# TMA4230 Functional analysis, Spring 2012

## Teachers

- Lecturer: Eduard Ortega.
- Teaching assistent: Eduard Ortega.

## Literature

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

## Schedule

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

### Lectures

### 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.

## Links

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

## 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:

time | student |
---|---|

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

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:

You can read more about functional analysis on Wikipedia.

## Semester plan

Week | Dates | Subjects | References | Exercises | Weekly | Solutions to exercises |
---|---|---|---|---|---|---|

2 | Jan 9-13 | Introduction/Review of TMA4145 | Chapter 1-3 | None | week 1 | |

3 | Jan 16-20 | Zorn's Lemma, Hahn-Banach theorems | Section 4.1-4.3 | 2.7.2, 2.8.9, 2.10.8, 2.10.13, 3.10.3, 3.10.4 | week 2 | |

4 | Jan 22-27 | Bounded linear functionals on C[a,b], Riesz's representations theorem, Hilbert-adjoint operators | Section 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+4.3.14 | week 3 | solution3.pdf |

5 | Jan 30 - Feb 3 | Adjoint operators, reflexives spaces, Baire's category theorem, uniform boundedness theorem | Section 4.5-4.7 | 3.8.5, 3.8,6, 3.8.8, 3.9.3, 3.9.10, 3.10.4 plus this exercise. | week 4 | solution4.pdf |

6 | Feb 6-10 | Strong and weak convergence, convergence of sequences of operators and functionals, open mapping theorem | Section 4.8-4.9 + 4.12 | 4.5.2, 4.5.9, 4.5.10, 4.6.4 and 4.6.7 plus two extra exercises | week 5 | solution5.pdf |

7 | Feb 11-15 | Closed linear operators, closed graph theorem, topology | Section 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 |

8 | Feb 20-24 | Topology, Compactness, Tychonoff ’s theorem | Page 39-43 + 52-54 of the notes | 4.12.5, 4.12.6, 4.12.8, 4.12.9 4.12.10, 4.13.11 and 4.13.14 | week 7 | solution7.pdf |

9 | Feb 27 - Mar 2 | Banach-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, 7.2 | 4 exercises that can be found here week 7. | week 8 | solution8.pdf |

10 | Mar 7-11 | spectral theory in finite dimensions and basic concepts of spectral theory. Spectral theory for Banach algebras | Section 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 | |

11 | Mar 11-16 | Spectral theory for Banach algebras, spectral properties of bounded self-adjoint linear operators | Section 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 | |

12 | Mar 19-23 | Spectral theory for Banach algebras, spectral properties of bounded self-adjoint linear operators | Section 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 | |

13 | Mar 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 | |

14 | Apr 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.