Kaisa_2012_3_photo by Veikko Somerpuro

Introduction to linear functional analysis

NB: Check out the details about the 2nd course exam below.

The goal of the course is to introduce fundamental concepts and techniques of linear functional analysis. We focus on the most common complete normed spaces, Banach and Hilbert spaces, and linear operators between them. The course provides the background to continue with applications and analysis involving infinite-dimensional Banach spaces such as sequence and function spaces.

The main reference for the course are the previous lecture notes which can be downloaded from the Materials section below. The notes are available only in Finnish but English textbooks and references will be suggested there, as well.

## Jani Lukkarinen

Published, 27.6.2019 at 19:27

1) In addition to standard writing equipment, you are allowed to bring in and consult *one handwritten* two-sided A4 sheet of personal notes during the exam. (as in partial exams)
2) The grading will be completed within 30 days but usually within 2 weeks.
3) Once you see your grade in Weboodi, you can contact the lecturer to check the detailed grading of the problems.

## Jani Lukkarinen

Published, 19.2.2019 at 20:21

The final and partial exams on 6.2.2019 have know been graded and sent to be registered in Oodi on 19.2. The participants may check the detailed grading by contacting the lecturer, e.g., during the office hours.

## Jani Lukkarinen

Published, 22.12.2018 at 16:48

Results from the course are now available under the Results section below (available for registered participants: you will need to login to read the PDF file).

## Jani Lukkarinen

Published, 12.12.2018 at 16:48

The second course exam will be on Tue 18.12. at *13:30*-16:00 in Exactum, D123. In addition to the Exercises 7-12, the material covered by the exam is listed below:

[Lecture notes (in Finnish)] Chapters 5-9.22 (latest version below, excluding sections marked with a "*")
[Kreyszig (in English)] The following parts of Chapters 2, 3, 4 and 7: Sections 2.6-2.10, 3.4-3.6, 3.8-3.10, 4.1-4.3, 4.7, 4.12-4.13, 7.6-7.7.

In addition to standard writing equipment, you are allowed to bring in and consult *one handwritten* two-sided A4 sheet of personal notes during the exam. (In particular, no textbooks or full lecture notes.)

## Jani Lukkarinen

Published, 3.12.2018 at 16:08

NB: The start of the 2. course exam has been postponed by 15 minutes from what was announced earlier: it wil be on Tue 18.12. at *13:30*-16:00 in Exactum, D123.

## Jani Lukkarinen

Published, 15.11.2018 at 14:53

Reminders: 1) 2nd partial exam will be on Tue 18.12. at 13-16 in Exactum, D123. 2) The lecture from Thu 22.11. is moved to Monday 19.11. at 12-14 in Exactum, B121.

## Jani Lukkarinen

Published, 12.11.2018 at 16:23

## Jani Lukkarinen

Published, 8.11.2018 at 11:32

Results from the first period are now available under the Results section below (available for registered participants: you will need to login to read the PDF file). Please contact the lecturer if you have any questions or comments about the listed results. You can check your detailed grading from the lecturer either during the office hours or during the lecture breaks.

## Jani Lukkarinen

Published, 17.10.2018 at 15:07

New course advert: If you are interested in learning more about advanced properties and uses of linear operators, check out the Operator theory course starting in the 2nd period, on 30.10.; course web page https://courses.helsinki.fi/en/MAST31906/125184524 (It should be possible to follow the course in parallel with the second half of the Functional analysis course.)

## Jani Lukkarinen

Published, 16.10.2018 at 18:33

The first course exam will be on Tue 23.10. at 13-16 in Exactum, D123. In addition to the Exercises 1-6, the material covered by the exam is listed below:

[Lecture notes (in Finnish)] Chapters 1-4
[Kreyszig (in English)] Chapters 1-3 + 5. In particular, we focus on the sections 1.1-1.5, 2.1-2.7, 2.10, 3.1-3.6, 5.1.

In addition to standard writing equipment, you are allowed to bring in and consult *one handwritten* two-sided A4 sheet of personal notes during the exam. (In particular, no textbooks or full lecture notes.)

## Pages

### Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Mon 3.9.2018
14:15 - 16:00
Thu 6.9.2018
12:15 - 16:00
Mon 17.9.2018
14:15 - 16:00
Thu 20.9.2018
12:15 - 14:00
Mon 24.9.2018
14:15 - 16:00
Thu 27.9.2018
12:15 - 14:00
Mon 1.10.2018
14:15 - 16:00
Thu 4.10.2018
12:15 - 14:00
Mon 8.10.2018
14:15 - 16:00
Thu 11.10.2018
12:15 - 14:00
Mon 15.10.2018
14:15 - 16:00
Thu 18.10.2018
12:15 - 14:00
Mon 29.10.2018
14:15 - 16:00
Thu 1.11.2018
12:15 - 14:00
Mon 5.11.2018
14:15 - 16:00
Thu 8.11.2018
12:15 - 14:00
Mon 12.11.2018
14:15 - 16:00
Thu 15.11.2018
12:15 - 14:00
Mon 19.11.2018
12:15 - 14:00
Mon 19.11.2018
14:15 - 16:00
Mon 26.11.2018
14:15 - 16:00
Thu 29.11.2018
12:15 - 14:00
Mon 3.12.2018
14:15 - 16:00
Mon 10.12.2018
14:15 - 16:00
Thu 13.12.2018
12:15 - 14:00

### Other teaching

13.09. - 18.10.2018 Thu 14.15-16.00
01.11. - 29.11.2018 Thu 14.15-16.00
13.12. - 13.12.2018 Thu 14.15-16.00
Jani Lukkarinen
Teaching language: English

### Material

The course will follow the lecture notes "Funktonaalianalyysin peruskurssi" (2012, last updated in 2017) by Kari Astala, Petteri Piiroinen & Hans-Olav Tylli. Minor updates are possible also during the course and the most current version for the Autumn 2018 course can be downloaded from below.

* The notes are available only in Finnish but an English textbook substitute, which is particularly suitable for those who wish to focus on applications, is

Erwin Kreyszig: Introductory functional analysis with applications (Wiley, 1989)

* Although perhaps not optimal as a textbook, a concise but solid reference book including also material for the courses Real Analysis I and Complex Analysis I is

W. Rudin: Real and Complex Analysis (Tata McGraw-Hill) [chapters 3-5 summarize the main material for this course]

* The following books contain related material (as well as much more), and may be used as supplement to the course notes/textbook:

Bollobas: Linear Analysis (Cambridge Mathematical Textbooks)
Werner: Funktionalanalysis (Springer-Lehrbuch) [in German]
W. Rudin: Functional Analysis (McGraw-Hill) [Part I generalizes the theory to topological vector spaces]

## Lecture material

• K. Astala, J. Lukkarinen, P. Piiroinen, H.-O. Tylli
Typos corrected for chapters 1-9, added section 6b and revised section 9 [typos corrected 17.12.2018, many thanks to Jaakko Sinko for a list of corrections]

#### Homework sets, second period

The exercise sessions will be on Thursdays 14:15-16:00 in Exactum, C129, on the date marked at the top of the exercise sheet. You can either come to present your solutions in the session, or contact the instructor Kalle Koskinen (kalle.koskinen@helsinki.fi) to make alternative arrangements if you cannot attend it.

Please make free use of the Ratkomo -tutorial sessions on the 3rd floor: the schedule is available at the web page https://blogs.helsinki.fi/ratkomo-solvery/aikataulu-timetable/ As mentioned during the lectures, the lecturer and/or instructor are often also answering questions in Ratkomo on Tuesdays between 13-15. In particular, you are welcome to ask questions from the lecturer during the office hours on Tue 14-15 at Exactum D335.

[Lecture notes: Chapter 4 (review), Kreyszig: Chapter 3 (review)]
[Lecture notes: Chapter 5.1-5.9, Kreyszig: Example 3.4-5, Sections 3.5-3.6.]
[Lecture notes: Chapter 6.1-6.8 & 5 (review), Kreyszig: 2.6, 2.7, 2.10]
[Lecture notes: Chapter 6, Kreyszig: 2.7, 5.4, 7.6, 7.7]
[New chapter 6b (see Materials) + Notes 7.1-7.8 / Kreyszig 3.8, 4.7]
[Notes: chapters 8 & 9, Kreyszig: 4.1-4.5, 4.7, 4.12-4.13]

[KK]
[KK]
[KK]
[KK]
[KK]
[KK]

#### Homework sets, first period

[chapters 1-2.10, Kreyszig: Chapters 1.1-1.4 and 2]
[Lecture notes: 2.1-2.30, Kreyszig: 1.2-1.4, 2.1, 2.2, 2.4-4, 2.5, 2.6, start of 2.7]
[Lecture notes: 2.24-2.31, 3.1-3.10, Kreyszig: 1.4-1.5, 2.6-2.7]
[Lecture notes: 3.11-3.32, Kreyszig: 2.2, 2.3, 2.10 + Bollobas: Ch.2, p.24- (for measure theory & series)]
[Lecture notes: 3.33-3.46, 4.1-4.9, Kreyszig: 5.1, 5.4, 3.1, 3.2]
[Lecture notes: 4.1-4.38, Kreyszig: 3.1-3.6]

[Kalle Koskinen]
[KK]
[KK]
[JL]
[KK]
[KK]

### Description

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

Belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Analysis I&II, Linear algebra I&II, Topology I

Elements of measure theory and complex analysis

Elements of linear functional analysis including Banach and Hilbert spaces and linear operators between them, three basic principles, and applications to differential equations.

Recommended time/stage of studies for completion: 1. year

Term/teaching period when the course will be offered: varying

Introduction to linear functional analysis including Banach and Hilbert spaces and linear operators between them; topology of normed spaces; examples of Banach spaces including sequence and function spaces; three basic principles; Fourier-series; Sobolev spaces; applications to differential equations.

Required: