Part II, Functional Analysis

For schedule, click here Links to an external site..

Lectures, 8-14 (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

More precise information will be given below in the file "homework". 

Preliminary planning of the lectures:

11/10  Lecture 1: L^p spaces, metric spaces, completion.  homework1 Download homework1 (updated 7/10)

8/11 Lecture 2: The Baire category theorem. Compactness and its consequences. homework2 Download homework2

15/11 Lecture 10: Normed spaces. Banach spaces. Subspaces. homework3 Download homework3

22/11 Lecture 11: Bounded linear operators. The Banach-Steinhaus theorem (principle of uniform boundedness). homework4 Download homework4

29/11 Lecture 12: The open mapping and closed graph theorems.  homework5 Download homework5

6/12 Lecture 13: The Hahn-Banach theorem and a bit about Hilbert spaces. homework6 Download homework6

13/12 Lecture 14: A bit more about Hilbert spaces and Applications. homework7 Download homework7    notes Download notes  (we will look at some of the applications there)


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 November 15.

Assignment 4: Due December 13.

The assessment of the homework assignements is as follows. 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.