Part II, Functional Analysis

Lectures, 8-14 (Fridays 3-5pm, on Campus): Alan Sola, Stockholm University (sola-at-math-dot-su-dot-se)

Exercise sessions (Wednesdays, 3-5pm): Scott Mason. 

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.

The material above is covered in chapters 3-4 in Friedman's book. See below for more detailed reading instructions.

Tentative schedule:

Lecture 8: L^p spaces and \ell^p spaces. Completeness of L^p (1\leq p \leq \infty).

                  Friedman, section 3.2.

                  Room Q15.

                  Notes: ARAIIl8.pdf Download ARAIIl8.pdf 

Lecture 9: The Baire category theorem. Compactness and its consequences.

                 Friedman, sections 3.4-3.5.

                 Room E32.

                Notes: ARAIl9.pdf Download ARAIl9.pdf 

Lecture 10: Normed spaces. Banach spaces. Subspaces.

                 Friedman, sections 4.1-4.3.

                Room D34.   

                Notes: ARAIl10.pdf Download ARAIl10.pdf 

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

                Friedman, sections 4.4-4.5

                Room Q15.

                Notes: ARAIl11.pdf Download ARAIl11.pdf 

Lecture 12: The open mapping and closed graph theorems.

                Friedman, section 4.6.

                Room E32.

                Notes: ARAIl12.pdf Download ARAIl12.pdf 

Lecture 13: The Hahn-Banach theorem and its consequences.

                Friedman, section 4.8

                Room E32.

                Notes: ARAIl13.pdf Download ARAIl13.pdf 

Lecture 14: Introduction to Hilbert spaces.

               Friedman, sections 6.1-6.2, 6.4

               Room D41.

               Notes: ARAIl14.pdf Download ARAIl14.pdf 

I intend to follow Friedman's book fairly closely, but in certain cases I may give alternative or modified proofs and include additional material. In the event of substantial deviations, I will post notes here or point you to appropriate sources.

Homework assignments

There will be two sets of homework assignments for this part of the course as well. By successfully completing a HW, you will earn 1 p on the final exam (to part A). The assignments will be posted here (one week before the due date).

Assignment 3: ARAIHW3-1.pdf Download ARAIHW3-1.pdf 

Assignment 4: ARAIHW4.pdf Download ARAIHW4.pdf 

The assessment of the homework assignments is as follows. On each (of the two) homework assignment ONE of the problems is picked at random; the same problem is selected for all students. This (and only this) problem is then graded: if it is correctly solved, with a complete solution, one gets 1p.

Solutions to the problems will be presented/discussed at the exercise sessions.