# TMA4230 Functional analysis, Spring 2011

## Teachers

- Lecturer: Toke Meier Carlsen.
- Teaching assistent: Tron Ånen Omland.

## Literature

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

## Schedule

We meet first time Thursday 13 January and continue according to the following schedule.

### Lectures

### Problem session

- Tuesday 14:15 - 15:00 in F4.

The last lecture will take place Friday 15 April.

## 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 Hance-Olsen covering some much used terms.

## The examination

The exam is oral and takes place Wednesday June 1 in room 734 and 822 in Sentralbygg 2. One hour, which includes time for the examiners to discuss the grade etc., will be scheduled for each student, so you should expect that your examination last for a maximum of 45 minutes. During these 45 minutes you will be asked questions from the syllabus (see below). You are not expected to remember every little details of every proof, but you should be able to state the main theorems and important definitions, give examples of applications of the main theorems and other important concepts, and tell something about the proofs of the main theorems. The schedule is as follows

Time | Name | Room |
---|---|---|

08:00 | Sigurd Storve, Christian G Frugone | 734 |

10:00 | Gizat Derebe Amare, Espen Sande | 822 |

13:00 | Anders Samuelsen Nordli, Karl Kristian Brustad | 822 |

15:00 | Axel Byberg Fosse, Erik Korsnes, Kalliopi Paolina Koutsaki | 822 |

You should be at the assigned room no later than at the time assigned to you. As you can see, two or more students are assigned for each time. This is to take in account that some might not show up.

## 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 Hance-Olsen's notes Assorted notes on functional analysis and the note about the Stone-Weierstrass Theorem I handed out in class (if you do not have this note, send me an email and I will then send you a pdf-file with the note, or see me in class or at my office if you prefer a paper version).

## 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 10-14 | Introduction/Review of TMA4145 | Chapter 1-3 | None | week1.pdf | |

3 | Jan 17-21 | 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 | week2.pdf | solution2.pdf |

4 | Jan 24-28 | 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 | week3.pdf | solution3.pdf |

5 | Jan 31 - Feb 4 | 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. | week4.pdf | solution4.pdf |

6 | Feb 7-11 | 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 which can be found here week3.pdf. | week5.pdf | solution5.pdf |

7 | Feb 14-18 | 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 week5.pdf. | week6.pdf | solution6.pdf |

8 | Feb 21-25 | Compactness, Tychonoff ’s theorem, Banach-Alaoglu 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 | week7.pdf | solution7.pdf |

9 | Feb 28 - Mar 4 | Stone-Weierstrass theorem, an application of Banach-Alaoglu's theorem to PDEs, spectral theory in finite dimensions and basic concepts of spectral theory | 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 week7.pdf. | week8.pdf | solution8.pdf |

10 | Mar 7-11 | 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 week8.pdf. | week9.pdf | solution9.pdf |

11 | Mar 14-18 | 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 | week10.pdf | solution10.pdf |

12 | Mar 21-25 | Spectral properties of bounded self-adjoint linear operators, positive operators | Section 9.2-9.3 | 7.3.9, 7.4.8, 7.4.9, 7.5.5, 7.5.9, 7.6.3 and 7.7.7. | week11.pdf | solution11.pdf |

13 | Mar 28 - Apr 1 | Square roots of a positive operators, projection operators, spectral families, spectral family of a bounded self-adjoint linear operator | Section 9.4-9.8 | 9.1.6, 9.2.9, 9.3.2, 9.3.9+10 and 9.3.11 | week12.pdf | solution12.pdf |

14 | Apr 4-8 | Introduction to C*-algebras and graph C*-algebras, guest lecture by Takeshi Katsura about semiprojectivity and graph C*-algebras | Slides | 9.4.8+9, 9.5.1, 9.5.3, 9.6.10+12+13 | week13.pdf | solution13.pdf |

15 | Apr 11-15 | Spectral family of a bounded self-adjoint linear operator, spectral representation of a bounded self-adjoint linear operator, extension of the spectral theorem to continuous functions, properties of the spectral family of a bounded self-adjoint linear operator | Section 9.8-9.11 | 9.8.1-9.8.4 plus two extra exercises which can be found here week13.pdf. | week14.pdf | solution14.pdf |

Unless mentioned otherwise the references above are to Kreyszig's book. "The notes" are Harald Hance-Olsen's notes Assorted notes on functional analysis.This plan is tentative and can (and probably will) be changed during the semester.

## 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.