be presented/discussed at the exercise sessions.
Questions for the oral exam
As described in the section about the exam there is a voluntary oral part, which mainly concerns the Functional Analysis part of the course. There will be two questions selected from the list that you can find below. These serve as a starting point for a conversation about content of the course.