TMA4230 Functional analysis, Spring 2013

• Lecturer: Antoine Julien.

• Erwin Kreyszig: Introductory functional analysis with applications, ISBN 0471504599.
• Harald Hance-Olsen: Assorted notes on functional analysis.

Results are available in studentweb.

• Generalities on Banach spaces, completeness.
• The Hahn–Banach Theorem.
• Duality, adjoint operator, canonical mapping in the bi-dual.
• Uniform boundedness theorem, open mapping theorem, closed graph theorem.
• Strong, weak, weak-* convergence.
• Topologies on B(H): strong operator topology, weak operator topology, norm topology.
• Banach–Alaoglu theorem (note: this is not an exam of topology, so the Banach Alaoglu theorem has to be known, but the details of the proof, and namely the Tychonoff's theorem won't be the focus here).
• Spectrum of operators (it is acceptable to restrict to operators defined on a Hilbert space, even though we have seen in class a more general definition of the spectrum).
• Resolvant operator, resolvant set, spectral radius, spectral mapping theorem for polynomials.
• Self-adjoint operators, positive operators.
• Square root of a positive operator.
• Definition of the spectral family of a self-adjoint operator, spectral theorem.
• The theorem of continuous functional calculus (just the statement, the proof does not need to be known).

Office hours before the exam are as follows:

• Thursday 30th of May from 10:00 to 12:00;
• Friday 31th of May from 10:00 to 12:00;
• Monday 3rd of June from 10:00 to 12:00;
• Wednesday 5th of June from 10:00 to 12:00.

My office is 929 in Sentralbygg II.

The exam will take place on the 7th of June, in room 734, Sentralbygg II. Each student is scheduled on a 45 minutes time slot. The oral exam itself will take around 35 minutes; the remaining time is left for deciding the grade.

The exam will always start by a question on the material (example : "can you state Hahn-Banach theorem?", or "can you define weak convergence?"). Then, follow-up questions, which will still be quite close to the material (ex: "can you be more specific about theorem … when the spaces are Hilbert, and not just Banach?"). Then, you will be given an exercise.

• 8h15-9h00: Tadesse Duguma,
• 9h00-9h45: Susane Solem,
• 9h45-10h30: Marius Lie Winger,
• 10h45-11h30: Finn Idar Grøtta,
• 11h30-12h15: Kristoffer Varholm,
• 13h30-14h15: Michael Kyei,
• 14h15-15h:
• 15h15-16h: Chouduri Hafsa,
• 16h-16h45: Ailo Aasen.

Thursday 25th and Monday 29th of April will be review sessions. Here are a few exercise sheets on which we will work:

• Sheet 1, with solutions.
• Sheet 2, with solutions

First class is Monday, January 14th. I won't be here during the first week, but lectures will be given by Toke Meier Carlsen.

• Mondays 12:15 - 14:00 in B22.
• Thursdays 8:15 - 10:00 in KJL3.

• Mondays 15:15 - 16:00 in KJL3.

There will be no problem session on the first week of class (Monday, January 14th).

• Week 5 (January 28th): prepare exercises 4.1.2, 4.2.3, 4.2.5, 4.2.6, 4.2.10, 2.8.12+4.3.14.
• Week 6 (February 4th): prepare exercises 3.8.5, 3.8,6, 3.8.8, 3.9.3, 3.9.10, 3.10.4.
• Week 7 (February 11th): we will work on ex. 4.5.2, 4.5.9, 4.5.10, 4.6.4, 4.6.7, 4.6.10.
• Week 8 (February 18th): exercises 4.7.6, 4.7.8, 4.8.2, 4.8.5+4.8.6, 4.9.5.
• Week 9 (February 25th): exercises 4.12.5, 4.12.6, 4.12.8, 4.12.9 4.12.10, 4.13.11 and 4.13.13.
• Week 10 (March 4th): these exercises, with an erratum compared with the first version.
• Week 11 (March 11th): 7.1.10, 7.1.14, 7.1.15, and these additionnal exercises.
• Week 12 (March 18th): 7.2.5, 7.3.4, 7.3.5, 7.4.4, 7.4.9, 7.7.8.
• Week 15 (April 8th): 9.1.9, 9.3.8, 9.3.9, 9.3.10, 9.3.12, 9.4.4, plus this exercise.
• Week 16: 9.2.10, 9.4.8, 9.4.9, 9.5.10, 9.6.9, 9.6.10, 9.6.12

This course will be taught in English. The course text and supplementary material are also written in English.

• The official course description.
• A brief English–Norwegian dictionary made by Harald Hance-Olsen covering some much used terms.

This course is in many ways a continuation of the course TMA4145 Linear Methods. The main subjects are complete normed vector spaces and bounded (continuous) linear operators on normed vector spaces. Highlights of the course include the following:

• The Hahn-Banach theorem.
• The open mapping theorem.
• The closed graph theorem.
• The Banach-Steinhaus theorem (the uniform boundedness principle).
• Dual spaces.
• Weak convergence.
• The Banach-Alaoglu theorem.
• The spectral theorem for bounded self-adjoint operators.

You can read more about functional analysis on Wikipedia.

• Week 3: Reviews on Banach spaces, completeness, bounded operators (chap. 2, mainly).
• Week 4: Hahn Banach theorems (chap 4.1, 4.2, 4.3).
• Week 5: Riesz representation theorem (4.4), review on Hilbert spaces and operators on Hilbert spaces (chap. 3), adjoint operator.

If you have any questions concerning the course, you are welcome to send me an or stop by my office. Here is my contact information.