TMA4230 Functional analysis, Spring 2012

Teachers

Literature

Schedule

We meet first time Tuesday 10 January and continue according to the following schedule.

Lectures

  • Monday 15:15 - 17:00 in KJL21.
  • Friday 12:15 - 14:00 in R4.

Problem session

  • Monday 14:15 - 15:00 in 1329.

Language

This course will be taught in English. The course text and supplementary material are also written in English. The exam will be in English or Norwegian at the choice of the student.

The examination

The exam is oral and will take place 22nd of may (we have changed the day) at room 656. One hour before your examination time you will pick randomly one of the above 9 exam, and then you have one hour to prepare it. It will be a blackboard presentation of the selected topic of about 30 minutes. You can give additional information and examples that complements your exposition. We don’t ask to give proofs of the theorems but at least you should prove that you understand all the ingredients that take part of it. After the exposition we can ask for further explanations or ask general questions of the subject. During the exposition you are allowed to bring some short paper notes.

Please during the next week tell me your preferences to the time you would prefer to do the exam. The starting time will be 9 am. We will make a break from 12 to 13 and then continue the whole afernoon.

You can make the exposition in English or Norwegian.

The order of the oral exam for TMA4230 is the following:

timestudent
9.00 - 10.00 Nikolai Werstad
10.00 - 11.00 Stein-Olav Davidsen
11.00 - 12.00 Ole Frederik Brevig
13.00 - 14.00 Mathias Nikolai Arnesen
14.00 - 15.00 Jørgen Endal
15.00 - 16.00
16.00 - 17.00 Espen
17.00 - 18.00 Daniel Wennberg

You should come to the room 656 one hour before to pick your topic.

Syllabus

You are expected to know and understand the contents of Section 4.1-4.9 and Section 4.12-4.13, Chapter 7 and Chapter 9 of Introductory functional analysis with applications by Erwin Kreyszig in addition to pp. 32-43 (excluding the section “Normal spaces and the existence of real continuous functions”), 52-54 (only the section “The Banach-Alaoglu theorem”) and 61-65 (excluding the section “Holomorphic functional calculus”) of Harald Hanche-Olsen's notes Assorted notes on functional analysis and the note about the Stone-Weierstrass Theorem I will hand out in class.

Contents of the course

Semester plan

WeekDatesSubjectsReferencesExercisesWeeklySolutions to exercises
2Jan 9-13Introduction/Review of TMA4145Chapter 1-3None week 1
3Jan 16-20Zorn's Lemma, Hahn-Banach theoremsSection 4.1-4.32.7.2, 2.8.9, 2.10.8, 2.10.13, 3.10.3, 3.10.4 week 2
4Jan 22-27Bounded linear functionals on C[a,b], Riesz's representations theorem, Hilbert-adjoint operatorsSection 4.4 + 3.8-3.9 4.1.2, 4.2.3, 4.2.5, 4.2.6, 4.2.10 and 2.8.12+4.3.14week 3 solution3.pdf
5Jan 30 - Feb 3Adjoint operators, reflexives spaces, Baire's category theorem, uniform boundedness theoremSection 4.5-4.73.8.5, 3.8,6, 3.8.8, 3.9.3, 3.9.10, 3.10.4 plus this exercise.week 4 solution4.pdf
6Feb 6-10Strong and weak convergence, convergence of sequences of operators and functionals, open mapping theoremSection 4.8-4.9 + 4.124.5.2, 4.5.9, 4.5.10, 4.6.4 and 4.6.7 plus two extra exercises week 5 solution5.pdf
7Feb 11-15Closed linear operators, closed graph theorem, topologySection 4.13 + page 32-39 of the notes 4.7.6, 4.8.1, 4.9.3 and 4.9.6 plus one extra exercise which can be found here week 6. week 6 solution6.pdf
8Feb 20-24Topology, Compactness, Tychonoff ’s theoremPage 39-43 + 52-54 of the notes4.12.5, 4.12.6, 4.12.8, 4.12.9 4.12.10, 4.13.11 and 4.13.14week 7solution7.pdf
9Feb 27 - Mar 2Banach-Alaoglu theorem, Stone-Weierstrass theorem, an application of Banach-Alaoglu's theorem to PDEs note on an application of Banach-Alaoglu's theorem to PDEs, section 7.1, 7.2 Notes on the Stone-Weierstrass theorem, note on an application of Banach-Alaoglu's theorem to PDEs, section 7.1, 7.24 exercises that can be found here week 7. week 8 solution8.pdf
10Mar 7-11spectral theory in finite dimensions and basic concepts of spectral theory. Spectral theory for Banach algebrasSection 7.3-7.7 + page 61–65 of the notes 7.1.10 ,7.1.15, 7.2.3 and 7.2.6 plus one extra exercise which can be found here week 8. week 9
11Mar 11-16Spectral theory for Banach algebras, spectral properties of bounded self-adjoint linear operatorsSection 7.3-7.7 + page 61–65 of the notes, Section 9.1-9.2 7.3.4-6, 7.4.4, 7.5.1, 7.7.4 and 7.7.5 week 10
12Mar 19-23Spectral theory for Banach algebras, spectral properties of bounded self-adjoint linear operatorsSection 7.3-7.7 + page 61–65 of the notes, Section 9.1-9.2 7.3.9, 7.4.8, 7.4.9, 7.5.5, 7.5.9, 7.6.3 and 7.7.7. week 11
13Mar 26-30 Spectral theory for bounded self-adjoint linear operators on Hilbert spaces Section 9.2-9.4 7.5.5, 7.5.9, 7.6.3, 7.7.7, 9.1.6, 9.2.9, 9.3.2, 9.3.9+10 and 9.3.11 week 12
14Apr 10-13 Projections on Hilbert spaces week 13

Questions

If you have any questions concerning the course, you are welcome to send me an email or stop by my office. You can find my contact information here.

2014-06-10, Stian Tamlagsrønning